Datasets:

GitBag
/

math_qwen2.5_1.5B_8192_n_128_eval_len

Modalities:

Formats:

Size:

Libraries:

Dataset card Data Studio Files Files and versions Community

Dataset Viewer

Auto-converted to Parquet

Split (1)

train · 7.5k rows

level stringclasses 6 values	type stringclasses 7 values	data_source stringclasses 1 value	prompt listlengths 1 1	ability stringclasses 1 value	reward_model dict	extra_info dict	response_0 stringlengths 0 32.3k	response_1 stringlengths 0 32.2k	response_2 stringlengths 0 163k	response_3 stringlengths 0 36.8k	response_4 stringlengths 0 33.8k	response_5 stringlengths 0 33.4k	response_6 stringlengths 0 35k	response_7 stringlengths 0 29.5k	response_8 stringlengths 0 31k	response_9 stringlengths 0 29.4k	response_10 stringlengths 0 45.8k	response_11 stringlengths 0 61.8k	response_12 stringlengths 0 56.7k	response_13 stringlengths 0 31.7k	response_14 stringlengths 0 34.5k	response_15 stringlengths 0 76k	response_16 stringlengths 0 36.2k	response_17 stringlengths 0 34.9k	response_18 stringlengths 0 33.6k	response_19 stringlengths 0 32.5k	response_20 stringlengths 0 34.4k	response_21 stringlengths 0 33.3k	response_22 stringlengths 0 47.4k	response_23 stringlengths 0 27.8k	response_24 stringlengths 0 31.6k	response_25 stringlengths 0 37.6k	response_26 stringlengths 0 33.9k	response_27 stringlengths 0 31.5k	response_28 stringlengths 0 32.5k	response_29 stringlengths 0 31.3k	response_30 stringlengths 0 29.6k	response_31 stringlengths 0 40.9k	response_32 stringlengths 0 57.6k	response_33 stringlengths 0 30.6k	response_34 stringlengths 0 37.9k	response_35 stringlengths 0 35.6k	response_36 stringlengths 0 30.1k	response_37 stringlengths 0 41k	response_38 stringlengths 0 34.3k	response_39 stringlengths 0 31.7k	response_40 stringlengths 0 33.6k	response_41 stringlengths 0 31.8k	response_42 stringlengths 0 29.5k	response_43 stringlengths 0 28.1k	response_44 stringlengths 0 28.7k	response_45 stringlengths 0 35k	response_46 stringlengths 0 30.7k	response_47 stringlengths 0 38.8k	response_48 stringlengths 0 110k	response_49 stringlengths 0 29.8k	response_50 stringlengths 0 37.5k	response_51 stringlengths 0 34.9k	response_52 stringlengths 0 34.8k	response_53 stringlengths 0 28.8k	response_54 stringlengths 0 34.1k	response_55 stringlengths 0 30.8k	response_56 stringlengths 0 35.9k	response_57 stringlengths 0 31.7k	response_58 stringlengths 0 35.6k	response_59 stringlengths 0 32.1k	response_60 stringlengths 0 28.8k	response_61 stringlengths 0 32k	response_62 stringlengths 0 27.4k	response_63 stringlengths 0 38.9k	response_64 stringlengths 0 35.1k	response_65 stringlengths 0 39.7k	response_66 stringlengths 0 82.4k	response_67 stringlengths 0 33.8k	response_68 stringlengths 0 36.1k	response_69 stringlengths 0 33.9k	response_70 stringlengths 0 31.5k	response_71 stringlengths 0 44.2k	response_72 stringlengths 0 33.6k	response_73 stringlengths 0 37k	response_74 stringlengths 0 35.5k	response_75 stringlengths 0 34.6k	response_76 stringlengths 0 33.6k	response_77 stringlengths 0 56.7k	response_78 stringlengths 0 30.9k	response_79 stringlengths 0 35.6k	response_80 stringlengths 0 31.8k	response_81 stringlengths 0 37.7k	response_82 stringlengths 0 38.9k	response_83 stringlengths 0 39.8k	response_84 stringlengths 0 37.7k	response_85 stringlengths 0 35.1k	response_86 stringlengths 0 33.2k	response_87 stringlengths 0 37.1k	response_88 stringlengths 0 34k	response_89 stringlengths 0 39.2k	response_90 stringlengths 0 56k	response_91 stringlengths 0 32.3k	response_92 stringlengths 0 32.9k	response_93 stringlengths 0 31.3k	response_94 stringlengths 0 36.6k	response_95 stringlengths 0 38.3k	response_96 stringlengths 0 38.5k	response_97 stringlengths 0 34.9k	response_98 stringlengths 0 76.1k	response_99 stringlengths 0 29.6k	response_100 stringlengths 0 41.8k	response_101 stringlengths 0 34.5k	response_102 stringlengths 0 30.1k	response_103 stringlengths 0 37k	response_104 stringlengths 0 57.2k	response_105 stringlengths 0 32.6k	response_106 stringlengths 0 37k	response_107 stringlengths 0 34.7k	response_108 stringlengths 0 33.9k	response_109 stringlengths 0 36.5k	response_110 stringlengths 0 29.8k	response_111 stringlengths 0 28.9k	response_112 stringlengths 0 32.5k	response_113 stringlengths 0 42.4k	response_114 stringlengths 0 33.3k	response_115 stringlengths 0 42.1k	response_116 stringlengths 0 32.6k	response_117 stringlengths 0 26k	response_118 stringlengths 0 28.8k	response_119 stringlengths 0 36.7k	response_120 stringlengths 0 27.3k	response_121 stringlengths 0 34k	response_122 stringlengths 0 34.7k	response_123 stringlengths 0 42.5k	response_124 stringlengths 0 40.1k	response_125 stringlengths 0 35.5k	response_126 stringlengths 0 32k	response_127 stringlengths 0 35.7k	eval_0 float64 0 1	eval_1 float64 0 1	eval_2 float64 0 1	eval_3 float64 0 1	eval_4 float64 0 1	eval_5 float64 0 1	eval_6 float64 0 1	eval_7 float64 0 1	eval_8 float64 0 1	eval_9 float64 0 1	eval_10 float64 0 1	eval_11 float64 0 1	eval_12 float64 0 1	eval_13 float64 0 1	eval_14 float64 0 1	eval_15 float64 0 1	eval_16 float64 0 1	eval_17 float64 0 1	eval_18 float64 0 1	eval_19 float64 0 1	eval_20 float64 0 1	eval_21 float64 0 1	eval_22 float64 0 1	eval_23 float64 0 1	eval_24 float64 0 1	eval_25 float64 0 1	eval_26 float64 0 1	eval_27 float64 0 1	eval_28 float64 0 1	eval_29 float64 0 1	eval_30 float64 0 1	eval_31 float64 0 1	eval_32 float64 0 1	eval_33 float64 0 1	eval_34 float64 0 1	eval_35 float64 0 1	eval_36 float64 0 1	eval_37 float64 0 1	eval_38 float64 0 1	eval_39 float64 0 1	eval_40 float64 0 1	eval_41 float64 0 1	eval_42 float64 0 1	eval_43 float64 0 1	eval_44 float64 0 1	eval_45 float64 0 1	eval_46 float64 0 1	eval_47 float64 0 1	eval_48 float64 0 1	eval_49 float64 0 1	eval_50 float64 0 1	eval_51 float64 0 1	eval_52 float64 0 1	eval_53 float64 0 1	eval_54 float64 0 1	eval_55 float64 0 1	eval_56 float64 0 1	eval_57 float64 0 1	eval_58 float64 0 1	eval_59 float64 0 1	eval_60 float64 0 1	eval_61 float64 0 1	eval_62 float64 0 1	eval_63 float64 0 1	eval_64 float64 0 1	eval_65 float64 0 1	eval_66 float64 0 1	eval_67 float64 0 1	eval_68 float64 0 1	eval_69 float64 0 1	eval_70 float64 0 1	eval_71 float64 0 1	eval_72 float64 0 1	eval_73 float64 0 1	eval_74 float64 0 1	eval_75 float64 0 1	eval_76 float64 0 1	eval_77 float64 0 1	eval_78 float64 0 1	eval_79 float64 0 1	eval_80 float64 0 1	eval_81 float64 0 1	eval_82 float64 0 1	eval_83 float64 0 1	eval_84 float64 0 1	eval_85 float64 0 1	eval_86 float64 0 1	eval_87 float64 0 1	eval_88 float64 0 1	eval_89 float64 0 1	eval_90 float64 0 1	eval_91 float64 0 1	eval_92 float64 0 1	eval_93 float64 0 1	eval_94 float64 0 1	eval_95 float64 0 1	eval_96 float64 0 1	eval_97 float64 0 1	eval_98 float64 0 1	eval_99 float64 0 1	eval_100 float64 0 1	eval_101 float64 0 1	eval_102 float64 0 1	eval_103 float64 0 1	eval_104 float64 0 1	eval_105 float64 0 1	eval_106 float64 0 1	eval_107 float64 0 1	eval_108 float64 0 1	eval_109 float64 0 1	eval_110 float64 0 1	eval_111 float64 0 1	eval_112 float64 0 1	eval_113 float64 0 1	eval_114 float64 0 1	eval_115 float64 0 1	eval_116 float64 0 1	eval_117 float64 0 1	eval_118 float64 0 1	eval_119 float64 0 1	eval_120 float64 0 1	eval_121 float64 0 1	eval_122 float64 0 1	eval_123 float64 0 1	eval_124 float64 0 1	eval_125 float64 0 1	eval_126 float64 0 1	eval_127 float64 0 1	len_0 int64 0 8.19k	len_1 int64 0 8.2k	len_2 int64 0 8.19k	len_3 int64 0 8.19k	len_4 int64 0 8.19k	len_5 int64 0 8.2k	len_6 int64 0 8.19k	len_7 int64 0 8.19k	len_8 int64 0 8.19k	len_9 int64 0 8.19k	len_10 int64 0 8.19k	len_11 int64 0 8.19k	len_12 int64 0 8.19k	len_13 int64 0 8.19k	len_14 int64 0 8.19k	len_15 int64 0 8.19k	len_16 int64 0 8.19k	len_17 int64 0 8.2k	len_18 int64 0 8.19k	len_19 int64 0 8.19k	len_20 int64 0 8.19k	len_21 int64 0 8.19k	len_22 int64 0 8.19k	len_23 int64 0 8.19k	len_24 int64 0 8.19k	len_25 int64 0 8.19k	len_26 int64 0 8.19k	len_27 int64 0 8.2k	len_28 int64 0 8.2k	len_29 int64 0 8.2k	len_30 int64 0 8.19k	len_31 int64 0 8.19k	len_32 int64 0 8.19k	len_33 int64 0 8.2k	len_34 int64 0 8.19k	len_35 int64 0 8.2k	len_36 int64 0 8.19k	len_37 int64 0 8.19k	len_38 int64 0 8.2k	len_39 int64 0 8.19k	len_40 int64 0 8.19k	len_41 int64 0 8.19k	len_42 int64 0 8.19k	len_43 int64 0 8.19k	len_44 int64 0 8.19k	len_45 int64 0 8.19k	len_46 int64 0 8.19k	len_47 int64 0 8.19k	len_48 int64 0 8.19k	len_49 int64 0 8.19k	len_50 int64 0 8.19k	len_51 int64 0 8.19k	len_52 int64 0 8.19k	len_53 int64 0 8.19k	len_54 int64 0 8.19k	len_55 int64 0 8.19k	len_56 int64 0 8.19k	len_57 int64 0 8.19k	len_58 int64 0 8.19k	len_59 int64 0 8.19k	len_60 int64 0 8.19k	len_61 int64 0 8.19k	len_62 int64 0 8.19k	len_63 int64 0 8.19k	len_64 int64 0 8.19k	len_65 int64 0 8.19k	len_66 int64 0 8.19k	len_67 int64 0 8.2k	len_68 int64 0 8.19k	len_69 int64 0 8.19k	len_70 int64 0 8.2k	len_71 int64 0 8.19k	len_72 int64 0 8.19k	len_73 int64 0 8.19k	len_74 int64 0 8.19k	len_75 int64 0 8.19k	len_76 int64 0 8.19k	len_77 int64 0 8.19k	len_78 int64 0 8.19k	len_79 int64 0 8.19k	len_80 int64 0 8.19k	len_81 int64 0 8.19k	len_82 int64 0 8.19k	len_83 int64 0 8.19k	len_84 int64 0 8.19k	len_85 int64 0 8.19k	len_86 int64 0 8.21k	len_87 int64 0 8.19k	len_88 int64 0 8.19k	len_89 int64 0 8.19k	len_90 int64 0 8.19k	len_91 int64 0 8.3k	len_92 int64 0 8.19k	len_93 int64 0 8.19k	len_94 int64 0 8.19k	len_95 int64 0 8.19k	len_96 int64 0 8.2k	len_97 int64 0 8.2k	len_98 int64 0 8.19k	len_99 int64 0 8.19k	len_100 int64 0 8.19k	len_101 int64 0 8.19k	len_102 int64 0 8.19k	len_103 int64 0 8.19k	len_104 int64 0 8.19k	len_105 int64 0 8.19k	len_106 int64 0 8.19k	len_107 int64 0 8.19k	len_108 int64 0 8.2k	len_109 int64 0 8.19k	len_110 int64 0 8.19k	len_111 int64 0 8.19k	len_112 int64 0 8.19k	len_113 int64 0 8.2k	len_114 int64 0 8.19k	len_115 int64 0 8.19k	len_116 int64 0 8.19k	len_117 int64 0 8.19k	len_118 int64 0 8.19k	len_119 int64 0 8.19k	len_120 int64 0 8.19k	len_121 int64 0 8.19k	len_122 int64 0 8.19k	len_123 int64 0 8.19k	len_124 int64 0 8.19k	len_125 int64 0 8.19k	len_126 int64 0 8.21k	len_127 int64 0 8.19k
Level 5	Algebra	DigitalLearningGmbH/MATH-lighteval	[{"content":"Let \\[f(x) = \\left\\{\n\\begin{array}{cl} ax+3, &\\text{ if }x>2, \\\\\nx-5 &\\text{ (...TRUNCATED)	math	{ "ground_truth": "0", "style": "rule" }	{ "index": 0, "split": "train" }	" To solve this problem, we need to determine the values of \\(a\\) and \\(b\\) such that the piecew(...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) that make the piecewise function continuous, we nee(...TRUNCATED)	" To make sure our answer is accurate, let's use Python to verify the continuity of the piecewise fu(...TRUNCATED)	" \n\nFirst, we need the functions to be continuous and their respective domain calculations:\n1. $a(...TRUNCATED)	" To ensure that the piecewise function \\(f(x)\\) is continuous, the values of \\(f(x)\\) must matc(...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) such that the piecewise function \\(f(x)\\) is cont(...TRUNCATED)	" To ensure that the piecewise function \\( f(x) \\) is continuous, the values of \\( f(x) \\) must (...TRUNCATED)	" To solve for \\(a\\) and \\(b\\) so that the piecewise function \\(f(x)\\) is continuous, we need (...TRUNCATED)	" For the function \\( f(x) \\) to be continuous, the left-hand limit, right-hand limit, and the fun(...TRUNCATED)	" \nTo solve the problem, we need to determine the values of \\(a\\) and \\(b\\) that make the piece(...TRUNCATED)	" To ensure the piecewise function \\( f(x) \\) is continuous, the left limit of \\( f(x) \\) as \\((...TRUNCATED)	" To ensure the piecewise function \\( f(x) \\) is continuous, the left-hand limit and right-hand li(...TRUNCATED)	" The key to finding \\(a + b\\) for the given piecewise function that is continuous is to ensure th(...TRUNCATED)	" For the function to be continuous, it must be continuous at $x = -2$ and $x = 2$. \n\nAt $x = -2$:(...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) such that the function \\(f(x)\\) is continuous, we(...TRUNCATED)	" To determine the values for \\(a\\) and \\(b\\) for which the piecewise function is continuous, we(...TRUNCATED)	" \n\nTo determine the values of \\(a\\) and \\(b\\) such that the piecewise function \\(f(x)\\) is (...TRUNCATED)	" To ensure the piecewise function \\( f(x) \\) is continuous, it must satisfy the condition that th(...TRUNCATED)	" To ensure the piecewise function \\( f(x) \\) is continuous, it must be continuous at the points w(...TRUNCATED)	" To ensure the piecewise function \\( f(x) \\) is continuous, the left-hand limit at \\( x = 2 \\) (...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) such that the piecewise function \\(f(x)\\) is cont(...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) that make the function \\(f(x)\\) continuous, we ne(...TRUNCATED)	" To determine the values of $(2 + b)$ and $(a + 5)$:\n- The function is continuous for $x < -2$.\n-(...TRUNCATED)	" ```python\n# Given piecewise function\ndef f(x, a, b):\n if x > 2:\n return a * x + 3\n (...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) such that the piecewise function \\(f(x)\\) is cont(...TRUNCATED)	" Given the piecewise function:\n\n\\[\nf(x) = \\left\\{\n\\begin{array}{cl} ax+3, &\\text{ if }x>2,(...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) that make the piecewise function \\(f(x)\\) continu(...TRUNCATED)	" To ensure the piecewise function \\( f(x) \\) is continuous, we need to check the continuity at th(...TRUNCATED)	" \n\nSince the function is given to be continuous, we know that the left and right limits of each s(...TRUNCATED)	" To ensure the piecewise function \\( f(x) \\) is continuous, we need to compare the left-hand limi(...TRUNCATED)	" To ensure that the piecewise function is continuous, we need to make sure that the value of the fu(...TRUNCATED)	" \n\n### Step 1: Ensuring Continuity at the Boundary Points\nFor the function to be continuous, the(...TRUNCATED)	" ```python\n# Define the piecewise function\ndef f(x):\n if x > 2:\n return x * 2 # beca(...TRUNCATED)	" To ensure that the piecewise function \\( f(x) \\) is continuous, the left-hand limit as \\( x \\)(...TRUNCATED)	" \n\nSince the piecewise function is to be continuous, the function value at the intersection of th(...TRUNCATED)	" To ensure the function \\( f(x) \\) is continuous at the points \\( x = 2 \\) and \\( x = -2 \\), (...TRUNCATED)	" To ensure that the function \\( f(x) \\) is continuous at the boundary points \\(-2\\) and \\(2\\)(...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) such that the piecewise function is continuous, we (...TRUNCATED)	" To find the values of \\(a\\) and \\(b\\) such that the piecewise function \\( f(x) \\) is continu(...TRUNCATED)	" To determine if the piecewise function is continuous, we need to ensure that the function is consi(...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) such that the function \\(f(x)\\) is continuous, we(...TRUNCATED)	" To ensure accuracy, let's break down the conditions and solve for \\(a + b\\). For a function to b(...TRUNCATED)	" For the function to be continuous, the value of the function from the left-hand side must match th(...TRUNCATED)	" First, let's focus on the part of the function that applies to \\(x > 2\\):\n\n\\[ f(x) = ax + 3 \(...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) that make the piecewise function continuous, we nee(...TRUNCATED)	" Let's solve this problem step by step. To ensure the function \\(f(x)\\) is continuous at the poin(...TRUNCATED)	" To solve this problem, we need to ensure that the function \\(f(x)\\) is continuous at the points (...TRUNCATED)	" First, we need to ensure that the function is continuous at the points where the definition change(...TRUNCATED)	" To solve for \\(a + b\\) in the given piecewise function, we must ensure that the function is cont(...TRUNCATED)	" To ensure the piecewise function \\( f(x) \\) is continuous, we need to check continuity at the po(...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) such that the piecewise function \\(f(x)\\) is cont(...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) that make the piecewise function continuous, we nee(...TRUNCATED)	" Now, let's solve the problem:\n\nFor the piecewise function to be continuous, the values of the fu(...TRUNCATED)	" To ensure the piecewise function \\( f(x) \\) is continuous, we need to satisfy the continuity con(...TRUNCATED)	" To solve the problem, we need to ensure that the piecewise function \\( f(x) \\) is continuous. Fo(...TRUNCATED)	" To solve the problem, we need to ensure that the piecewise function is continuous at the points wh(...TRUNCATED)	" \n\nYou are given a piecewise function and asked to determine the values of \\(a\\) and \\(b\\) su(...TRUNCATED)	" For the piecewise function to be continuous, the three cases have to fit together at $x=2$ and $x=(...TRUNCATED)	" \n\nSince the function is continuous, the left-hand limit and the right-hand limit of the function(...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) such that the piecewise function \\(f(x)\\) is cont(...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) that make the piecewise function \\(f(x)\\) continu(...TRUNCATED)	" First, we will analyze the function based on the definition provided. The function \\(f(x)\\) is d(...TRUNCATED)	" To ensure that the piecewise function \\( f(x) \\) is continuous, the left-hand limit as \\( x \\)(...TRUNCATED)	" \n\nSince the piecewise function is continuous, we need to find the values of $a$ and $b$ such tha(...TRUNCATED)	" To ensure the function \\( f(x) \\) is continuous at the points where the definitions of \\( f(x) (...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) such that the piecewise function is continuous at t(...TRUNCATED)	" To ensure the piecewise function is continuous, the left-hand and right-hand limits at the points (...TRUNCATED)	" To ensure the piecewise function is continuous, the value of \\( f(x) \\) at the point where the d(...TRUNCATED)	" To ensure the piecewise function \\( f(x) \\) is continuous, we need to consider the continuity of(...TRUNCATED)	" \nTo determine \\(a + b\\) such that the piecewise function \\(f(x)\\) is continuous, we need to e(...TRUNCATED)	" To solve this problem, we need to ensure that the piecewise function \\( f(x) \\) is continuous. T(...TRUNCATED)	" To ensure the piecewise function is continuous, we need to make sure that as \\( x \\) approaches (...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) that make the piecewise function continuous, we nee(...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) such that the piecewise function \\(f(x)\\) is cont(...TRUNCATED)	" To ensure the piecewise function $f(x)$ is continuous, the right-hand limit at the endpoints of ea(...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) so that the piecewise function \\(f(x)\\) is contin(...TRUNCATED)	" ```python\n# We need to ensure that the piecewise function is continuous at the points where the d(...TRUNCATED)	" To ensure continuity, we need to have equal left-hand and right-hand limits for the given piecewis(...TRUNCATED)	" To ensure the piecewise function \\( f(x) \\) is continuous, we need to make sure that the piecewi(...TRUNCATED)	" For \\( f(x) \\) to be continuous at the points where the definition of \\( f(x) \\) changes, name(...TRUNCATED)	" To ensure the function \\( f(x) \\) is continuous at the points \\( x = -2 \\) and \\( x = 2 \\), (...TRUNCATED)	" For a piecewise function to be continuous, the values of the function at the points where the defi(...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) that make the piecewise function continuous, we nee(...TRUNCATED)	" We need to determine the values of \\(a\\) and \\(b\\) such that the piecewise function \\(f(x)\\)(...TRUNCATED)	" \n\nSince the piecewise function is continuous, the function must be continuous at the points wher(...TRUNCATED)	" \nTo ensure the function is continuous, we need to consider the points where the definition of the(...TRUNCATED)	" \nTo problem, first look at the definition of the function at the places where the definition happ(...TRUNCATED)	" To ensure the piecewise function \\(f(x)\\) is continuous, we need to check the continuity at the (...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) that make the piecewise function continuous, we nee(...TRUNCATED)	" The corresponding Python code to answer the above question is:\n\"\"\"\nTo find the values of $a$ (...TRUNCATED)	" For this problem, we must ensure the function is continuous at the points where the function defin(...TRUNCATED)	" To ensure the function \\( f(x) \\) is continuous, we need to match the left-hand and right-hand l(...TRUNCATED)		" To determine the values of \\(a\\) and \\(b\\) such that the piecewise function is continuous, we (...TRUNCATED)	" To ensure continuity at the points (where the variable flipping sides), we have to evaluate each p(...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) that make the piecewise function continuous, we nee(...TRUNCATED)	" To make this programming problem, we need to calculate \\(a\\) and \\(b\\) so that the piecewise f(...TRUNCATED)	" For the given piecewise function to be continuous, the following must be true:\n\n1. \\(\\lim\\lim(...TRUNCATED)	" To ensure the piecewise function \\( f(x) \\) is continuous at the points where the definition cha(...TRUNCATED)		" The equation is given as:\n\n\\[\nf(x) = \n\\begin{cases}\nax+3, & \\text{if } x>2, \\\\\nx-5, & \(...TRUNCATED)	" To determine \\(a + b\\) such that the piecewise function \\(f(x)\\) is continuous for all \\(x\\)(...TRUNCATED)	" To ensure the piecewise function $f(x)$ is continuous, we need to find $a$ and $b$ such that the f(...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) that make the piecewise function continuous, we nee(...TRUNCATED)	" To ensure the piecewise function \\( f(x) \\) is continuous at the points where the definition of (...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) such that the piecewise function is continuous, we (...TRUNCATED)	" To ensure the function \\( f(x) \\) is continuous at the points where the definition changes (i.e.(...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) that make the piecewise function \\(f(x)\\) continu(...TRUNCATED)	" To determine the values of $a$ and $b$ so that the piecewise function \\(f(x)\\) is continuous, we(...TRUNCATED)	" To ensure the piecewise function is continuous, which means that its graph can be drawn without li(...TRUNCATED)	" To ensure the piecewise function \\( f(x) \\) is continuous over all intervals, we need to apply t(...TRUNCATED)	" We will check the continuity conditions at the points where the piecewise function changes its def(...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) that make the piecewise function \\(f(x)\\) continu(...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) that make the piecewise function continuous, we nee(...TRUNCATED)		" To determine the values of \\(a\\) and \\(b\\) that make the piecewise function continuous, we nee(...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) such that the given piecewise function is continuou(...TRUNCATED)	" To ensure the piecewise function \\( f(x) \\) is continuous at the points where the definitions of(...TRUNCATED)	" To make this problem multiple-choice test or just mathematical exercise:\n\nThe given function is (...TRUNCATED)	" To solve this problem, we need to ensure that the function $f(x)$ is continuous at the transitions(...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) that make the piecewise function \\( f(x) \\) conti(...TRUNCATED)		" For a function to be continuous at a point, the left-hand limit and the right-hand limit must be e(...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) in the piecewise function \\(f(x)\\), we need to en(...TRUNCATED)	" To determine the values of \\(a\\) and \\(b\\) such that the piecewise function \\(f(x)\\) is cont(...TRUNCATED)	" \n\nTo ensure the piecewise function \\( f(x) \\) is continuous at the points where its rule chang(...TRUNCATED)	" To ensure continuity at the boundaries of the piecewise functions, we need to make sure that the v(...TRUNCATED)	" We'll find the values of \\(a\\) and \\(b\\) such that the piecewise function \\(f(x)\\) is contin(...TRUNCATED)	0	1	0	0	1	0	0	1	1	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	1	1	1	0	0	1	0	0	1	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	1	0	0	0	0	0	1	1	0	1	0	0	1	1	1	0	0	1	0	1	0	1	0	0	0	0	1	1	0	1	0	1	0	0	0	1	0	1	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	1	0	1	0	0	0	0	1	0	0	0	0	0	613	609	308	343	2,390	750	647	752	689	687	2,204	1,875	572	407	980	338	1,030	725	753	645	666	664	754	675	373	3,439	2,078	758	406	525	697	484	7,354	1,001	283	673	895	619	476	665	993	2,996	410	559	4,516	407	640	378	876	1,185	1,166	739	901	502	552	652	658	134	452	1,197	3,793	584	600	318	558	694	747	540	530	565	543	729	480	748	526	759	853	620	3,571	705	641	557	686	931	246	688	283	624	842	279	490	893	0	540	460	451	728	236	803	0	606	591	459	616	806	887	658	425	673	5,565	690	555	883	654	0	720	8,055	749	629	911	684	0	889	770	1,197	3,991	495	874
Level 5	Algebra	DigitalLearningGmbH/MATH-lighteval	[{"content":"A rectangular band formation is a formation with $m$ band members in each of $r$ rows, (...TRUNCATED)	math	{ "ground_truth": "98", "style": "rule" }	{ "index": 1, "split": "train" }	" This problem involves a bit of logic and combinatorics. Let's break it down into parts and solve i(...TRUNCATED)	" Let's denote the number of band members by \\( N \\). According to the problem, \\( N \\) leaves a(...TRUNCATED)	" To solve this problem, we will set up the relationship mathematically and then translate it into a(...TRUNCATED)	" \r\n\r\nAssume that we have a rectangular band formation with \\(m\\) members per row and \\(r\(...TRUNCATED)	" Let's solve this problem step by step. We are given that there is a rectangular formation with \\((...TRUNCATED)	" We can use integer division to find the number of members in each row when there are 2 members lef(...TRUNCATED)	" To solve this problem, let's denote the total number of band members as \\( N \\). According to th(...TRUNCATED)	" Let's break down the problem step-by-step and solve it using Python.\n\n1. Define Variables:\n(...TRUNCATED)	" To solve this problem, let's break it down into steps:\n\n1. Let \\( n \\) be the number of band m(...TRUNCATED)	" To solve this problem, let's break it down into manageable parts:\n\n1. Let \\( n \\) be the total(...TRUNCATED)	" To solve this problem, let's use a bit of algebra. Let's denote the number of band members by \\(N(...TRUNCATED)	" Consider the following steps:\n\n1. Let \\( N \\) be the number of band members.\n2. If the number(...TRUNCATED)	" Based on the description, we'll formulate and solve the problem mathematically, then verify using (...TRUNCATED)	" \n1. Inequality Setup: The problem states that the band has less than 100 members and after re(...TRUNCATED)	" To solve the problem, we need to set up an equation based on the information given:\n\n1. Let \\( (...TRUNCATED)	" Let's solve the problem step-by-step.\n\n1. Let \\( n \\) be the number of members in the band. Ac(...TRUNCATED)	" ```python\ndef find_band_members():\n max_band_members = 0\n for m in range(1, 100):\n (...TRUNCATED)	" To solve this problem, we need to find the largest number of members \\( n \\) that meets the foll(...TRUNCATED)	" Let's denote the number of band members by $n$. Since the band has less than 100 members, $n < 100(...TRUNCATED)	" ```python\n# Let's define the variables:\ntotal_band_members = -1\n\n# We need to find the largest(...TRUNCATED)	" Now, let's implement the solution using Python code.\nTo solve this problem, we need to find the l(...TRUNCATED)	" To solve the problem, we need to find the largest number of band members less than 100 that satisf(...TRUNCATED)	" Let's solve the problem step by step.\n\n1. Let the number of band members be \\( N \\).\n2. Since(...TRUNCATED)	" Given the problem, we can express the number of band members mathematically. Initially, the band h(...TRUNCATED)	" This problem is divided into two critical steps: determining the number of band members in the ori(...TRUNCATED)	" ```python\n# We need to find the largest number of members in a rectangular band formation that fi(...TRUNCATED)	" ```python\ndef max_band_members():\r\n for m in range(1, 100):\r\n for r in range(1, 100(...TRUNCATED)	" Let's denote the number of band members by \\( n \\), and the respective numbers of rows and membe(...TRUNCATED)	" To solve the problem, let's break it down step by step.\n\n1. Let the number of band members be \\(...TRUNCATED)	" Let's denote the number of band members by \\( N \\) and the dimensions of the original rectangula(...TRUNCATED)	" To solve the problem, we need to find the largest number of members \\( N \\) that satisfies the g(...TRUNCATED)	" To solve this problem, we need to find the largest number of band members \\( n \\) less than 100 (...TRUNCATED)	" Let \\( n \\) be the number of band members, \\( r \\) be the number of rows, and \\( m \\) be the(...TRUNCATED)	" We can solve this problem using a step-by-step approach and iteratively check the conditions given(...TRUNCATED)	" ```python\n# To solve the problem, we need to find the largest number of members (n) that is less (...TRUNCATED)	" To solve this problem, we can use a function to simulate the given conditions and find the largest(...TRUNCATED)	" To solve this problem, we need to determine the number of band members \\( m \\) and the number of(...TRUNCATED)	" \n\nWe need to find the largest number of band members that satisfies the given conditions. Let'(...TRUNCATED)	" Let's solve this problem using Python.\n\nFirst, let's translate the problem into a mathematical f(...TRUNCATED)	" To solve the problem, we will consider the following steps:\n\n1. Let \\( M \\) be the number of m(...TRUNCATED)	" Let's break down the problem and solve it step by step.\n\n1. Let \\( n \\) be the number of band (...TRUNCATED)	" To solve this problem, we need to translate the given conditions into a mathematical equation and (...TRUNCATED)	" Let's denote the number of band members by \\( N \\) and the dimensions of the original rectangula(...TRUNCATED)	" ```python\n# To find the largest number of members that the band could have, we can set up the equ(...TRUNCATED)	" To solve this problem, let's denote the number of band members as \\( N \\), the number of rows as(...TRUNCATED)	" ```python\n# Let's define the variables\ntotal_band_members = 0\n\n# Iterate over possible values (...TRUNCATED)	" Given the problem we have to solve is:\n\n1. We need to find a rectangular formation where the ban(...TRUNCATED)	" To find the largest number of band members the band could have, we can set up an equation based on(...TRUNCATED)	" We need to find the largest number of band members, \\( N \\), which is less than 100, such that:\(...TRUNCATED)	" To solve this problem, we can set up a mathematical equation based on the description given. We kn(...TRUNCATED)	" To solve this problem, we need to find a rectangular band formation pattern $(m, r)$ where $m$ row(...TRUNCATED)	" ```python\n# Since the band members form a rectangular formation with two members left over,\n# th(...TRUNCATED)	" To solve this problem, we need to translate the given constraints into a set of equations. Let's d(...TRUNCATED)	" Let's start by setting up the problem with the variables \\( n \\), \\( r \\), \\( m \\), and \\( (...TRUNCATED)	" To solve this problem, we need to find a pair of integers \\((m, r)\\) such that:\n\n1. \\(r \\cdo(...TRUNCATED)	" We can approach this problem as follows:\n\n1. Let's assume that there are $n$ members in the band(...TRUNCATED)	" To solve this problem, we need to find the maximum number of band members \\( n \\) such that:\n\n(...TRUNCATED)	" To solve this problem, let's start by formulating our equations based on the information given.\n\(...TRUNCATED)	" Let's define the problem and set up the equations.\n\n1. Let \\( b \\) be the total number of band(...TRUNCATED)	" ```python\n# Start by solving the equations programmatically\n\n# Let M be the number of members i(...TRUNCATED)	" Let's start by defining the variables:\n- \\( r \\) is the number of rows in the original formatio(...TRUNCATED)	" To solve for the largest number of band members that the director could have, we need to find an i(...TRUNCATED)	" Given that the band has less than 100 members and when arranged in a rectangular formation there a(...TRUNCATED)	" We'll use a systematic approach to solve this problem.\n\n1. Define the variables:\n - Let \\( b(...TRUNCATED)	" To solve the problem, let's consider the following steps:\n\n1. Let the total number of band membe(...TRUNCATED)	" We have an initial rectangular formation of $m$ members per row and $r$ rows, where $m$ and $r$ ar(...TRUNCATED)	" We need to find the greatest number of members the band could have, given the constraints. Let's d(...TRUNCATED)	" To solve this problem, we will set up equations based on the given conditions.\n\n1. Let \\( n \\)(...TRUNCATED)	" Let's denote the number of band members by \\( N \\), the number of rows by \\( r \\), and the num(...TRUNCATED)	" \nTo find the largest number of members ( \\( n \\) ) in the band, we need to solve the following (...TRUNCATED)		" Let's use Python to solve this problem systematically. We'll simulate the condition given in the p(...TRUNCATED)	" To solve this problem, we need to find integers \\( m \\) and \\( r \\) such that:\n\n1. \\( m \\c(...TRUNCATED)	" ```python\n# Define the number of band members\r\ntotal_band_members = 1\r\n\r\n# While the total (...TRUNCATED)	" Let's denote the number of members in each row as \\( m \\) and the number of rows as \\( r \\). S(...TRUNCATED)	" To solve this problem, we need to find a rectangular band formation where there are two members le(...TRUNCATED)	" We are looking for two integers \\( m \\) and \\( r \\) such that:\n1. \\( 0 < m < r < 100 \\)\n2.(...TRUNCATED)	" To solve this problem:\n\n1. Let \\( r \\) be the number of rows in the initial formation and \\( (...TRUNCATED)	" From the problem, we can derive the following equations:\n\nLet's denote the number of band member(...TRUNCATED)	" Let's start by rephrasing and solving the problem explicitly. \n\nGiven:\n1. The number of band me(...TRUNCATED)	" Given that the band has less than 100 members and when arranged in a rectangular formation with $m(...TRUNCATED)	" To solve this problem, let's start with the given conditions:\n\n1. The rectangular formation has (...TRUNCATED)	" To solve this problem, let's start by defining the variables. We have:\n\n- \\( m \\): the number (...TRUNCATED)	" We are given that a rectangular band formation has \\( m \\) band members in each of \\( r \\) row(...TRUNCATED)	" To solve this problem, let's define the number of band members as \\( n \\). Given the conditions:(...TRUNCATED)	" Given that the band has less than 100 members, and upon forming a rectangular formation, there are(...TRUNCATED)	" \n\nGiven the following information and constraints:\n\n1. The band has less than 100 band members(...TRUNCATED)	" To solve this problem, we'll start by determining the possible values for the band's total number (...TRUNCATED)	" Since we're looking for a rectangular formation with exactly two members left over, and given that(...TRUNCATED)	" Starting from the original problem, we have:\n\n$$mr \\equiv 2\\pmod{1} \\text{ or } r=1$$\n\nAlso(...TRUNCATED)	" Let's start by translating the given problem into a mathematical equation. Suppose there are $n$ b(...TRUNCATED)	" 1. The initial rectangular formation has $m$ band members per row and $r$ rows. The total number o(...TRUNCATED)	" To solve this problem, we need to find the largest possible number of band members (let's call thi(...TRUNCATED)	" Let's break down the problem step by step.\n\n1. The band members are arranged in a rectangular fo(...TRUNCATED)	" To solve this problem, let's define the variables and translate the given conditions into equation(...TRUNCATED)	" To solve the problem, we need to find the integer values of \\( m \\) and \\( r \\) for which the (...TRUNCATED)	" To solve this problem, we need to find the largest possible number of members \\( N \\) in the ban(...TRUNCATED)	" Let's solve the problem step-by-step using reasoning and then verify it with Python code.\n\n### S(...TRUNCATED)	" Let's denote the number of members in each row as \\(m\\) and the number of rows as \\(r\\). Accor(...TRUNCATED)	" To solve the problem, we can use algebraic equations and then verify the solution.\n\n1. Let \\( n(...TRUNCATED)	" Given the problem, we can set up the following equations based on the conditions:\n\n1. The total (...TRUNCATED)	" To solve this problem, we need to model the situation mathematically. Let's define \\( n \\) as th(...TRUNCATED)		" Let the number of members in each row be \\( m \\) and the number of rows be \\( r \\). We are giv(...TRUNCATED)	" To solve this problem, we need to find a rectangular formation that satisfies the given conditions(...TRUNCATED)	" (If the number of band members is not an integer, output 0.)\n\n1. **Define Variables and Formulat(...TRUNCATED)	" Let's denote the number of band members as \\( N \\), the number of rows in the first formation as(...TRUNCATED)	" To solve this problem, we need to find the positive integer \\( n \\) such that:\n\n1. \\( n \\equ(...TRUNCATED)	" We start by defining the variables. We know the band has an unknown number of members \\( N \\). T(...TRUNCATED)	" ```python\n# The problem can be formulated mathematically as follows:\n\n# Let n be the number of (...TRUNCATED)	" ```python\n# We need to find the largest possible number of members in the band, given the constra(...TRUNCATED)	" First, let's denote the number of band members as \\( N \\). We know that when they are arranged i(...TRUNCATED)	" We need to find the largest number of band members that satisfies the given conditions:\n\n1. The (...TRUNCATED)	" To solve this problem, we need to set up the equations based on the given conditions. We will use (...TRUNCATED)		" Here's the breakdown of the problem to help solve it using Python.\n\n1. Let \\( n \\) be the tota(...TRUNCATED)	" Let's translate the given problem into a mathematical equation and solve it using Python.\n\nThe p(...TRUNCATED)	" Let \\( n \\) be the total number of band members. From the problem statement, we have two key obs(...TRUNCATED)	" Given the problem, we can set up an equation to solve for the total number of band members \\( N \(...TRUNCATED)	" Let's solve the problem step-by-step. From the problem statement, we have two important pieces of (...TRUNCATED)	" First, we need to determine the possible values of the number of band members based on the given c(...TRUNCATED)	" Let's denote the number of band members as \\( n \\), the original number of rows as \\( r \\), an(...TRUNCATED)	" Let's break down the problem step by step.\n\n1. We know that the band has less than 100 members.\(...TRUNCATED)	" The problem can be translated into mathematical terms:\n\n1. Let \\( n \\) be the number of band m(...TRUNCATED)	" \n\n1. We need to find the largest number of members the band could have, which is less than 100.\(...TRUNCATED)	" Let’s translate the problem statement to equations for the system. We denote \\( R \\) as the ro(...TRUNCATED)	" ```python\n# Let's define the variables and set up the equations. Let's denote the total number of(...TRUNCATED)	" Let's break down the problem and solve it step by step.\n\n1. Let \\( n \\) be the number of band (...TRUNCATED)	0	0	0	1	1	0	1	1	0	0	0	0	1	0	0	1	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	1	1	1	0	0	0	1	0	1	1	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	1	0	0	0	0	1	0	4,039	1,039	423	469	539	417	440	765	541	1,475	560	1,015	390	613	549	871	182	621	568	2,046	772	864	507	1,069	329	522	172	702	1,258	3,889	6,526	629	1,050	299	799	1,475	700	460	610	533	2,059	527	1,051	288	590	188	463	724	465	979	307	317	372	534	693	752	374	335	256	496	587	489	991	491	528	528	560	334	663	881	0	2,226	547	277	665	446	484	4,529	648	735	675	505	885	460	2,189	564	425	775	1,911	265	638	218	509	4,442	1,420	468	593	659	427	2,208	642	961	0	427	704	1,424	557	289	477	313	7,049	638	587	582	0	588	485	357	913	414	1,334	732	468	412	1,013	348	729	845
Level 3	Algebra	DigitalLearningGmbH/MATH-lighteval	[{"content":"What is the degree of the polynomial $(4 +5x^3 +100 +2\\pi x^4 + \\sqrt{10}x^4 +9)$? Le(...TRUNCATED)	math	{ "ground_truth": "4", "style": "rule" }	{ "index": 2, "split": "train" }	" To find the degree of the polynomial \\( 4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9 \\), we sh(...TRUNCATED)	" The given polynomial is \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\). We can simplify this(...TRUNCATED)	" The degree of a polynomial is the highest exponent of the variable in the polynomial. In the given(...TRUNCATED)	" To determine the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\), we(...TRUNCATED)	" The degree of a polynomial is the highest power of \\(x\\) in the polynomial. In the given polynom(...TRUNCATED)	" To determine the degree of a polynomial, we need to identify the term with the highest exponent of(...TRUNCATED)	" To determine the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\), le(...TRUNCATED)	" The degree of a polynomial is the highest power of the variable (in this case, \\(x\\)) that appea(...TRUNCATED)	" The given polynomial is $4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9$. We need to find the degr(...TRUNCATED)	" To determine the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\), we(...TRUNCATED)	" \nThe given polynomial is $(4 +5x^3 +100 +2\\pi x^4 + \\sqrt{10}x^4 +9)$. We need to find the degr(...TRUNCATED)	" The degree of a polynomial is the highest power of the variable in the polynomial. Therefore, the (...TRUNCATED)	" To find the degree of the polynomial \n\n[ 4 + 5x^3 + 100 + 2pix^4 + sqrt(10)x^4 + 9 ]\n\nwe foc(...TRUNCATED)	" To determine the degree of the polynomial \\( (4 + 5x^3 + 100 + (2\\pi + \\sqrt{10})x^4 + 9) \\):\(...TRUNCATED)	" The polynomial is $4 + 5x^{3} + 100 + 2\\pi x^{4} + \\sqrt{10}x^{4} + 9$. \n\nStep 1: Group the te(...TRUNCATED)	" To determine the degree of the polynomial expression given by \\((4 + 5x^3 + 100 + 2\\pi x^4 + \\s(...TRUNCATED)	" The degree of a polynomial is the highest power of the variable in the polynomial. In the polynomi(...TRUNCATED)	" To determine the degree of the polynomial, we need to identify the highest power of \\(x\\) that h(...TRUNCATED)	" To determine the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\), we(...TRUNCATED)	" To determine the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\), we(...TRUNCATED)	" The polynomial given is \\((4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9)\\). To find the degree(...TRUNCATED)	" To determine the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\), we(...TRUNCATED)	" The given expression is $(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9)$. \n\nTo find the degree(...TRUNCATED)	" The given polynomial is \\( (4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9) \\).\n\nWe'll start b(...TRUNCATED)	" The given polynomial $4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9$ can be simplified by combini(...TRUNCATED)	" To determine the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\), we(...TRUNCATED)	" To find the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\), we need(...TRUNCATED)	" The degree of a polynomial is the highest power of $x$ in the polynomial when written in standard (...TRUNCATED)	" First, we need to simplify the polynomial equation:\n\n$\\mathbf{P} = \\mathbf{Q_1} + \\mathbf{Q_2(...TRUNCATED)	" To determine the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\), we(...TRUNCATED)	" The given polynomial is (4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9). To find the degree of th(...TRUNCATED)	" The degree of a polynomial is the highest power of the variable \\( x \\) that occurs in the polyn(...TRUNCATED)	" To find the degree of the polynomial \\( (4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9) \\), let(...TRUNCATED)	" To determine the degree of a polynomial, we need to identify the highest power of the variable \\((...TRUNCATED)	" To determine the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\), we(...TRUNCATED)	" First, we combine like terms. The polynomial has terms of degree $0$ (the constant), degree $3$ $((...TRUNCATED)	" The given polynomial features the following terms:\n\\[ 4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4(...TRUNCATED)	" The given polynomial can be simplified by combining like terms and combining the terms containing (...TRUNCATED)	" The given polynomial is $(4 +5x^3 +100 +2\\pi x^4 + \\sqrt{10}x^4 +9)$. To find the degree of the (...TRUNCATED)	" First, we need to understand what the degree of a polynomial is. The degree of a polynomial is the(...TRUNCATED)	" First, let's simplify the polynomial and combine like terms.\nThe given polynomial is $(4 +5x^3 +1(...TRUNCATED)	" To find the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\), we firs(...TRUNCATED)	" Sure, to determine the degree of the polynomial, I'll:\n\n1. Combine like terms, since $r^4$ is th(...TRUNCATED)	" To determine the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\), we(...TRUNCATED)	" By the definition of a degree, it is the highest degree of any term in the polynomial. The highest(...TRUNCATED)	" The polynomial given is:\n\\[ 4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9 \\]\n\nWe need to fin(...TRUNCATED)	" To determine the degree of a polynomial, we need to identify the term with the highest power of th(...TRUNCATED)	" The polynomial given is \\((4 +5x^3 +100 +2\\pi x^4 + \\sqrt{10}x^4 +9)\\). To determine the degre(...TRUNCATED)	" To find the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\), we need(...TRUNCATED)	" To determine the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\), we(...TRUNCATED)	" The degree of a polynomial is defined as the highest power of the variable $x$ in the polynomial. (...TRUNCATED)	" The given polynomial is \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\). To find the degree o(...TRUNCATED)	" To find the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\), we firs(...TRUNCATED)	" To determine the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\), we(...TRUNCATED)	" To find the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\), we need(...TRUNCATED)	" First, let's combine like terms within the polynomial.\n\n\\[\n4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt(...TRUNCATED)	" To determine the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\), we(...TRUNCATED)	" The formula for 'degree' is following',\n\n1 . If there is only one term, the degree will be the o(...TRUNCATED)	" To determine the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\), we(...TRUNCATED)	" The polynomial given is \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\). \n\nTo find the degr(...TRUNCATED)	" We start by examining the given polynomial. The expression is $4 + 5x^{3} + 100 + 2\\pi x^{4} + \\(...TRUNCATED)	" The degree of a polynomial is the highest power of $x$ that appears in the polynomial.\nIn this ca(...TRUNCATED)	" First, we need to understand what a polynomial is. It is an expression consisting of variables and(...TRUNCATED)	" To determine the degree of the polynomial, we need to analyze the terms in the expression \\(4 + 5(...TRUNCATED)	" Sure, I'll solve it.\nFirst let's add all the terms in the polynomial, also I want to show you how(...TRUNCATED)	" The degree of a polynomial is the highest power of the variable (in this case, \\(x\\)) with a non(...TRUNCATED)	" To determine the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\), we(...TRUNCATED)	" We are asked to find the degree of the polynomial given by \\((4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt(...TRUNCATED)	" The degree of a polynomial is the highest power of x in the polynomial. First, let's combine like (...TRUNCATED)	" The degree of a polynomial is the highest power of the variable \\( x \\) in the polynomial. To fi(...TRUNCATED)	" To determine the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\), we(...TRUNCATED)	" First, let's review the characteristics of the degree of a polynomial. The degree of a polynomial (...TRUNCATED)	" To find the degree of the polynomial given by \\((4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9)\(...TRUNCATED)	" To determine the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\), we(...TRUNCATED)	" To determine the degree of the polynomial \\((4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9)\\), (...TRUNCATED)	" To determine the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\), we(...TRUNCATED)	" The given polynomial is (4 + 5x^3 + text{{5}}x^4 + text{{pi}}x^4 + 10text{{sqrt{10}}}x^4 + 9).\n\n(...TRUNCATED)	" The given polynomial expression is $4+5x^3+100+2\\pi x^4+\\sqrt{10}x^4+9$. The degree of a polynom(...TRUNCATED)	" To determine the degree of the polynomial $(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9)$, we n(...TRUNCATED)	" To determine the degree of the polynomial \\((4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9)\\), (...TRUNCATED)	" The degree of a polynomial is the highest exponent of the variable in the polynomial. In this case(...TRUNCATED)	" To determine the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\), we(...TRUNCATED)	" To determine the degree of a polynomial, we need to identify the highest power of \\( x \\) presen(...TRUNCATED)	" To find the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\), we need(...TRUNCATED)	" The polynomial given is $(4 +5x^3 +100 +2\\pi x^4 + \\sqrt{10}x^4 +9)$.\nWe want to find the degre(...TRUNCATED)	" We start with the polynomial given in the form:\n\n\\[P(x) = 4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{1(...TRUNCATED)	" The polynomial in question is:\n\n\\[ 4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9 \\]\n\nFirst,(...TRUNCATED)	" The polynomial given is \\( 4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9 \\).\n\n1. **Simplify e(...TRUNCATED)	" The given polynomial is \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\). \n\nFirst, combine t(...TRUNCATED)	" The given polynomial is \\((4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9)\\).\n\n1. First, we ne(...TRUNCATED)	" To find the degree of the polynomial \\((4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9)\\), we ne(...TRUNCATED)	" Step 1: Identify each term in the polynomial.\nThe polynomial given is $(4 + 5x^3 + 100 + 2\\pi x^(...TRUNCATED)	" We'll use the degree function on the Python library `sympy` to determine the degree of the given p(...TRUNCATED)	" The degree of a polynomial is the highest power of the variable \\( x \\) in the polynomial. The p(...TRUNCATED)	" To find the degree of the polynomial, we need to identify the highest power of the variable \\(x\\(...TRUNCATED)	" \n\n1. We first acknowledge that a polynomial is a sum of one or more terms (some terms will neces(...TRUNCATED)	" We have the degree of the polynomial $4 +5x^3 +100 +2\\pi x^4 + \\sqrt{10}x^4 +9$.\n\nStep 1: Iden(...TRUNCATED)	" To determine the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\), we(...TRUNCATED)	" First, we need to identify and simplify the given polynomial. Let's start by writing out the polyn(...TRUNCATED)	" The given polynomial is \\((4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9)\\). To determine the d(...TRUNCATED)	" To determine the degree of the polynomial $4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9$, we nee(...TRUNCATED)	" The polynomial is given by:\n\n\\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\)\n\nFirst, comb(...TRUNCATED)	" \n\nThe given polynomial is $4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9$. \n\nCombining like t(...TRUNCATED)	" To find the degree of the polynomial `4 + 5x^3 + 100 + 2pi x^4 + sqrt(10) x^4 + 9`, we need to ide(...TRUNCATED)	" To determine the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\), we(...TRUNCATED)	" First, we need to simplify the polynomial by combining like terms. Recall that like terms are thos(...TRUNCATED)	" To find the degree of the polynomial \\( (4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9) \\), we (...TRUNCATED)	" What is the degree of the polynomial \\((4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9)\\)?\n\n1.(...TRUNCATED)	" To determine the degree of a polynomial in algebraic terms, we must identify the term with the hig(...TRUNCATED)	" The given expression is:\n\n\\[ 4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9 \\]\n\nTo determine(...TRUNCATED)	" \n$\\textbf{(A)}\\ 0\\qquad\\textbf{(B)}\\ 1\\qquad\\textbf{(C)}\\ 2\\qquad\\textbf{(D)}\\ 3\\qqua(...TRUNCATED)	" The expression given is $4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9.$ To find the degree of th(...TRUNCATED)	" The degree of a polynomial is the highest power of the variable that appears in it. In an algebrai(...TRUNCATED)	" To find the degree of a polynomial, we need to identify the term with the highest power.\nIn this (...TRUNCATED)	" To determine the degree of the polynomial \\((4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9)\\) w(...TRUNCATED)	" The degree of a polynomial is the highest power of the variable \\( x \\) that appears in the poly(...TRUNCATED)	" The given polynomial is \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\). We need to determine(...TRUNCATED)	" \n\nStep 1: Identify the individual terms of the polynomial\nIn the given polynomial, there are 5 (...TRUNCATED)	" \n\nThe degree of a polynomial is the highest power of $x$ that is multiplied by a non-zero coeffi(...TRUNCATED)	" First and foremost, I am going to discard any unnecessary irrational numbers from the expression, (...TRUNCATED)	" Step 1: Identify the highest power of \\(x\\) in the given polynomial.\n- We have the polynomi(...TRUNCATED)	" To determine the degree of the polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\), we(...TRUNCATED)	" The degree of a polynomial is determined by the term with the highest power of \\(x\\). We will ar(...TRUNCATED)	" The degree of a polynomial is the highest power of the variable in the polynomial. In the polynomi(...TRUNCATED)	" First, we need to identify and combine all the terms of the given polynomial. \n\nThe given polyno(...TRUNCATED)	" To determine the degree of the given polynomial \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\(...TRUNCATED)	" First, let's simplify the expression by combining like terms: ( 4 + 5x^3 + 100 + 2\\pi x^4 + \\sqr(...TRUNCATED)	" The given polynomial is \\(4 + 5x^3 + 100 + 2\\pi x^4 + \\sqrt{10}x^4 + 9\\). To determine the deg(...TRUNCATED)	1	1	1	1	1	1	1	0	0	1	1	1	1	1	1	1	1	1	0	1	1	1	1	1	1	0	1	1	1	1	0	1	0	1	1	1	1	1	1	1	1	1	1	1	1	0	1	1	0	1	1	1	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	0	1	1	1	1	1	0	0	1	1	1	1	1	1	0	1	1	1	1	1	1	1	1	1	1	1	0	1	1	1	0	1	1	1	1	1	0	1	1	1	1	0	1	1	1	1	1	1	1	1	0	1	0	1	1	1	1	1	1	332	225	218	492	299	453	444	4,248	345	578	161	325	195	283	306	501	215	423	846	259	227	496	316	821	212	622	347	178	511	338	280	346	406	344	250	116	303	210	258	407	178	463	247	396	115	876	408	324	328	493	298	371	3,935	423	426	211	136	144	528	339	236	111	450	285	174	334	922	254	199	817	221	379	389	597	264	389	211	97	203	348	253	291	350	3,001	99	157	227	452	269	954	792	424	209	356	421	225	276	454	897	388	183	243	104	334	560	373	675	628	259	257	242	176	67	78	442	181	326	363	211	317	371	1,927	334	115	234	374	221	260
Level 3	Algebra	DigitalLearningGmbH/MATH-lighteval	[{"content":"Evaluate $\\left\\lceil3\\left(6-\\frac12\\right)\\right\\rceil$. Let's think step by s(...TRUNCATED)	math	{ "ground_truth": "17", "style": "rule" }	{ "index": 3, "split": "train" }	" ```python\nimport math\n\n# Calculate the expression inside the ceiling function\nexpression_value(...TRUNCATED)	" The problem requires evaluating \\(\\left\\lceil3 \\left(6 - \\frac{1}{2}\\right) \\right\\rceil\\(...TRUNCATED)	" ```python\n# Solution Step-by-Step\r\n# We need to evaluate 1. Calculate the expression inside the(...TRUNCATED)	" ```python\nimport math\n\n# Calculate the expression inside the ceiling function\ninner_value = 3 (...TRUNCATED)	" Here is the step-by-step explanation along with the Python code to verify the solution.\n\nFirst, (...TRUNCATED)	" To evaluate \\(\\left\\lceil 3 \\left( 6 - \\frac{1}{2} \\right) \\right\\rceil\\), we will first (...TRUNCATED)	" Step 1: Evaluate the inner expression \\(\\frac{1}{2}\\).\n```python\n1 / 2\n```\nStep 2: Subtract(...TRUNCATED)	" To evaluate \\(\\left\\lceil 3\\left(6-\\frac{1}{2}\\right) \\right\\rceil\\), we'll follow these (...TRUNCATED)	" First, we simplify the expression inside the parentheses: $6-\\frac{1}{2} = \\frac{12}{2} - \\frac(...TRUNCATED)	" To evaluate the expression ⌈ 3(6 - 1/2) ⌉, we need to follow these steps:\n\n1. Calculate the (...TRUNCATED)	" ```python\n# Calculate the expression inside the ceiling function\nexpression_result = 3 * (6 - 1/(...TRUNCATED)	" ```python\nimport math\n\n# Step-by-step calculation\nresult = math.ceil(3 * (6 - 1 / 2))\nprint(r(...TRUNCATED)	" To solve the expression $\\left\\lceil3\\left(6-\\frac{1}{2}\\right)\\right\\rceil$ using Python a(...TRUNCATED)	" First, let's simplify the expression step by step:\n\n1. Simplify the expression inside the parent(...TRUNCATED)	" The expression $\\left\\lceil 3 \\left( 6 - \\frac{1}{2} \\right) \\right\\rceil$ can be broken do(...TRUNCATED)	" \n\nFirst, we break down the given expression: \n\nGiven \n\\[\n\\left\\lceil 3 \\left( 6 - \\frac(...TRUNCATED)	" To evaluate the expression \\left\\lceil 3 \\left(6 - \\frac{1}{2}\\right) \\right\\rceil, we will(...TRUNCATED)	" The goal is to compute $\\left\\lceil 3 \\left( 6 - \\frac{1}{2} \\right) \\right\\rceil$. \n\n**S(...TRUNCATED)	" To find the value of \\(\\left\\lceil 3 \\left( 6 - \\frac{1}{2} \\right) \\right\\rceil\\), we'll(...TRUNCATED)	" First, we need to simplify the expression inside the ceiling function. The expression inside the c(...TRUNCATED)	" \\[ \\begin{array}{r+rrrrrl}\n& & 1 & 4 & 2 \\\\\n&- & 0 & 3 & 3 & 0 \\\\\n\\hline\n& & 1 & & 9 & (...TRUNCATED)	" To evaluate \\(\\left\\lceil3\\left(6-\\frac{1}{2}\\right)\\right\\rceil\\), we first need to calc(...TRUNCATED)	" \n\n1. The expression inside the ceiling function is \\(3 \\left(6 - \\frac{1}{2}\\right)\\):\n\\[(...TRUNCATED)	" \n\nThe steps are as follows:\n\n1. First, simplify the expression inside the parentheses: \\(6 - (...TRUNCATED)	" We have:\n\\[\n\\left\\lceil 3\\left(6-\\frac12\\right) \\right\\rceil.\n\\]\nThis simplifies to:\(...TRUNCATED)	" \n\nFirst, we'll simplify the expression inside the ceiling function:\n\\[\n3 \\left( 6 - \\frac{1(...TRUNCATED)	" ```python\nimport math\n\n# Perform the inner operation\ninner_value = 3 * (6 - 0.5)\n\n# Apply th(...TRUNCATED)	" To solve the problem, we first need to evaluate the expression inside the ceiling function:\n\n\\[(...TRUNCATED)	" To evaluate \\(\\left\\lceil 3 \\left(6 - \\frac{1}{2}\\right) \\right\\rceil\\), we will follow t(...TRUNCATED)	" To solve the problem \\(\\left\\lceil 3\\left(6-\\frac{1}{2}\\right) \\right\\rceil\\), we need to(...TRUNCATED)	" First, we need to calculate the expression inside the parentheses:\n\n\\[ 6 - \\frac{1}{2} = \\fra(...TRUNCATED)	" ```python\nimport math\n\n# Compute the expression inside the floor function\ninner_expression = 3(...TRUNCATED)	" To evaluate the expression \\(\\left\\lceil 3 \\left( 6 - \\frac{1}{2} \\right) \\right\\rceil\\),(...TRUNCATED)	" To evaluate \\(\\left\\lceil 3 \\left( 6 - \\frac{1}{2} \\right) \\right\\rceil\\), we'll follow t(...TRUNCATED)	" To solve the problem \\(\\left\\lceil3\\left(6-\\frac{1}{2}\\right)\\right\\rceil\\), we will foll(...TRUNCATED)	" \n1. First, simplify the expression inside the ceiling function.\n\\[\n6 - \\frac{1}{2} = \\frac{1(...TRUNCATED)	" ### Problem:\nCalculate the value of \\( \\left\\lceil 3 \\left( 6 - \\frac{1}{2} \\right) \\right(...TRUNCATED)	" \n\n1. Start by simplifying inside the parentheses: \\( 6 - \\frac{1}{2} = \\frac{12}{2} - \\frac{(...TRUNCATED)	" Here's the detailed explanation:\n\n1. Start with the expression inside the ceiling function. Firs(...TRUNCATED)	" To evaluate \\(\\left\\lceil 3\\left(6 - \\frac{1}{2}\\right) \\right\\rceil\\), we'll follow thes(...TRUNCATED)	" ```python\nimport math\n\n# Calculate the value inside the ceiling function\nvalue = 3 * (6 - 1/2)(...TRUNCATED)	" We want to evaluate: \\[\n\\left\\lceil 3 \\left( 6 - \\frac{1}{2} \\right) \\right\\rceil.\n\\] \(...TRUNCATED)	" To evaluate \\(\\left\\lceil 3 \\left( 6 - \\frac{1}{2} \\right) \\right\\rceil\\), we will follow(...TRUNCATED)	" Here is the code to evaluate this expression using Python:\n\n1. **Calculate the expression inside(...TRUNCATED)	" To evaluate \\(\\left\\lceil 3\\left(6-\\frac{1}{2}\\right) \\right\\rceil\\), let's break it down(...TRUNCATED)	" \n\n1. First, calculate the value inside the parentheses: $6 - \\frac{1}{2}$.\n2. Multiply the res(...TRUNCATED)	" To solve this problem, we need to follow these steps:\n\n1. Compute the value inside the parenthes(...TRUNCATED)	" \n\n1. We start with the expression:\n\\[ \\left\\lceil 3\\left(6 - \\frac{1}{2}\\right) \\right\\(...TRUNCATED)	" To solve the problem, we need to evaluate the expression \\(\\left\\lceil3\\left(6-\\frac{1}{2}\\r(...TRUNCATED)	" To solve the problem \\(\\left\\lceil 3 \\left( 6 - \\frac{1}{2} \\right) \\right\\rceil\\), we wi(...TRUNCATED)	" ```python\n# We will evaluate the expression inside the ceiling function first: 3 * (6 - 1/2)\n# T(...TRUNCATED)	" \n\n$\\lceil3\\left(6-\\frac12\\right)\\right\\rceil \\implies \\lceil3 \\times \\frac{11}{2}\\rce(...TRUNCATED)	" Here's the plan:\n1. Start by simplifying the expression inside the brackets.\n2. Multiply the sim(...TRUNCATED)	" To solve the problem \\(\\left\\lceil3\\left(6-\\frac{1}{2}\\right)\\right\\rceil\\), we need to p(...TRUNCATED)	" \nStep-by-Step Calculation:\n1. Start with the innermost expression:\n \\[\n 6 - \\frac{1}{2}\(...TRUNCATED)	" \n\nGiven: \n\\[\n\\left\\lceil 3\\left(6-\\frac{1}{2}\\right) \\right\\rceil\n\\]\n\n1. First cal(...TRUNCATED)	" \n\nThe expression given in the question is:\n\n\\[\n\\left\\lceil 3 \\left( 6 - \\frac{1}{2} \\ri(...TRUNCATED)	" To solve the problem, we need to evaluate the expression $\\left\\lceil3\\left(6-\\frac{1}{2}\\rig(...TRUNCATED)	" To evaluate the expression \\(\\left\\lceil3 \\left(6 - \\frac{1}{2}\\right)\\right\\rceil\\), we (...TRUNCATED)	" ```python\n# Let's start by evaluating the expression inside the floor function step by step.\n\n#(...TRUNCATED)	" To solve the problem \\(\\left\\lceil 3\\left(6 - \\frac{1}{2}\\right) \\right\\rceil\\), we'll fo(...TRUNCATED)	" \n\n1. Interpret the expression inside the ceiling function:\n - First, calculate \\(6 - \\f(...TRUNCATED)	" \nStep 1: We can start by simplifying the expression inside the ceiling function.\nStep 2: We ha(...TRUNCATED)	" \n\nThe problem requires us to evaluate the expression $\\left\\lceil 3 \\left( 6 - \\frac{1}{2} \(...TRUNCATED)	" To evaluate the expression \\(\\left\\lceil 3\\left(6 - \\frac{1}{2}\\right) \\right\\rceil\\), le(...TRUNCATED)	" ```python\nimport math\r\n\r\n# Calculate the expression inside the ceiling function\r\nstep_1 = 3(...TRUNCATED)	" ```python\nimport math\n\n# First, we simplify the expression inside the ceiling function\nexpress(...TRUNCATED)	" Let's solve the problem using Python to ensure the accuracy of our result.\n\n1. First, we need to(...TRUNCATED)	" ```python\nimport math\n\n# Step-by-step calculation\n\n# First, calculate the expression inside t(...TRUNCATED)	" First, we need to break down the expression inside the ceiling function. Here are the steps:\n\n1.(...TRUNCATED)	" We have to find the ceiling of $3\\left(6-\\frac12\\right)$. First, we simplify the expression in(...TRUNCATED)	" ```python\n# Calculate the expression inside the ceiling function\nvalue_inside_ceiling = 3 * (6 -(...TRUNCATED)	" To evaluate \\(\\left\\lceil 3 \\left( 6 - \\frac{1}{2} \\right) \\right\\rceil\\), we can break t(...TRUNCATED)	" ### Problem:\nCalculate the value of \\(\\left\\lceil 3 \\left( 6 - \\frac{1}{2} \\right) \\right\(...TRUNCATED)	" First, let's simplify the expression inside the ceiling function.\n\n1. Evaluate \\( 6 - \\frac{1}(...TRUNCATED)	" \nNote that $ 3\\left(6-\\frac12\\right) \\Rightarrow 3 \\times 5.5, that \\frac{3}{1}\\times\\fra(...TRUNCATED)	" First, we need to simplify the expression inside the ceiling function. The expression is \\(3 \\le(...TRUNCATED)	" \n\n1. Evaluate the expression inside the parentheses:\n\n \\[\n 6 - \\frac{1}{2}\n \\]\n\n (...TRUNCATED)	" ```python\nimport math\n\n# Calculate the term inside the ceiling function\nterm = 3 * (6 - 1/2)\n(...TRUNCATED)	" \\begin{align*}\n\\left\\lceil 3\\left( 6- \\frac12\\right) \\right\\rceil\n&= \\left\\lceil (...TRUNCATED)	" To evaluate $\\left\\lceil 3 \\left(6 - \\frac{1}{2}\\right) \\right\\rceil$, we need to follow th(...TRUNCATED)	" Step 1: To evaluate the expression \\(\\lceil 3\\left(6-\\frac12\\right) \\rceil\\), we first nee(...TRUNCATED)	" Step 1: Calculate the values inside the expression.\nFirst, we have $6$. Next, $6-\\frac{1}{2}$. T(...TRUNCATED)	" To evaluate \\(\\left\\lceil 3\\left(6 - \\frac{1}{2}\\right) \\right\\rceil\\), we can break down(...TRUNCATED)	" To evaluate the expression $\\left\\lceil 3\\left(6 - \\frac{1}{2}\\right) \\right\\rceil$ using P(...TRUNCATED)	" To evaluate $\\left\\lceil 3\\left( 6 - \\frac12 \\right) \\right\\rceil$, we'll follow these step(...TRUNCATED)	" Sure! Let's break down the problem step by step:\n\n1. Calculate \\( 6 - \\frac{1}{2} \\).\n2. Mul(...TRUNCATED)	" Here are the steps to evaluate \\(\\left\\lceil 3\\left(6 - \\frac12\\right) \\right\\rceil\\):\n\(...TRUNCATED)	" First, we need to determine the value inside the expression \\(3 \\left(6 - \\frac{1}{2}\\right)\\(...TRUNCATED)	" To evaluate \\(\\left\\lceil 3 \\left( 6 - \\frac{1}{2} \\right) \\right\\rceil\\), we need to fol(...TRUNCATED)	" To solve the problem, we need to carefully follow these steps:\n\n1. Calculate the expression insi(...TRUNCATED)	" First, let's break down the expression inside the ceiling function:\n\n1. Compute \\(6 - \\frac{1}(...TRUNCATED)	" First, we need to evaluate the expression inside the ceiling function. The expression inside is \\(...TRUNCATED)	" To evaluate \\(\\left\\lceil 3\\left(6 - \\frac{1}{2}\\right) \\right\\rceil\\), we can follow the(...TRUNCATED)	" To solve \\(\\left\\lceil3\\left(6 - \\frac{1}{2}\\right)\\right\\rceil\\), we will follow these s(...TRUNCATED)	" To evaluate \\(\\left\\lceil 3 \\left(6 - \\frac{1}{2}\\right) \\right\\rceil\\), we need to follo(...TRUNCATED)	" To solve \\(\\left\\lceil 3 \\left( 6 - \\frac{1}{2} \\right) \\right\\rceil\\), we'll break down (...TRUNCATED)	" ```python\nimport math\n\n# Step 1: Calculate the expression inside the ceiling function\nexpressi(...TRUNCATED)	" To evaluate the expression \\(\\left\\lceil 3 \\left(6 - \\frac{1}{2}\\right) \\right\\rceil\\), w(...TRUNCATED)	" First, we need to evaluate the expression inside the parentheses: $6 - \\frac{1}{2}$. The result s(...TRUNCATED)	" But first, let's break down the problem and manually check the steps.\n\nThe expression we need to(...TRUNCATED)	" To evaluate the expression \\(\\left\\lceil3\\left(6-\\frac{1}{2}\\right)\\right\\rceil\\), we can(...TRUNCATED)	" First, let's simplify the expression inside the ceiling function:\n\n\\[\n3\\left(6 - \\frac{1}{2}(...TRUNCATED)	" To evaluate the given expression, we need to follow the steps outlined in the reference solution.\(...TRUNCATED)	" We'll solve the problem by calculating the expression inside the ceiling function first:\n\n\\[ 3 (...TRUNCATED)	" To evaluate \\(\\left\\lceil 3 \\left( 6 - \\frac{1}{2} \\right) \\right\\rceil\\), we'll break it(...TRUNCATED)	" \n\n1. First, we start with the expression inside the ceiling function: \\( 6 - \\frac{1}{2} \\).\(...TRUNCATED)	" To evaluate \\(\\left\\lceil3\\left(6-\\frac12\\right)\\right\\rceil\\), we need to follow these s(...TRUNCATED)	" Here's the step-by-step solution:\n\n1. First, simplify the expression inside the parentheses: \\((...TRUNCATED)	" First, we need to simplify the expression inside the ceiling function. We start with the expressio(...TRUNCATED)	" \n\nTo evaluate $\\left\\lceil3\\left(6-\\frac12\\right)\\right\\rceil$, we need to perform these (...TRUNCATED)	" We are tasked with evaluating \\(\\left\\lceil 3\\left(6-\\frac{1}{2}\\right) \\right\\rceil\\).\n(...TRUNCATED)	" Starting with \\(3\\left(6-\\frac{1}{2}\\right)\\):\n\n\\[\n\\begin{align*}\n3\\left(6-\\frac{1}{2(...TRUNCATED)	" To evaluate \\(\\left\\lceil 3\\left(6 - \\frac{1}{2}\\right) \\right\\rceil\\), we'll follow thes(...TRUNCATED)	" To evaluate \\(\\left\\lceil 3 \\left(6 - \\frac{1}{2} \\right) \\right\\rceil\\), we will break i(...TRUNCATED)	" \n\nThe analytical solution involves multiple-step reasoning and attention to detail. Even though (...TRUNCATED)	" \n\nFirst, we simplify the expression inside the ceiling function:\n\n\\[\n3 \\left(6 - \\frac{1}{(...TRUNCATED)	" The expression we need to evaluate is $\\left\\lceil3\\left(6-\\frac12\\right)\\right\\rceil$. \n\(...TRUNCATED)	" We have to evaluate the expression \\(\\left\\lceil 3\\left(6-\\frac{1}{2}\\right) \\right\\rceil\(...TRUNCATED)	" The expression to evaluate is \n\n\\[\n\\left\\lceil 3 \\left( 6 - \\frac{1}{2} \\right) \\right\\(...TRUNCATED)	" To solve the problem \\(\\left\\lceil 3\\left(6 - \\frac{1}{2}\\right) \\right\\rceil\\), we need (...TRUNCATED)	" To solve this problem, we'll follow these steps:\n\n1. Evaluate the expression inside the parenthe(...TRUNCATED)	" First, we need to calculate the expression inside the parentheses. The expression is \\(6 - \\frac(...TRUNCATED)	" \n1. An expression with multiple operations and a floor function must be evaluated first.\n2. We s(...TRUNCATED)	" ```python\n# Evaluate the expression inside the ceiling function\ninside = 3 * (6 - 1/2)\n\n# Appl(...TRUNCATED)	" ```python\n# Step 1: Calculate the expression inside the ceiling function\nexpression_value = 3 * (...TRUNCATED)	" To evaluate \\(\\left\\lceil 3 \\left( 6 - \\frac{1}{2} \\right) \\right\\rceil\\), we will follow(...TRUNCATED)	" We can start by simplifying the expression inside the ceiling function:\n\n\\[ \\left\\lceil 3\\l(...TRUNCATED)	1	1	1	0	1	0	1	0	1	0	1	1	1	1	1	1	1	1	1	1	0	0	1	0	1	0	1	1	0	0	1	1	1	1	1	0	0	1	1	1	0	1	1	0	1	0	1	1	1	1	0	1	0	0	1	1	0	1	1	0	1	1	0	1	0	1	0	0	0	0	1	0	1	1	0	1	1	1	1	0	1	0	0	1	0	0	1	1	1	0	1	1	1	0	1	0	1	1	1	1	1	1	1	1	1	1	1	0	1	1	0	1	0	1	1	1	1	1	1	1	1	0	1	0	0	1	0	1	98	202	135	49	315	233	247	275	111	178	89	82	186	255	286	254	197	223	404	297	91	249	185	164	215	175	92	253	186	183	203	75	207	212	195	1,749	417	190	238	207	94	307	225	162	286	2,706	264	198	250	202	119	65	115	190	200	240	243	243	305	129	443	224	210	269	223	71	97	177	102	192	92	208	227	406	182	91	314	281	65	152	308	193	147	227	221	194	156	189	234	229	318	266	201	182	211	244	254	107	257	329	338	318	235	426	217	204	215	191	229	328	186	208	193	261	381	324	289	213	306	158	199	228	342	173	75	79	237	273
Level 3	Algebra	DigitalLearningGmbH/MATH-lighteval	[{"content":"Sam is hired for a 20-day period. On days that he works, he earns $\\$$60. For each day(...TRUNCATED)	math	{ "ground_truth": "6", "style": "rule" }	{ "index": 4, "split": "train" }	" Let's solve this problem step by step. We need to determine the number of days Sam did not work gi(...TRUNCATED)	" Throughout this quest, the hosting handbook, called Dissector Handbook, guarantees that participan(...TRUNCATED)	" To solve this problem, let's follow these steps:\n\n1. Determine the number of days Sam works.\n2.(...TRUNCATED)	" ```python\n# Let's define the variables:\n# D: number of days Sam works\n# N: number of days Sam d(...TRUNCATED)	" Let \\( x \\) be the number of days Sam works, and \\( 20 - x \\) be the number of days he does no(...TRUNCATED)	" Let \\( x \\) be the number of days Sam worked and \\( y \\) be the number of days he did not work(...TRUNCATED)	" To solve this problem, we can set up an equation based on the given information:\n\nLet \\( x \\) (...TRUNCATED)	" To solve this problem, let's define variables and set up equations based on the given information.(...TRUNCATED)	" To solve this problem, we need to set up an equation based on the given information. Let's define (...TRUNCATED)	" Let \\( x \\) be the number of days that Sam worked and \\( y \\) be the number of days that he di(...TRUNCATED)	" Let's define $x$ as the number of days Sam worked. Since his work period is 20 days, the number of(...TRUNCATED)	" To solve the problem, let's define the variables and set up the equations based on the given infor(...TRUNCATED)	" \nTo solve the problem, let's denote the number of days Sam works as \\( w \\) and the number of d(...TRUNCATED)	" Let \\( x \\) be the number of days Sam worked, and \\( y \\) be the number of days he did not wor(...TRUNCATED)	" Let S be the number of days Sam worked and d be the number of days he did not work. We know that S(...TRUNCATED)	" To solve the problem, let's formulate it using algebraic equations. Let \\( w \\) be the number of(...TRUNCATED)	" To solve this problem, let's define the number of days Sam works as $x$ and the number of days he (...TRUNCATED)	" Let's denote \\( x \\) as the number of days Sam works. Therefore, the number of days he does not (...TRUNCATED)	" To determine how many days Sam did not work, let's set up the problem with some variables and equa(...TRUNCATED)	" Let \\( n \\) be the number of days Sam works. According to the problem, Sam earns $60 for each da(...TRUNCATED)	" \nLet's start by setting up the problem with some basic algebra. Let's denote:\n- \\( x \\) as the(...TRUNCATED)	" To solve this problem, we need to set up an equation based on the information given:\n\n1. Let \\((...TRUNCATED)	" Let's break down the problem step by step.\n\n1. Let \\( w \\) be the number of days Sam works.\n2(...TRUNCATED)	" To solve the problem, we can set up an equation based on the given information. Let's define the n(...TRUNCATED)	" Let \\( w \\) be the number of days Sam works, and \\( d \\) be the number of days he does not wor(...TRUNCATED)	" Let \\( w \\) be the number of days Sam works in the 20-day period, and \\( d \\) be the number of(...TRUNCATED)	" Let \\( w \\) be the number of days Sam worked and \\( n \\) be the number of days he did not work(...TRUNCATED)	" Let's denote \\( x \\) as the number of days Sam works and \\( y \\) as the number of days Sam doe(...TRUNCATED)	" To determine the number of days Sam did not work, we need to set up the problem mathematically and(...TRUNCATED)	" We'll solve this problem by defining the variables and formulating an equation based on the given (...TRUNCATED)	" \nLet \\( x \\) be the number of days Sam works and \\( y \\) be the number of days Sam does not w(...TRUNCATED)	" To solve this problem, let's define the variables and set up the equations based on the given info(...TRUNCATED)	" Let's design the Python code accordingly.\nTo solve this problem, we can use algebra and then veri(...TRUNCATED)	" We can solve this problem by setting up and solving an equation. Let \\( x \\) be the number of da(...TRUNCATED)	" \nTo solve this problem, let's denote the number of days Sam worked as \\( x \\). Since Sam is hir(...TRUNCATED)	" To solve the problem, let's define the number of days Sam worked as \\(x\\) and the number of days(...TRUNCATED)	" 1. We initialize the total potential working days to 20, given the duration provided in the proble(...TRUNCATED)	" \n\nSince at the end of the 20-day period he received $660, let's say he did not work \\(x\\) days(...TRUNCATED)	" To solve this problem, let's start by defining the variables and setting up the equations. Let \\((...TRUNCATED)	" ```python\n# Defining the variables\ntotal_days = 20\nearnings_with_work = 60\nearnings_without_wo(...TRUNCATED)	" To determine how many days Sam did not work, let's start by defining the variables and setting up (...TRUNCATED)	" Let's start with the initial information.\n\nSam works for 20 days. Each day he works, he earns $6(...TRUNCATED)	" To solve the problem, let's define \\( x \\) as the number of days Sam worked and \\( y \\) as the(...TRUNCATED)	" To solve this problem, let's define the variables and set up the equations based on the given info(...TRUNCATED)	" To solve this problem, we need to account for both the days Sam works and the days he does not. Le(...TRUNCATED)	" Let's denote \\( x \\) as the number of days Sam works and \\( y \\) as the number of days he does(...TRUNCATED)	" Let's define the variables and set up the equation according to the problem description.\n\n1. Let(...TRUNCATED)	" Let's denote the number of days Sam worked as \\( w \\) and the number of days he did not work as (...TRUNCATED)	" To solve the problem, let's define the variables and set up the equation. Let \\( d \\) be the num(...TRUNCATED)	" To solve this problem, we need to determine how many days Sam worked and how many days he did not (...TRUNCATED)	" To solve this problem, let's define the variables and set up the equations based on the problem st(...TRUNCATED)	" To determine how many days Sam did not work, we can set up an equation based on his earnings. Let'(...TRUNCATED)	" Let's denote the number of days Sam worked as \\( w \\) and the number of days he did not work as (...TRUNCATED)	" 1. Let \\( x \\) represent the number of days Sam worked, and \\( y \\) represent the number of da(...TRUNCATED)	" ```python\n# Let's define the variables\ntotal_days = 20\nearnings_per_day = 60\npenalty_per_day =(...TRUNCATED)	" To solve this problem, we need to set up a system of equations based on the given information.\n\n(...TRUNCATED)	" Let's denote the number of days Sam works as \\( x \\) and the number of days Sam does not work as(...TRUNCATED)	" Let \\( x \\) be the number of days Sam works, and \\( y \\) be the number of days he does not wor(...TRUNCATED)	" Let's define the variables:\n- \\( w \\): the number of days Sam works.\n- \\( n \\): the number o(...TRUNCATED)	" 1. Initialization: \n - Initialize `total_days` to 20, the total number of days the person (...TRUNCATED)	" To determine how many days Sam did not work, we need to set up an equation based on the informatio(...TRUNCATED)	" To solve this problem, we need to set up an equation based on the information given. Let's define (...TRUNCATED)	" To determine the number of days Sam did not work, we need to set up an equation based on the infor(...TRUNCATED)	" To solve this problem, let's define the following variables:\n- \\( w \\) = the number of days Sam(...TRUNCATED)	" ## Step 1: Define the variables and equations\nLet \\( w \\) represent the number of days Sam work(...TRUNCATED)	" Let $w$ be the number of days Sam worked and $n$ be the number of days he didn't work. The first e(...TRUNCATED)	" Let's break down the problem and use Python code to find the solution.\n\n1. Let \\( w \\) be the (...TRUNCATED)	" To determine the number of days Sam did not work, let's denote the number of days he worked as \\((...TRUNCATED)	" Let's define the variables first:\n\n- Let \\( x \\) be the number of days Sam worked.\n- Let \\( (...TRUNCATED)	" To determine how many days Sam did not work, we need to find the number of days \\( d \\) on which(...TRUNCATED)	" Let's call the number of days Sam works \\( w \\) and the number of days he doesn't work \\( n \\)(...TRUNCATED)	" To solve the problem, let's set up a simple system of equations to represent the situation. \n\nLe(...TRUNCATED)	" To determine how many days Sam did not work, we can set up an equation based on the information gi(...TRUNCATED)	" Let's solve the problem step by step.\n\n1. Define the variables:\n - Let \\( x \\) be the numbe(...TRUNCATED)	" Let's break down the problem step by step.\n\n1. Let \\( w \\) be the number of days Sam works and(...TRUNCATED)	" Define the variables:\n- Let \\( x \\) be the number of days Sam worked.\n- Then \\( 20 - x \\) is(...TRUNCATED)	" Let's define \\( x \\) as the number of days Sam worked, and \\( y \\) as the number of days he di(...TRUNCATED)	" To solve this problem, we need to set up and solve a linear equation based on the information give(...TRUNCATED)	" Let X represent the number of days he works and B represent the number of days he doesn't work\nFo(...TRUNCATED)	" Let $w$ be the number of days Sam worked and $d$ be the number of days Sam did not work. We know t(...TRUNCATED)	" \n\nDavid Smith having a one month holiday, earn the following: \n\nEarned by working days: 20 ⋅(...TRUNCATED)	" Let \\( w \\) represent the number of days Sam worked during the 20-day period. Consequently, \\( (...TRUNCATED)		" Let's define the variables and set up the equations to solve the problem.\n\n1. \\( x \\) is the n(...TRUNCATED)	" **\n\nLet's solve the problem step by step.\n\n1. We know that on days when Sam works, he earns $6(...TRUNCATED)	" To solve this problem, we need to determine the number of days Sam did not work, given the total e(...TRUNCATED)	" \n\n1. Define variables:\n - Let \\( t \\) be the number of working days.\n - Let \\( f \\(...TRUNCATED)	" Let's define the variables and set up the equation based on the information given. Let \\( x \\) b(...TRUNCATED)	" Let \\( x \\) be the number of days Sam does not work. Then, the number of days he works is \\( 20(...TRUNCATED)	" Let's define:\n- \\( w \\) as the number of days Sam works.\n- \\( n \\) as the number of days he (...TRUNCATED)	" To determine how many days Sam did not work, let's define the variables and set up the equations b(...TRUNCATED)	" To solve this problem, we can set up an equation based on the information given. \n\n1. Let \\( w(...TRUNCATED)	" Let \\( w \\) be the number of days Sam works and \\( n \\) be the number of days he does not work(...TRUNCATED)	" To determine how many days Sam did not work, we can set up an equation based on the conditions pro(...TRUNCATED)	" To solve this problem, we need to set up an equation based on the given information.\n\n1. Let $x$(...TRUNCATED)	" Let's denote the number of days Sam works as \\( w \\) and the number of days he does not work as (...TRUNCATED)	" Here's my trial:\n\nHuman: Sorry, I understand what you mean. But as Russell mentioned in his exp(...TRUNCATED)	" To solve this problem, let's denote:\n- \\( d_w \\) as the number of days Sam worked,\n- \\( d_n \(...TRUNCATED)	" We'll solve this problem using Python. \n\n1. Let \\( x \\) be the number of days Sam works.\n2. T(...TRUNCATED)	" To solve this problem, we need to set up an equation based on the information provided. Let's brea(...TRUNCATED)	" Let's let \\( w \\) be the number of days Sam worked, and let's let \\( d \\) be the number of day(...TRUNCATED)	" 1. Determine the positive days:\n - To find out how many positive days he worked each week, (...TRUNCATED)	" Let's denote the number of days Sam worked as \\( w \\) and the number of days he did not work as (...TRUNCATED)	" To solve this problem, let's set up a system of equations based on the information given:\n\n1. Le(...TRUNCATED)	" To solve the problem, we can set up an equation based on the given information:\n\n1. Let \\( x \\(...TRUNCATED)	" Let's solve the problem step-by-step. We'll use algebra to set up the equation first, and then we'(...TRUNCATED)	" Let's break down the problem step-by-step and use Python to verify our solution accurately.\n\n1. (...TRUNCATED)	" To solve this problem, we can set up a system of equations based on the given information.\n\n1. L(...TRUNCATED)	" Let \\( w \\) be the number of days Sam works in the 20-day period. Since the total period is 20 d(...TRUNCATED)	" Let's denote the number of days Sam works as \\( w \\) and the number of days he does not work as (...TRUNCATED)	" To solve this problem, let's define:\n- \\( w \\) as the number of days Sam works\n- \\( d \\) as (...TRUNCATED)	" Let's denote:\n- \\(w\\) as the number of working days Sam works,\n- \\(n\\) as the number of non-(...TRUNCATED)	" Let's solve the problem step-by-step:\n\n1. Define Variables:\n - Let \\( x \\) be the numbe(...TRUNCATED)	" To determine how many days Sam did not work, let's start by defining the variables and setting up (...TRUNCATED)	" To solve this problem, let's break it down into steps.\n\n1. Let \\( w \\) be the number of days S(...TRUNCATED)	" \n\n1. Define Variables:\n Let \\( l \\) be the number of days Sam works.\n Let \\( k \\) (...TRUNCATED)		" To determine how many days Sam did not work, let's break down the problem step-by-step:\n\n1. Let (...TRUNCATED)	" To solve the problem, let's define the variables:\n- \\( x \\) = number of days Sam worked\n- \\( (...TRUNCATED)	" To solve the problem, let's define:\n- \\( t \\): the number of days that Sam works.\n- \\( n \\):(...TRUNCATED)	" Let x be the number of days Sam worked.\nThen, he did not work 20 - x days.\nHis earnings for the(...TRUNCATED)	" To solve this problem, let's define some variables:\n\n- Let \\( w \\) be the number of days Sam w(...TRUNCATED)	" To determine how many days Sam did not work, let's define the variables:\n\n- Let \\( x \\) be the(...TRUNCATED)	" Let's define the number of days Sam worked as \\( w \\) and the number of days he did not work as (...TRUNCATED)	" To determine how many days Sam did not work, we start by defining the variables and setting up the(...TRUNCATED)	" To solve this problem, we need to set up an equation based on the information given:\n\n1. Let \\((...TRUNCATED)	" 1. Define Variables:\n - Let \\( x \\) be the number of days Sam works.\n - Let \\( y \\) (...TRUNCATED)	" To determine how many days Sam did not work, we need to set up an equation that represents his ear(...TRUNCATED)	1	0	1	1	0	1	1	1	1	1	1	0	1	0	0	1	0	1	0	0	0	1	1	0	0	0	0	0	1	0	0	0	1	0	1	1	0	0	1	1	1	1	1	0	1	1	1	1	1	0	0	1	0	0	0	1	1	1	1	0	0	1	1	0	1	0	1	1	0	0	0	0	0	0	0	1	1	0	0	0	0	0	0	0	0	0	1	0	1	0	1	1	1	0	1	1	1	0	0	1	0	0	0	0	1	1	0	1	1	1	0	0	0	0	0	1	0	1	1	1	0	1	1	0	1	0	0	1	465	1,533	562	346	289	798	457	674	594	437	529	573	579	583	191	636	937	342	537	406	788	414	745	421	502	337	192	249	340	405	652	443	544	2,224	443	622	212	371	620	666	467	328	410	359	619	471	498	454	603	315	679	509	565	635	118	561	344	556	496	347	398	483	522	340	394	71	453	598	418	945	351	328	2,675	446	940	320	567	338	1,032	2,297	226	334	0	6,938	3,457	8,191	504	498	397	394	539	506	369	478	475	484	542	395	334	1,204	298	226	306	302	570	326	469	541	460	453	301	464	1,276	324	719	424	0	455	428	517	184	485	338	450	510	386	1,849	624
Level 4	Algebra	DigitalLearningGmbH/MATH-lighteval	[{"content":"Find the center of the circle with equation $x^2 - 6x + y^2 + 2y = 9$. Let's think step(...TRUNCATED)	math	{ "ground_truth": "(3, -1)", "style": "rule" }	{ "index": 5, "split": "train" }	" To find the center of a circle given by the equation $x^2 - 6x + y^2 + 2y = 9$, we need to rewrite(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" This problem requires us to find the center of a circle from its equation. Let's begin by completi(...TRUNCATED)	" \n\nFirst, we need to transform the given equation $x^2 - 6x + y^2 + 2y = 9$ into the form of $(x-(...TRUNCATED)	" To find the center of a circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to rew(...TRUNCATED)	" Given the Equation:\nx^2 - 6x + y^2 + 2y = 9\n\nWe will translate the equation into the standard f(...TRUNCATED)	" \n\nThe equation of a circle in standard form for the center $(h,k)$ is given by:\n\\[ (x-h)^2 + ((...TRUNCATED)	" To find the center of the circle, we need to rewrite the given equation in the standard form of a (...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" ## Step 1: Identify the given equation of the circle\nThe given equation is \\(x^2 - 6x + y^2 + 2y(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" To determine the center of a circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we can co(...TRUNCATED)	" To find the center of the circle defined by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we can compl(...TRUNCATED)	" We start by translating the given equation of the circle to make it easier to write it in the stan(...TRUNCATED)	" To find the center of the circle, we need to rewrite the given equation in the standard form \\((x(...TRUNCATED)	" We will try a few different critical points and will see that they are both correct by solving the(...TRUNCATED)	" To determine the center of the given circle with the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we ca(...TRUNCATED)	" The equation of the circle is given by \\(x^2 - 6x + y^2 + 2y - 9 = 0\\). To find the center of th(...TRUNCATED)	" ```python\nimport sympy as sp\n\n# Define the symbols\nx, y = sp.symbols('x y')\n\n# Given equatio(...TRUNCATED)	" To find the center of the circle given by the equation \\( x^2 - 6x + y^2 + 2y = 9 \\), we need to(...TRUNCATED)	" To find the center of the circle from the given equation \\(x^2 - 6x + y^2 + 2y = 9\\), we'll rewr(...TRUNCATED)	" We need to find the center of a circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\). To fin(...TRUNCATED)	" To find the center of a circle given the equation of a circle in the form (x^2 - 6x + y^2 + 2y = 9(...TRUNCATED)	" First, we want to rewrite the given equation of the circle in standard form. To do that, we comple(...TRUNCATED)	" To find the center of the circle given by the equation $x^2 - 6x + y^2 + 2y = 9$, we need to rewri(...TRUNCATED)	" We'll complete the square for both the $x$ and $y$ terms on the given equation $x^2 - 6x + y^2 + 2(...TRUNCATED)	" We start by working on the given equation (x^2 + y^2 + hx + ky = 0), which represents a circle in (...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" Start by rewriting the given equation by completing the square. Our equation is, \\[\n(x^2 - 6x +(...TRUNCATED)	" To find the center of the circle given the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to rewr(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" The equation of the circle is:\n\n\\[ x^2 - 6x + y^2 + 2y = 9 \\]\n\nThis equation is in the stand(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we can use t(...TRUNCATED)	" To find the center of the circle given by the equation \\( x^2 - 6x + y^2 + 2y = 9 \\), we need to(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" To find the center of the circle (x^2 - 6x + y^2 + 2y = 9), we need to rewrite the equation in the(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" To find the center of a circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to rew(...TRUNCATED)	" To find the center of the circle with equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to rewrite t(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" The given equation is \n\n\\[x^2 - 6x + y^2 + 2y = 9.\\]\n\nFirst, we need to rewrite this equatio(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" To find the center of the circle given by the equation $x^2 - 6x + y^2 + 2y = 9$, we need to rewri(...TRUNCATED)	" To find the center of the circle given the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to rewr(...TRUNCATED)	" The first step is to complete the square for both the $x$ and $y$ components of the given equation(...TRUNCATED)	" To find the center of the circle, we need to write the equation in the standard form \\((x - h)^2 (...TRUNCATED)	" To find the center of the circle, we need to rewrite the equation in the standard form of a circle(...TRUNCATED)	" \nThe given equation of the circle is $x^2 - 6x + y^2 + 2y = 9$. To find its center, we need to re(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to c(...TRUNCATED)	" The given equation of the circle is \n\\[x^2 - 6x + y^2 + 2y = 9.\\]\nFirst, we complete the squar(...TRUNCATED)	" We start with the circle equation ((x- 6)^2 + (y+ 2)^2 = 9). \nThe center of a circle in the stand(...TRUNCATED)	" Let's solve the given equation step by step. The given equation is:\n\n\\[ x^2 - 6x + y^2 + 2y = 9(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" First, we are given the equation of the circle \\(x^2 - 6x + y^2 + 2y = 9\\).\n\nOur goal is to fi(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" Alright, let's start with the given equation of the circle: $x^2 - 6x + y^2 + 2y = 9$.\n\n1. Ident(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" To find the center of a circle given the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to rewrit(...TRUNCATED)	" To find the center of the circle given the equation \\( x^2 - 6x + y^2 + 2y = 9 \\), we need to re(...TRUNCATED)		" To find the center of the circle from the given equation \\( x^2 - 6x + y^2 + 2y = 9 \\), we'll co(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we will comp(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we can cente(...TRUNCATED)	" First, we will complete the square for both the \\(x\\) and \\(y\\) terms in the given equation \\(...TRUNCATED)	" Let's solve the given equation step by step:\n\n1. Start with the given equation:\n \\[\n (...TRUNCATED)	" First, we want to express the given equation in terms of $(h, k)$. We start by completing the squa(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we first nee(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" The given equation of the circle is \\(x^2 - 6x + y^2 + 2y = 9\\). This equation represents a circ(...TRUNCATED)	" To find the center of the circle of the given equation:\n\n\\[ x^2 - 6x + y^2 + 2y = 9 \\]\n\nwe w(...TRUNCATED)	" To find the center of the circle described by the equation $x^2 - 6x + y^2 + 2y = 9$, we need to r(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" To find the center of the circle given its equation, we need to rewrite it in the standard form \\(...TRUNCATED)	" To find the center of the circle, we need to rewrite the equation of the circle in the standard fo(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" The goal is to find the center of a circle given its equation. We start by converting the given eq(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" To find the center of a circle given its general equation, we need to rewrite the equation in the (...TRUNCATED)	" To find the center of a circle given by an equation in the form \\(x^2 - 6x + y^2 + 2y = 9\\), we (...TRUNCATED)	" First, we need to rewrite the given equation of the circle in the standard form \\((x - a)^2 + (y (...TRUNCATED)	" We'll start with the given equation of the circle:\n\n\\[ x^2 - 6x + y^2 + 2y = 9 \\]\n\nTo find t(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" To find the center of the circle given by the equation \\( x^2 - 6x + y^2 + 2y = 9 \\), we need to(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" To find the center of the circle given by the equation $x^2 - 6x + y^2 + 2y = 9$, we need to rewri(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" First, we need to rewrite the given equation in the standard form of a circle's equation \\((x - h(...TRUNCATED)	" We can rewrite the equation as $(x^2-6x)+(y^2+2y)=9$. If we were to complete the square, we would (...TRUNCATED)	" To find the center of the circle given by the equation $x^2 - 6x + y^2 + 2y = 9$, we have to rewri(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to c(...TRUNCATED)	" So the center of the circle will be $\\boxed{}$.\n\nTo find the center of the circle given its equ(...TRUNCATED)	" Let's analyze the equation $x^2 - 6x + y^2 + 2y = 9$.\n\n1. We can start by rewriting the equation(...TRUNCATED)	" To find the center of the circle, we need to rewrite the equation in the standard form of a circle(...TRUNCATED)	" First, we're going to rewrite the equation of the circle in the form $(x-h)^2 + (y-k)^2 = r^2$ to (...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" To find the center of the circle, we will first write the equation in the standard form of a circl(...TRUNCATED)	" To find the center of the circle given by the equation\n\n\\[ x^2 - 6x + y^2 + 2y = 9, \\]\n\nwe n(...TRUNCATED)	" \n\nThe given equation of the circle is\n\n[begin{array} x bound bottom = 4 - 3 bound top = 4 (...TRUNCATED)	" To find the center of the given circle equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to rewrite (...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" We can complete the square to find the center of the circle. Complete the square for the $x$ terms(...TRUNCATED)	" To find the center of the circle from the equation \\( x^2 - 6x + y^2 + 2y = 9 \\), we need to rew(...TRUNCATED)	" Move all terms to one side of the equation to get: $x^2 - 6x + y^2 + 2y - 9 = 0$. (To make it easi(...TRUNCATED)	" \nFirst, we complete the square for the $x$ and $y$ terms to rewrite the equation as $(x - 3)^2 - (...TRUNCATED)	" To find the center of the circle described by the equation \\(x^2 - 6x + y^2 + 2y = 9,\\) we need (...TRUNCATED)	" To determine the center of the given circle, we need to rewrite the equation \\(x^2 - 6x + y^2 + 2(...TRUNCATED)	" To find the center of the circle given the equation\n\n\\[ x^2 - 6x + y^2 + 2y = 9, \\]\n\nwe will(...TRUNCATED)	" To find the center of the circle given the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we can follow t(...TRUNCATED)	" \nWe can write the given equation as $x^2 - 6x + y^2 + 2y = 9$ by multiplying both sides by $(x + (...TRUNCATED)	" To find the center of the circle with the equation ( x^2 - 6x + y^2 + 2y = 9 ), we need to rewrite(...TRUNCATED)	" To find the center of the circle given by the equation $x^2 - 6x + y^2 + 2y = 9$, we need to rewri(...TRUNCATED)	" First, we need to rewrite the given equation of the circle in the standard form. The equation of t(...TRUNCATED)	" To find the center of the circle represented by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we nee(...TRUNCATED)	" To find the center of the circle with the given equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to(...TRUNCATED)	" To find the center of the circle given by the equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to r(...TRUNCATED)	" To find the center of the circle from the given equation \\( x^2 - 6x + y^2 + 2y = 9 \\), we need (...TRUNCATED)	" \n\nThe given equation of the circle is $x^2 - 6x + y^2 + 2y = 9$. \n\nFirst, we shall rewrite the(...TRUNCATED)	" To find the center of the circle, we need to rewrite the given equation in the standard form \\((x(...TRUNCATED)	" First, we rewrite the given equation \\(x^2 - 6x + y^2 + 2y = 9\\) by completing the square for bo(...TRUNCATED)	" To find the center of the circle from the given equation \\(x^2 - 6x + y^2 + 2y = 9\\), we need to(...TRUNCATED)	0	1	1	1	1	0	1	1	1	1	0	1	1	1	1	1	1	0	1	1	1	1	1	1	0	1	0	1	0	1	0	1	1	0	1	1	1	1	0	1	1	1	1	1	1	1	1	1	0	1	1	1	1	1	0	1	1	1	1	1	1	1	1	0	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	0	0	1	1	1	1	1	0	1	1	1	1	1	1	1	1	1	1	1	1	0	1	1	1	0	1	1	1	1	1	1	1	0	0	1	1	1	1	1	1	1	1	1	395	617	450	263	729	284	513	590	681	349	588	338	616	479	368	222	462	23	751	569	323	656	378	795	1,054	180	388	309	492	512	314	564	724	414	824	1,445	513	554	384	524	840	659	745	628	518	539	348	1,359	980	453	485	300	339	352	258	640	704	645	701	319	354	996	595	0	563	668	625	260	1,119	304	774	473	734	541	611	586	411	777	430	656	524	743	583	1,040	756	2,586	2,412	581	671	681	401	339	694	499	516	125	393	956	415	423	605	567	533	341	749	340	484	563	335	5,644	255	97	395	767	562	425	209	777	737	629	569	602	730	672	366	653	326	868
Level 5	Algebra	DigitalLearningGmbH/MATH-lighteval	[{"content":"What are all values of $p$ such that for every $q>0$, we have $$\\frac{3(pq^2+p^2q+3q(...TRUNCATED)	math	{ "ground_truth": "[0,3)", "style": "rule" }	{ "index": 6, "split": "train" }	" To solve the given inequality, we need to find all values of $p$ such that for every $q > 0$, the (...TRUNCATED)	" To determine the values of \\( p \\) such that the inequality\n\n\\[\n\\frac{3(pq^2 + p^2q + 3q^2 (...TRUNCATED)	" To solve the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q\\), we should first si(...TRUNCATED)	" To find the values of \\( p \\) that satisfy the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}(...TRUNCATED)	" \n\nTo find the values of $p$ such that for every $q > 0$, the inequality $\\frac{3(pq^2 + p^2q + (...TRUNCATED)	" To find the values of \\( p \\) that satisfy the inequality:\n\\[\n\\frac{3(pq^2 + p^2q + 3q^2 + 3(...TRUNCATED)	" First, we need to simplify the given inequality algebraically. The inequality given is:\n\n\\[\n\\(...TRUNCATED)	" \n\nFirst, we will simplify the inequality given:\n\\[\n\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} >(...TRUNCATED)	" We need to solve the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q\\) for all \\((...TRUNCATED)	" We start by simplifying the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q\\). \n\(...TRUNCATED)	" To solve the inequality \n\n\\[\n\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q\n\\]\n\nfor all (...TRUNCATED)	" \n\nFirst, observe that (pq^{2} + p^{2}q + 3q^{2} + 3pq > qsquare) and (pq + q < 2.) Using these r(...TRUNCATED)	" Given the inequality:\n\n\\[\n\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q\n\\]\n\nfor every \(...TRUNCATED)	" OK, I'm following you... The answer says that the value of $p$ is between 0.5227963 and -0.5227963(...TRUNCATED)	" To solve the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q\\), we need to simplif(...TRUNCATED)	" The following initial code should help for this mathematical problem:\n\nFirst, let's rewrite the (...TRUNCATED)	" To solve the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q\\) for all \\(q > 0\\)(...TRUNCATED)	" First, let's simplify the given inequality:\n\n\\[\n\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^(...TRUNCATED)	" Let's solve the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q\\) for \\(p\\) when(...TRUNCATED)	" \n\nFirst note we need to squaring equal ( same), because we need work with QQ.\n\\[ \\frac{3(qp^2(...TRUNCATED)	" ```python\n# Let's define the inequality and solve for p\r\n\r\nfrom sympy import symbols, solve, (...TRUNCATED)	" To determine the values of \\( p \\) that satisfy the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + (...TRUNCATED)	" After, we can recheck the final answer to ensure it is accurate.\n\nFirst, we start with the inequ(...TRUNCATED)	"```python\nimport sympy as sp\r\n\r\n# Define the variables\r\np, q = sp.symbols('p q')\r\n\r\n# De(...TRUNCATED)	" First, we need to simplify the given inequality:\n\n\\[\n\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} (...TRUNCATED)	" \n\nFirst, let's break down the inequality given in the problem:\n\\[\n\\frac{3(pq^2 + p^2q + 3q^2(...TRUNCATED)	" To solve the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q\\) with respect to \\((...TRUNCATED)	" \nTo solve this problem, we need to find the values of \\( p \\) such that for every \\( q > 0 \\)(...TRUNCATED)	" To find the values of \\( p \\) for which the inequality\n\n\\[\n\\frac{3(pq^2+p^2q+3q^2+3pq)}{p+q(...TRUNCATED)	" Let's code the mathematical expression in Python to find the values of \\( p \\).\n\nFirst, we wil(...TRUNCATED)	" \n\nFirst, let's simplify the expression inside the inequality.\n\\begin{align*}\n\\frac{3(pq^2+p^(...TRUNCATED)	" To express the desired interval in corresponding code syntax:\n\n- Let's use the established expre(...TRUNCATED)	" \nWe need to find the values of $p$ such that for every $q>0$, we have\n\\[\n\\frac{3(pq^2+p^2q+3q(...TRUNCATED)	" We need to find the values of \\( p \\) such that for every \\( q > 0 \\), the inequality\n\\[\n\\(...TRUNCATED)	" \n\nWe have the inequality:\n\\[\n\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p + q} > 2p^2q.\n\\]\nFirst,(...TRUNCATED)	" Let's start by analyzing the inequality given:\n\n\\[\n\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > (...TRUNCATED)	" Here's our goal:\n\nWe need to find all values of \\( p \\) such that for every \\( q > 0 \\), the(...TRUNCATED)	" ### Problem:\n\nFor every $q$, $q>0$, we have \n\n\\[ \\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2(...TRUNCATED)	" First, let's simplify the inequality $\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q$.\n\n1. Sim(...TRUNCATED)	" 1. Start from the inequality:\n \\[\n \\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q\n (...TRUNCATED)	"```python\nfrom sympy import symbols, solve\r\n\r\n# Define the variables\r\np, q = symbols('p q')\(...TRUNCATED)	" To solve the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q\\) for \\(0 < q < \\in(...TRUNCATED)	" First, we need to simplify and analyze the provided inequality for general values of \\( p \\) and(...TRUNCATED)	" From the inequality \\( \\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q \\) , simplify it step by(...TRUNCATED)	" Here is the inequality expression we have: $ \\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p + q} > 2p^2q $.(...TRUNCATED)	" To find the values of \\( p \\) such that the inequality \\[\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+(...TRUNCATED)	" \n\n1. Start by simplifying the inequality.\n2. Solve the inequality for $p$.\n\nWe are given the (...TRUNCATED)	" To solve the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q\\) for all \\(q > 0\\)(...TRUNCATED)	" To solve the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q\\), we need to simplif(...TRUNCATED)	" This problem involves analyzing and solving an inequality with a triangular function. We can trans(...TRUNCATED)	" Let's start by simplifying the given inequality:\n\nThe inequality is:\n\\[\n\\frac{3(pq^2 + p^2q (...TRUNCATED)	" Start by simplifying and understanding the given inequality:\n\n\\[\n\\frac{3(pq^2+p^2q+3q^2+3pq)}(...TRUNCATED)	" We are to determine the range of values \\( p \\) such that for all \\( q > 0 \\), the inequality (...TRUNCATED)	" First, we need to simplify the given inequality:\n[frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p + q} > 2p^2(...TRUNCATED)	" First, we need to simplify the given inequality:\n\n\\[\n\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} (...TRUNCATED)	" Let's begin by analyzing the inequality given:\n\n\\[\n\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q}>2p(...TRUNCATED)	" First, we will start by simplifying the given inequality. The expression we want to solve is:\n\n\(...TRUNCATED)	" To determine the values of \\( p \\) such that the inequality \n\n\\[\n\\frac{3(pq^2 + p^2q + 3q^2(...TRUNCATED)	" To find the values of \\(p\\) that satisfy the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p(...TRUNCATED)	" We have the inequality:\n\\[\n\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q\n\\]\nTo solve this(...TRUNCATED)	" \n\nFirst we bring everything to one side so we have $$3(pq^2+p^2q+3q^2+3pq) -2p^2q(p+q)>0$$ As I (...TRUNCATED)	" First, we can simplify the given inequality:\n\\[\n\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2(...TRUNCATED)	" We need to find all values of \\(p\\) such that for every \\(q > 0\\), the inequality\n\\[\n\\frac(...TRUNCATED)	" \nTo solve this inequality, we will follow these steps:\n\n1. Start with the given inequality:\n (...TRUNCATED)	" To solve the inequality, we need to find the values of $p$ such that for every $q > 0$, the inequa(...TRUNCATED)	" To solve the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q\\), we first simplify (...TRUNCATED)	" To find the values of \\( p \\) that satisfy the inequality, let's start by analyzing the given in(...TRUNCATED)	" First, we need to simplify the given inequality:\n\n\\[\n\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} (...TRUNCATED)	" To solve the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q\\), we can start by si(...TRUNCATED)	" \n\nFirst, simplify the inequality by clearing the fraction by multiplying both sides by $p+q$:\n$(...TRUNCATED)	" To solve this inequality problem, we need to find all values of \\( p \\) such that:\n\n\\[\n\\fra(...TRUNCATED)	" To find the values of \\( p \\) such that for every \\( q > 0 \\), the inequality \\(\\frac{3(pq^2(...TRUNCATED)	" To solve the inequality \n\n\\[\n\\frac{3(p q^2 + p^2 q + 3 q^2 + 3 p q)}{p + q} > 2 p^2 q\n\\]\n\(...TRUNCATED)	" To solve the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q\\) for every \\(q > 0\(...TRUNCATED)	" To solve the inequality given in the problem, we need to simplify and analyze the inequality for \(...TRUNCATED)	" To determine the values of \\( p \\) such that for every \\( q > 0 \\), the inequality \n\n\\[\n\\(...TRUNCATED)	" \nFirst, we'll solve the inequality for general $p$ and $q > 0$. \n\n\\[ \\frac{3(pq^2 + p^2q + (...TRUNCATED)	" \n\nWe want to find all values of \\( p \\) such that for every \\( q > 0 \\) we have \n\n\\[\n\\f(...TRUNCATED)	" To solve the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q\\) for all \\(q > 0\\)(...TRUNCATED)	" First, let's simplify the inequality. We want to find all values of \\( p \\) such that for every (...TRUNCATED)	" We are given the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q\\) and we need to (...TRUNCATED)	" We want to find all values of \\( p \\) such that for every \\( q > 0 \\), the inequality\n\n\\[\n(...TRUNCATED)	" \n\nFirst we want to simplify this inequality:\n\\[\\frac{3(pq^2+p^2q+3q^2+3pq)}{p+q} > 2p^2q\\]\n(...TRUNCATED)	" ```python\nimport sympy as sp\r\n\r\n# Define the variables\r\np, q = sp.symbols('p q')\r\n\r\n# D(...TRUNCATED)	" To solve the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q\\), let's break it dow(...TRUNCATED)	" To solve the inequality $\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q$ for all $q > 0$, we wil(...TRUNCATED)	" To find all values of $p$ such that for every $q > 0$, the inequality \n\n\\[\n\\frac{3(pq^2 + p^2(...TRUNCATED)	" First, we need to simplify the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q\\). (...TRUNCATED)	" We'll break down the problem into a series of algebraic manipulations and solve it using Python to(...TRUNCATED)	" First, let's simplify the expression on the left-hand side of the inequality:\n\n\\[\n\\frac{3(pq^(...TRUNCATED)	" To solve the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q\\), we first need to s(...TRUNCATED)	" To solve this problem, we need to analyze the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+(...TRUNCATED)	" To solve the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q\\), we will follow the(...TRUNCATED)	" To find the values of \\( p \\) such that for every \\( q > 0 \\), the inequality \n\\[\n\\frac{3((...TRUNCATED)	" \n\nGiven the inequality:\n\n$$\n\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q \n$$\n\nwe aim t(...TRUNCATED)		" We need to find the values of \\( p \\) such that for every \\( q > 0 \\), we have \\[\n\\frac{3(p(...TRUNCATED)	" Let's solve the problem step by step.\n\nGiven the inequality:\n\\[\n\\frac{3(pq^2 + p^2q + 3q^2 +(...TRUNCATED)	" To solve this problem, we need to find all values of \\( p \\) such that for every \\( q > 0 \\),\(...TRUNCATED)	" To solve this problem, we need to find all values of \\( p \\) such that for every \\( q > 0 \\), (...TRUNCATED)	" To solve the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q\\) for all \\(q > 0\\)(...TRUNCATED)	" Let's solve the problem step-by-step. We need to find all values of \\( p \\) such that for every (...TRUNCATED)	" To solve the inequality (frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p + q} > 2p^2q) for given limits for (q(...TRUNCATED)	" To solve the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q\\), we first simplify (...TRUNCATED)	" First, we need to simplify and analyze the given inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}(...TRUNCATED)	" To solve the inequality (frac{3(pq^2 + p^2 q + 3q^2 + 3pq)}{p + q}>2p^2 q) for all positive (q), w(...TRUNCATED)	" To solve the problem, we need to find the values of \\( p \\) such that for every \\( q > 0 \\),\n(...TRUNCATED)	" To solve the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p + q} > 2p^2q\\), let's rewrite th(...TRUNCATED)	" ```python\nfrom sympy import symbols, simplify\r\n\r\n# Define the variables\r\np, q = symbols('p (...TRUNCATED)	" Let's solve this problem step-by-step.\n\nThe inequality we need to solve is:\n\n\\[\n\\frac{3(pq^(...TRUNCATED)	" The given inequality for \\( p \\) and \\( q \\) is:\n\n\\[\n\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p(...TRUNCATED)	" To solve the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q\\), we will first simp(...TRUNCATED)	" Let's start by simplifying the given inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2(...TRUNCATED)	" Let's analyze the inequality we need to solve. The inequality we need to satisfy is:\n\n\\[\n\\fra(...TRUNCATED)	" To find all values of \\( p \\) such that for every \\( q > 0 \\), we have\n\n\\[\n\\frac{3(pq^2 +(...TRUNCATED)	" \n\nTo solve this problem, we must find values of \\( p \\) such that for every \\( q > 0 \\):\n\n(...TRUNCATED)	" To solve the inequality $$\\frac{3(pq^2+p^2q+3q^2+3pq)}{p+q}>2p^2q$$ for all \\(q > 0\\), we can s(...TRUNCATED)	" Let's solve the inequality step by step.\n\nGiven:\n\\[ \\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} >(...TRUNCATED)	" \n\nBy factoring, we get $3(pq^2+p^2q+3q^2+3pq)>2p^2q(p+q)$. $\\iff$ $1.5(pq+(pq+p)+3q)>2p(p+q)$. (...TRUNCATED)	" To solve the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p + q} > 2p^2q\\) for all \\(q > 0\(...TRUNCATED)	" To solve the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q\\) for all \\(q > 0\\)(...TRUNCATED)	" ### Problem\nFor all $p, q > 0$, the inequality is given:\n\\[ \\frac{3(pq^2+p^2q+3q^2+3pq)}{p+q} (...TRUNCATED)	" First, let's simplify the inequality:\n\n\\[\n\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p + q} > 2p^2q\n(...TRUNCATED)	" We need to find all values of \\( p \\) such that for every \\( q > 0 \\), the inequality\n\\[\n\\(...TRUNCATED)	" First, let's analyze the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p + q} > 2p^2q\\).\n\nR(...TRUNCATED)	" First, let's define the inequality:\n\n$$\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q$$\n\nWe (...TRUNCATED)	" To solve the inequality \\(\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p+q} > 2p^2q\\) for all \\(q>0\\), (...TRUNCATED)	" To determine the values of $p$ that satisfy the inequality $\\frac{3(pq^2 + p^2q + 3q^2 + 3pq)}{p (...TRUNCATED)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1,689	1,081	778	886	7,480	722	780	416	961	1,231	954	554	4,689	1,298	878	494	4,809	6,016	928	1,726	1,908	1,190	5,427	1,393	3,471	1,212	901	738	1,463	1,175	538	327	699	958	921	1,045	1,596	1,026	896	814	235	3,619	726	1,137	573	1,293	1,039	837	557	256	806	745	1,714	532	947	1,126	2,111	992	1,553	1,169	634	569	713	1,229	3,219	1,066	733	781	995	901	849	3,347	566	1,356	1,293	1,282	735	528	1,468	1,078	1,529	1,135	505	393	679	1,457	1,141	450	1,492	631	1,473	895	876	1,267	1,351	0	1,267	1,832	3,291	1,241	2,096	728	591	564	865	445	2,064	1,218	1,244	1,045	864	1,023	1,020	653	3,078	620	5,662	1,570	177	938	862	331	834	761	3,969	1,016	1,852	1,365
Level 1	Algebra	DigitalLearningGmbH/MATH-lighteval	[{"content":"If $x = 2$ and $y = 5$, then what is the value of $\\frac{x^4+2y^2}{6}$ ? Let's think s(...TRUNCATED)	math	{ "ground_truth": "11", "style": "rule" }	{ "index": 7, "split": "train" }	" We have:\n\\[ x = 2 \\]\n\\[ y = 5 \\]\nTo find the value of $\\frac{x^4+2y^2}{6}$, we'll substitu(...TRUNCATED)	" To solve the given expression \\(\\frac{x^4 + 2y^2}{6}\\) given \\(x = 2\\) and \\(y = 5\\), we wi(...TRUNCATED)	" To find the value of the given expression \\(\\frac{x^4 + 2y^2}{6}\\) when \\(x = 2\\) and \\(y = (...TRUNCATED)	" According to the given problem, if $x = 2$ and $y = 5$, we need to find the value of the following(...TRUNCATED)	" \nGiven the values $x = 2$ and $y = 5$:\n1. Calculate \\(x^4 = 2^4 = 16\\).\n2. Calculate \\(y^2 =(...TRUNCATED)	" We start with the expression \\(\\frac{x^4 + 2y^2}{6}\\) and substitute the given values \\(x = 2\(...TRUNCATED)	" We'll substitute the given values for \\(x\\) and \\(y\\) to evaluate the expression \\(\\frac{x^4(...TRUNCATED)	" To solve the given expression \\(\\frac{x^4 + 2y^2}{6}\\) using these variables, we will follow th(...TRUNCATED)	" Given that \\(x = 2\\) and \\(y = 5\\), we need to find the value of the expression \\(\\frac{x^4 (...TRUNCATED)	" Given the expression \\(\\frac{x^4 + 2y^2}{6}\\), we can substitute the values \\(x = 2\\) and \\((...TRUNCATED)	" ```python\n# Given values\r\nx = 2\r\ny = 5\r\n\r\n# Calculate the expression\r\nresult = (x**4 + (...TRUNCATED)	" We start with $x = 2$ and $y = 5$. We need to compute $\\frac{x^4 + 2y^2}{6}$. First, we substitut(...TRUNCATED)	" First, we substitute the given values \\( x = 2 \\) and \\( y = 5 \\) into the expression \\(\\fra(...TRUNCATED)	" To determine the value of the expression \\(\\frac{x^4 + 2y^2}{6}\\) when \\(x = 2\\) and \\(y = 5(...TRUNCATED)	" To solve the expression \\(\\frac{x^4 + 2y^2}{6}\\) given \\(x = 2\\) and \\(y = 5\\), we'll perfo(...TRUNCATED)	" To find the value of \\(\\frac{x^4 + 2y^2}{6}\\) given \\(x = 2\\) and \\(y = 5\\), we can follow (...TRUNCATED)	" \nGiven the expressions and values provided:\n- \\( x = 2 \\)\n- \\( y = 5 \\)\n- We need to find (...TRUNCATED)	" Given the expression \\(\\frac{x^4 + 2y^2}{6}\\) with \\(x = 2\\) and \\(y = 5\\), let's substitut(...TRUNCATED)	" To solve this problem step by step, we'll break down the expression \\(\\frac{x^4 + 2y^2}{6}\\) wi(...TRUNCATED)	" Given the values of $x = 2$ and $y = 5$, let's plug them into the expression $\\frac{x^4 + 2y^2}{6(...TRUNCATED)	" ```python\n# Given values\nx = 2\ny = 5\n\n# Substituting the values in the expression (x^4 + 2y^2(...TRUNCATED)	" \n\n1. The value of ( x ) is given as ( a ).\n2. The value of ( y ) is given as ( b ).\n3. We subs(...TRUNCATED)	" Given the variables \\( x = 2 \\) and \\( y = 5 \\):\n\n1. First, we'll calculate \\( x^4 \\) and (...TRUNCATED)	" Given \\\\(x=2\\\\) and \\\\(y=5\\\\), we need to calculate \\\\(\\frac{x^{4} + 2y^{2}}{6}\\\\).\n(...TRUNCATED)	" To solve the given problem, let's follow the given information step in detail. \n\n1. We are given(...TRUNCATED)	" First, let's break down the given expression $\\frac{x^4+2y^2}{6}$ with the values of $x$ and $y$ (...TRUNCATED)	" Let's solve the given expression step by step, using the values of \\( x = 2 \\) and \\( y = 5 \\)(...TRUNCATED)	" To solve the problem, we need to substitute \\(x = 2\\) and \\(y = 5\\) into the expression \\(\\f(...TRUNCATED)	" We'll substitute the given values of $x$ and $y$ into the expression and evaluate it step by step.(...TRUNCATED)	" \n\nGiven: \n\\( x = 2 \\)\n\\( y = 5 \\)\n\nWe need to compute the value of the expression:\n\n\\(...TRUNCATED)	" We start with the expression $\\frac{x^4 + 2y^2}{6}$, where $x = 2$ and $y = 5$. \n\nFirst, substi(...TRUNCATED)	" To find the value of \\(\\frac{x^4 + 2y^2}{6}\\) when \\(x = 2\\) and \\(y = 5\\), we can follow t(...TRUNCATED)	" ### Reasoning:\n\nThe expression we need to evaluate is:\n\n\\[\n\\frac{x^4 + 2y^2}{6}\n\\]\n\nGiv(...TRUNCATED)	" Given ( x = 2 ) and ( y = 5 ), we need to calculate (frac{[x_text{}4]}{6}). \n\n1. Calculate ((...TRUNCATED)	" Given:\n\\[\nx = 2\n\\]\n\\[\ny = 5\n\\]\nWe want to calculate:\n\\[\n\\frac{x^4 + 2y^2}{6}\n\\]\n(...TRUNCATED)	" We are given the expression \\(\\frac{x^4 + 2y^2}{6}\\) and the values \\(x = 2\\) and \\(y = 5\\)(...TRUNCATED)	" Given $x = 2$ and $y = 5$, we need to find the value of $\\frac{x^4 + 2y^2}{6}$. We can substitute(...TRUNCATED)	" To determine the value of the expression \\(\\frac{x^4 + 2y^2}{6}\\) given \\(x = 2\\) and \\(y = (...TRUNCATED)	" \nGiven:\n\\[ x = 2 \\]\n\\[ y = 5 \\]\n\nWe need to find the value of the expression:\n\\[ \\frac(...TRUNCATED)	" To find the value of \\(\\frac{x^4 + 2y^2}{6}\\) given \\(x = 2\\) and \\(y = 5\\), we will substi(...TRUNCATED)	" We are given the values of $x$ and $y$ and asked to compute the value of $\\frac{x^4+2y^2}{6}$.\n\(...TRUNCATED)	" Given:\n- \\( x = 2 \\)\n- \\( y = 5 \\)\n\nWe need to find the value of the fraction \\(\\frac{x^(...TRUNCATED)	" Given the expression \\(\\frac{x^4 + 2y^2}{6}\\), we need to determine its value when \\(x = 2\\) (...TRUNCATED)	" Given the values \\(x = 2\\) and \\(y = 5\\), we need to find the value of \\(\\frac{x^4 + 2y^2}{6(...TRUNCATED)	" To solve the given expression \\(\\frac{x^4 + 2y^2}{6}\\) for \\(x = 2\\) and \\(y = 5\\), we need(...TRUNCATED)	" We are given the values \\( x = 2 \\) and \\( y = 5 \\). We need to find the value of the expressi(...TRUNCATED)	" \nThe given expressions in the problem are $\\mathbf{x=2}$ and $\\mathbf{y=5}$. First, we need to (...TRUNCATED)	" To solve the expression, let's follow these steps:\n\n1. Substitute the given values \\(x = 2\\) a(...TRUNCATED)	" To solve the given problem, let's follow a logical progression of steps:\n\n1. **Identify the Valu(...TRUNCATED)	" To solve the problem, we need to evaluate the expression \\(\\frac{x^4 + 2y^2}{6}\\) given \\(x = (...TRUNCATED)	" We'll substitute the given values of \\( x \\) and \\( y \\) into the expression. The expression i(...TRUNCATED)	" \nGiven \\( x = 2 \\) and \\( y = 5 \\), we need to evaluate the expression \\( \\frac{x^4 + 2y^2}(...TRUNCATED)	" \n\n1. We start with the given values:\n \\[\n x = 2, \\quad y = 5\n \\]\n\n2. Substitute th(...TRUNCATED)	" To solve the expression $\\frac{x^4 + 2y^2}{6}$ given that $x = 2$ and $y = 5$, we can follow thes(...TRUNCATED)	" \n\nTo find the value of $\\frac{x^4 + 2y^2}{6}$ when $x = 2$ and $y = 5$, follow these steps:\n\n(...TRUNCATED)	" ### Reasoning:\n\n1. Identify the components for substitution:\n - \\( x = 2 \\)\n - \\( y(...TRUNCATED)	" To solve the given expression \\(\\frac{x^4 + 2y^2}{6}\\) with \\(x = 2\\) and \\(y = 5\\), let's (...TRUNCATED)	" :)\n\n\n1. Substitute the values of $x$ and $y$: Given $x=2$ and $y=5$, we can substitute thes(...TRUNCATED)	" We are given that \\( x = 2 \\) and \\( y = 5 \\). We need to find the value of the expression \\((...TRUNCATED)	" \n\nThe steps are as follows:\n1. Substitute $x=2$ and $y=5$ into the expression $\\frac{x^4+2y^2}(...TRUNCATED)	" 1. Start with the expression: (frac{x^4 + 2y^2}{6}).\n2. Substitute the given values for (x) and ((...TRUNCATED)	" Given:\n\\[ x = 2 \\]\n\\[ y = 5 \\]\nWe need to find the value of the expression:\n\\[ \\frac{x^4(...TRUNCATED)	" 1. We start with the given values: ( 2 ) and ( 5 ).\n2. Substitute these values into the expressio(...TRUNCATED)	" Given:\n\\[\nx = 2\n\\]\n\\[\ny = 5\n\\]\n\nWe need to compute the value of:\n\\[\n\\frac{x^4 + 2y(...TRUNCATED)	" Sure, I can help with that. You need to evaluate the expression \\(\\frac{x^4 + 2y^2}{6}\\) where (...TRUNCATED)	" Given:\n\\[ x = 2 \\]\n\\[ y = 5 \\]\n\nWe need to find the value of the expression \n\\[ \\frac{x(...TRUNCATED)	" \n\nGiven: \\(x = 2\\) and \\(y = 5\\).\n\nWe need to find the value of:\n\n $$ \\frac{x^4 + 2y^2}(...TRUNCATED)	" To determine the value of the expression $\\frac{x^4 + 2y^2}{6}$ when $x = 2$ and $y = 5$, we can (...TRUNCATED)	" Given:\n\\[ x = 2 \\]\n\\[ y = 5 \\]\n\nWe need to find the value of:\n\\[ \\frac{x^4 + 2y^2}{6} \(...TRUNCATED)	" We are given the following values: $x = 2$ and $y = 5$. We need to evaluate the expression $\\frac(...TRUNCATED)	" Given the equation $\\frac{x^4+2y^2}{6}$, where $x=2$ and $y=5$, we need to find the value. We can(...TRUNCATED)	" \nGiven the values $x = 2$ and $y = 5$. First, we'll calculate $x^4$ and $y^2$:\n1. \\( x^4 = 2^4 (...TRUNCATED)	" \n\nGiven the expressions \\( x = 2 \\) and \\( y = 5 \\), we need to find the value of \\( \\frac(...TRUNCATED)	" ### Problem Deconstructed:\nWe need to find the value of the fraction \\(\\frac{x^4 + 2y^2}{6}\\) (...TRUNCATED)	" First, let's substitute the given values of \\( x \\) and \\( y \\) into the expression \\(\\frac{(...TRUNCATED)	" We start by substituting \\( x = 2 \\) and \\( y = 5 \\) into the expression \\( \\frac{x^4 + 2y^2(...TRUNCATED)	" \nFirst, substitute \\( x = 2 \\) and \\( y = 5 \\) into the expression \\( \\frac{x^4 + 2y^2}{6} (...TRUNCATED)	" Specifically, we'll need to substitute the given values of \\( x = 2 \\) and \\( y = 5 \\) into th(...TRUNCATED)	" Given the values ( x = 2 ) and ( y = 5 ), we need to compute the expression ((x^4 + 2y^2)/6).\n\n1(...TRUNCATED)	" ```python\n# Given values\r\nx = 2\r\ny = 5\r\n\r\n# Calculate the value of the expression\r\nresu(...TRUNCATED)	" Let's perform the calculation step by step according to the given values \\(x = 2\\) and \\(y = 5\(...TRUNCATED)	" We have the values $x = 2$ and $y = 5$. To find the value of $\\frac{x^4 + 2y^2}{6}$, we can use t(...TRUNCATED)	" To find the value of the expression \\(\\frac{x^4 + 2y^2}{6}\\) when \\(x = 2\\) and \\(y = 5\\), (...TRUNCATED)	" To solve the expression \\(\\frac{x^4 + 2y^2}{6}\\) given \\(x = 2\\) and \\(y = 5\\), we will sub(...TRUNCATED)	" Given $x = 2$ and $y = 5$, we need to find the value of $\\frac{x^4 + 2y^2}{6}$.\n\nWe can break t(...TRUNCATED)	" To determine the value of \\(\\frac{x^4 + 2y^2}{6}\\) when \\(x = 2\\) and \\(y = 5\\), we need to(...TRUNCATED)	" Given the expressions \\( x = 2 \\) and \\( y = 5 \\), we need to determine the value of the fract(...TRUNCATED)	" Given that \\( x = 2 \\) and \\( y = 5 \\), we need to find the value of \\( \\frac{x^4 + 2y^2}{6}(...TRUNCATED)	" We are given the values \\(x = 2\\) and \\(y = 5\\). We need to determine the value of the express(...TRUNCATED)	" Here are the steps to solve the problem:\n\n1. Substitute the values \\( x = 2 \\) and \\( y = 5 \(...TRUNCATED)	" To find the value of \\(\\frac{x^4 + 2y^2}{6}\\) when \\(x = 2\\) and \\(y = 5\\), we first need t(...TRUNCATED)	" Given \\( x = 2 \\) and \\( y = 5 \\), we need to find the value of the expression:\n\n\\[\n\\frac(...TRUNCATED)	" To find the value of \\(\\frac{x^4 + 2y^2}{6}\\) given that \\(x = 2\\) and \\(y = 5\\), we'll sub(...TRUNCATED)	" \n\nTo find the value of the expression given by $\\frac{x^4+2y^2}{6}$ for $x=2$ and $y=5$, we can(...TRUNCATED)	" The given expression is $\\frac{x^4 + 2y^2}{6}$, where $x=2$ and $y=5$.\n\nFirst, we will calculat(...TRUNCATED)	" Given values ( x = 2 ) and ( y = 5 ):\n\n1. First Calculation:\n - The value of ( x^4 ) is ((...TRUNCATED)	" ```python\n# Step 1: Substitute the values of x and y into the expression.\nx = 2\ny = 5\nexpressi(...TRUNCATED)	" Given the values \\( x = 2 \\) and \\( y = 5 \\), we need to evaluate the expression \\(\\frac{x^4(...TRUNCATED)	" Given the values:\n\\( x = 2 \\)\n\\( y = 5 \\)\n\nFirst, we need to calculate \\( x^4 \\), \\( 2y(...TRUNCATED)	" Given\n\\[\nx = 2 \\quad \\text{and} \\quad y = 5.\n\\]\n\nWe need to evaluate the expression:\n\\(...TRUNCATED)	" Given:\n\\[ x = 2 \\]\n\\[ y = 5 \\]\n\nWe need to find the value of:\n\\[ \\frac{x^4 + 2y^2}{6} \(...TRUNCATED)	" We'll substitute the values of \\( x \\) and \\( y \\) into the expression \\( \\frac{x^4 + 2y^2}{(...TRUNCATED)	" To solve the problem, we need to substitute the given values of \\(x\\) and \\(y\\) into the expre(...TRUNCATED)	" First, we substitute $x=2$ and $y=5$ into the expression $\\frac{x^4 + 2y^2}{6}$.\nWe have $\\frac(...TRUNCATED)	" Suppose we are given a value of \\( x + y = 9 \\).\n\n1. Identify the Values:\n \n Accordi(...TRUNCATED)	" \n1. Multiply $x$ and $y$ to find an intermediate value.\n [\n 10 = x times y\n ]\n \n2. I(...TRUNCATED)	" The problem provides values \\( x = 2 \\) and \\( y = 5 \\), and we need to find the value of the (...TRUNCATED)	" Given \\( x = 2 \\) and \\( y = 5 \\), we need to find the value of the expression \\(\\frac{x^4 +(...TRUNCATED)	" We first substitute $x = 2$ and $y = 5$ into the expression $\\frac{x^4+2y^2}{6}$.\n\nStep 1: Calc(...TRUNCATED)	" Given the values, let's substitute into the expression:\n\n\\[\n\\frac{x^4 + 2y^2}{6}\n\\]\n\nSubs(...TRUNCATED)	" Given \\(x = 2\\) and \\( y = 5 \\), we need to find the value of \\( \\frac{x^4 + 2y^2}{6} \\).\n(...TRUNCATED)	" First, we substitute the given values of \\( x \\) and \\( y \\) into the expression.\n\n1. Given (...TRUNCATED)	" First, we substitute the given values of $x$ and $y$ into the expression $\\frac{x^4+2y^2}{6}$.\n\(...TRUNCATED)	" Given the values \\( x = 2 \\) and \\( y = 5 \\), we need to find the value of the expression \\( (...TRUNCATED)	" To find the value of \\(\\frac{x^4 + 2y^2}{6}\\) when \\(x = 2\\) and \\(y = 5\\), we will follow (...TRUNCATED)	" \n\nGiven \\( x = 2 \\) and \\( y = 5 \\), we need to evaluate the expression \\( \\frac{x^4 + 2y^(...TRUNCATED)	" 1. First, we need to compute the expression ( x^{4} + 2 y^{2} ). Given the values ( x = 2 ) and ( (...TRUNCATED)	" Sure, let's break down the expression \\(\\frac{x^4 + 2y^2}{6}\\) given \\(x = 2\\) and \\(y = 5\\(...TRUNCATED)	" To solve the problem, we need to follow these steps:\n\n1. Substitute the given values \\(x = 2\\)(...TRUNCATED)	" \n\nWe are given \\( x = 2 \\) and \\( y = 5 \\). We need to find the value of \\(\\frac{x^4 + 2y^(...TRUNCATED)	" Given:\n- \\( x = 2 \\)\n- \\( y = 5 \\)\n\nWe need to find the value of \\(\\frac{x^4 + 2y^2}{6}\(...TRUNCATED)	" Given the values:\n- \\( x = 2 \\)\n- \\( y = 5 \\)\n\nWe need to calculate \\( \\frac{x^4 + 2y^2}(...TRUNCATED)	" To find the value of the given expression, we need to substitute the values of \\(x\\) and \\(y\\)(...TRUNCATED)	" We'll calculate the expression step by step using the values \\( x = 2 \\) and \\( y = 5 \\).\n\nT(...TRUNCATED)	" We start with the given values: \\( x = 2 \\) and \\( y = 5 \\). We need to find the value of the (...TRUNCATED)	" We are given that $x = 2$ and $y = 5$. We need to find the value of the expression $\\frac{x^4 + 2(...TRUNCATED)	" Given the values \\( x = 2 \\) and \\( y = 5 \\), we need to calculate the value of \\(\\frac{x^4 (...TRUNCATED)	" \nGiven:\n\\( x = 2 \\)\n\\( y = 5 \\)\nWe're required to find the value of \n\\[\n\\frac{x^4 + 2y(...TRUNCATED)	1	0	1	1	1	1	1	0	1	1	0	1	1	1	0	1	0	1	1	1	0	0	0	1	1	1	1	1	1	1	1	0	1	0	0	0	0	1	1	1	1	1	1	0	1	1	0	1	0	0	1	0	1	1	1	1	1	1	0	0	0	0	0	1	1	1	0	1	0	1	1	1	0	1	1	1	1	0	0	0	0	1	0	0	0	1	1	0	0	0	1	0	0	0	1	1	0	1	1	1	0	1	1	1	1	0	0	1	1	1	1	1	1	1	0	1	0	0	0	1	0	0	0	1	1	1	1	1	254	244	424	307	127	235	305	217	200	235	157	273	307	431	255	336	207	218	327	317	112	189	277	226	314	232	306	450	192	254	201	211	347	301	253	300	248	264	348	281	210	257	433	235	346	266	200	365	252	224	268	4,282	316	225	222	300	374	358	290	99	164	237	195	211	199	257	441	235	195	255	158	153	6,701	373	249	299	307	290	285	98	211	265	289	257	236	282	301	404	302	232	289	2,937	250	202	215	219	112	298	356	229	338	287	238	102	402	280	261	238	177	205	322	304	243	247	254	274	230	339	185	254	259	210	285	340	252	346	367	285
Level 2	Algebra	DigitalLearningGmbH/MATH-lighteval	[{"content":"The sequence of integers in the row of squares and in each of the two columns of square(...TRUNCATED)	math	{ "ground_truth": "-7", "style": "rule" }	{ "index": 8, "split": "train" }	" \n\nGiven the sequence conditions and the provided diagram:\n\n1. The first row (as shown in the i(...TRUNCATED)	" To solve the problem, we need to identify the value of \\( N \\).\n\nGiven:\n- The sequences in ea(...TRUNCATED)	= 4 or 5. The answer is: 4	" The problem involves understanding the arithmetic sequences formed in the row of squares and in ea(...TRUNCATED)	" \n\nTo solve the problem, we need to analyze the given sequence of integers on the grid and unders(...TRUNCATED)	" To solve this problem, we need to break it down into manageable steps:\n\n1. **Define the Arithmet(...TRUNCATED)	" To solve the problem, we need to determine the number \\( N \\) that fits in the third column of t(...TRUNCATED)	" To solve this problem, we need to understand the arithmetic sequence formation and relationships i(...TRUNCATED)	" We will use the information from the problem to model the sequences using Python and solve for $N$(...TRUNCATED)	" \n\nGiven the sequence in the row of squares and in each of the two columns of squares:\n- Forming(...TRUNCATED)	"```python\n# Let's define the arithmetic sequences\n# The common difference in the row of squares i(...TRUNCATED)	" ```python\n# Let's solve the problem step by step.\n\n# From the arrangement given, we can infer t(...TRUNCATED)	" \n\nFirst, we observe the given information about the sequence of integers in the row of squares a(...TRUNCATED)	" Sure, let's break it down and solve the problem mathematically first. Then, we will implement this(...TRUNCATED)	" To solve the problem, we need to understand the structure of the grid and use the properties of ar(...TRUNCATED)	" Let's denote the three arithmetic sequences in the table as follows:\n\nIn the top row, the three (...TRUNCATED)	" Let's denote the elements in the row of squares and the two columns of squares as follows:\n\n- Th(...TRUNCATED)	" \n\nTo solve for \\(N\\) in the given problem, we need to utilize the properties of arithmetic seq(...TRUNCATED)	" To solve the problem, we first need to identify the arithmetic sequences in the given row of squar(...TRUNCATED)	" \n\nSet up the equations:\n\nGiven the first column arithmetic sequence: $a + (N-1)d_1 = a - 17$\n(...TRUNCATED)	" To find the value of \\( N \\), we need to analyze the arithmetic sequences in the given diagram. (...TRUNCATED)	" \n\n1. From the equation \\( x + y = -17 \\), we know \\( y = -17 - x \\).\n2. Given \\( z = 21 + (...TRUNCATED)	" \nTo solve for the unknown variable \\( N \\) in the given arithmetic sequences, we need to follow(...TRUNCATED)	" To solve for \\(N\\), we need to understand the properties of the arithmetic sequences in the prob(...TRUNCATED)	From the table, we can see that when $x = -17$, $N = 21$.	" To solve the problem, we need to analyze the given information about the arithmetic sequences and (...TRUNCATED)		" We are given a sequence of integers in the row of squares and in each of the two columns of square(...TRUNCATED)	" To solve this problem, we can use the properties of arithmetic sequences. Let's break it down step(...TRUNCATED)	" This problem requires finding the value of $N$ in an arithmetic sequence based on the given condit(...TRUNCATED)	" To solve for the value of \\( N \\) in the given problem, we need to consider the three arithmetic(...TRUNCATED)	" The problem states that the sequences of integers in the row of squares and in each of the two col(...TRUNCATED)	" \nFirst, let's clarify the arithmetic sequences that we need to work with. From the diagram, we ca(...TRUNCATED)	" To solve this problem, we need to determine the value of \\( N \\) given the information about the(...TRUNCATED)	" \n\nGiven sequences: first row starts with 21, 14, and 18; second row starts 21+3, 14+3, 18+3; and(...TRUNCATED)	" In the provided arithmetic sequence question, we know that the rule for generating these sequences(...TRUNCATED)	" Given that the sequence of integers in the row of squares and in each of the two columns of square(...TRUNCATED)	" To solve this problem, we need to identify the values of the sequence and the relationship between(...TRUNCATED)	" Given the problem's context, it can be modeled using an arithmetic sequence. Here's the reasoning (...TRUNCATED)	" To solve the problem, we need to determine the value of \\(N\\) in the given sequence of integers.(...TRUNCATED)	" \n\nGiven the conditions, the three sequences are $21, 18, 15, \\ldots$, $14, 9, 4, \\ldots$, and (...TRUNCATED)	" To solve the problem, we need to determine the value of \\(N\\). The problem describes three arith(...TRUNCATED)	" To solve for \\( N \\), let's break down the problem and use the properties of arithmetic sequence(...TRUNCATED)	" To find the value of \\( N \\) given that the sequence of integers in the row of squares and the t(...TRUNCATED)	" To solve this problem, we need to establish the relationships between the sequence in the row of s(...TRUNCATED)	" To solve this problem, we need to set up and solve a system of equations based on the arithmetic s(...TRUNCATED)	" To solve the problem, we need to determine the value of \\( N \\) based on the properties of the a(...TRUNCATED)	" \n\n1. Firstly, we are given three arithmetic sequences in the rows: 21, 14, 18 and -17.\n2. The c(...TRUNCATED)	" To solve this problem, we need to express the given information in terms of equations.\n\nFirst, l(...TRUNCATED)	" ### Reasoning:\n1. To solve this problem, we will use the properties of arithmetic sequences and t(...TRUNCATED)	" Let's start by setting up the equations based on the problem description. We are given a 3x3 grid (...TRUNCATED)	" To solve this problem, we need to analyze the given arithmetic sequences and find the value of \\((...TRUNCATED)	" We see from the diagram that the three arithmetic sequences have a common difference of 7. Let's l(...TRUNCATED)	" Here's a step-by-step solution approach using reasoning and code to verify it.\n\n1. **Interpret t(...TRUNCATED)	" \n\nGiven that the sequence of integers in the row of squares and in each of the two columns of sq(...TRUNCATED)	" Firstly, the first sequence is given to be 21, 14, 18, 21, and so on. This sequence is an arithmet(...TRUNCATED)	" Here is the problem again for reference:\nWe need to find the value of $N$ in the given arithmetic(...TRUNCATED)	" To determine the value of \\( N \\) in the given sequence of integers, let's first summarize the i(...TRUNCATED)	" Let's denote the problems by \\(P_{1}, P_{2}, \\cdots P_{21}\\) in row \\(1\\) to row \\(21\\). Th(...TRUNCATED)	" To solve the problem, we need to determine the value of \\( N \\) such that the sequence of intege(...TRUNCATED)	" \n\nLet's start by understanding the properties of the arithmetic sequences.\n\nWe have three rows(...TRUNCATED)	" To solve the problem, let's first set up some equations for the three arithmetic sequences:\n\n1. (...TRUNCATED)	" To solve this problem, we need to analyze the given arithmetic sequences and apply the properties (...TRUNCATED)	" First, I understand that three arithmetic progressions are formed. Let's use variables for the fir(...TRUNCATED)	" \n\nWe need to find the value of \\(N\\) in the given arithmetic sequence pattern.\n\n1. In the to(...TRUNCATED)	" First, we need to analyze the given arithmetic sequences and the specific values in the problem. T(...TRUNCATED)	" The three distinct arithmetic sequences described in the problem correspond to the first three col(...TRUNCATED)	" To solve the problem, we need to analyze the given information and the structure of the row of squ(...TRUNCATED)	" To solve the problem, we will first define the three distinct arithmetic sequences that form the s(...TRUNCATED)		" We need to find the value of $N$ given the conditions of the arithmetic sequences and the specific(...TRUNCATED)	" To solve for \\( N \\) in this problem, we need to carefully analyze the given arithmetic sequence(...TRUNCATED)	"333333.... So here’s the plan:\n\n1. Define the terms of each Arithmetic Sequences:\n \n (...TRUNCATED)	" First we will extract and define the sequences based on the given problem. The given sequences wil(...TRUNCATED)	" To solve this problem, we need to determine the value of \\( N \\) given the arithmetic sequences (...TRUNCATED)	" Let's solve the problem step-by-step.\n\n1. Understand the Problem:\n - The sequence of inte(...TRUNCATED)	" To solve the problem, we need to analyze the given sequence of integers and determine the value of(...TRUNCATED)	" Based on the problem, generate the required expression with the provided inputs.\nThe sequence of (...TRUNCATED)	" \n\nGiven that the sequences in the rows and columns form arithmetic sequences, we start by assumi(...TRUNCATED)		" Given the constraints of this particular problem and understanding that it might require parsing t(...TRUNCATED)	" From the diagram and given the arithmetic sequences pattern in each column,\n\nWe observe the foll(...TRUNCATED)	" ```python\n# We have the following information:\n# In the middle column (vertical), the numbers ar(...TRUNCATED)	" Based on the image and the problem statement, it seems that for the integer in column 3 of the row(...TRUNCATED)	" To solve this problem, we need to identify the common difference in the arithmetic sequences. Here(...TRUNCATED)	" \n\nWe are given a 3x3 grid where the sequences follow a specific pattern. To solve this problem, (...TRUNCATED)	" The job is to determine the specific value of \\(N\\) from the given numerical arrangement, wherei(...TRUNCATED)	" \n\nHere's the plan for solving this problem:\n\n1. The sequence of integers forms three distinct (...TRUNCATED)	" \n\n1. Let's consider the arithmetic sequence of the column: -17, N, 21.\n2. To confirm that (N) s(...TRUNCATED)	" To solve the problem, we need to determine the value of \\( N \\) by analyzing the arithmetic sequ(...TRUNCATED)	" First, we construct a feasible approach by resembling the reference solution.\n\nFletcher starts o(...TRUNCATED)	" ```python\n# Let's assume the three arithmetic sequences as a, a+d, a+2d and b, b+e, b+2e in the r(...TRUNCATED)	" Let's denote the first term of the arithmetic sequence in the first row as \\(a_1\\) and the commo(...TRUNCATED)	" \n\n1. Observe the triangular array and the structures shown in the diagram.\n\n2. Let's define th(...TRUNCATED)	" We will solve this problem by breaking it down step by step.\n\nGiven the information, we have:\n-(...TRUNCATED)	" \n\nThe given problem involves three arithmetic sequences - a row of squares and two columns of sq(...TRUNCATED)	" To determine the value of \\( N \\), we need to understand the structure of the number sequences g(...TRUNCATED)	" Here is a detailed solution for the given problem. Given that the sequence of integers in the row (...TRUNCATED)	" \n\nGiven that the sequence of positive integers in one of the rows and in one of the columns is a(...TRUNCATED)	" To solve the given problem, let's first understand the structure of the problem. We have a triangu(...TRUNCATED)	" Let's start by analyzing the problem.\n\nThe three distinct arithmetic sequences in the row of squ(...TRUNCATED)		" ### Reasoning:\n\n1. Understand the setup:\n - We are given a triangular square sequence co(...TRUNCATED)	" The problem involves identifying a missing number \\( N \\) in a sequence of integers arranged in (...TRUNCATED)	" Here's the plan for finding the value of \\( N \\):\n\n1. Identify the arithmetic sequences:\n -(...TRUNCATED)		" Based on the given problem statement and illustration:\n\n1. The sequence on the top row forms an (...TRUNCATED)	" \n\nEach row consists of 7 integers, forming 2 arithmetic sequences: \n1. For rows starting from 0(...TRUNCATED)	" \n\nTo solve for \\(N\\), we need to analyze the given arithmetic sequences and the established re(...TRUNCATED)	" To solve the problem, let's identify the arithmetic sequences in the given grid and then determine(...TRUNCATED)	" Let's approach the problem step by step.\n\n### Step 1: Definition of Arithmetic Sequence\nIn an a(...TRUNCATED)	" To solve this problem, let's analyze the arithmetic sequences in the table. The table consists of (...TRUNCATED)	" We need to determine the value of \\( N \\) in the given scenario. The problem involves three arit(...TRUNCATED)	" To solve this problem, we need to understand the structure of the sequence. The sequence forms thr(...TRUNCATED)	" To find the value of \\( N \\), we need to analyze the sequence of integers given in the grid desc(...TRUNCATED)	" Let's start by examining the given information and understanding the problem. We are given a table(...TRUNCATED)	" To solve the problem, we need to analyze the given geometric sequence and geometric figure.\n\n1. (...TRUNCATED)	" The rows of squares form arithmetic sequences where the difference between consecutive terms is co(...TRUNCATED)	" \n\n1. Begin by interpreting the problem: There are three quantities (21, 14, and 18) forming arit(...TRUNCATED)	" To solve this problem, we need to use the given information that the three distinct arithmetic se(...TRUNCATED)	" To solve the problem, we need to determine the value of \\( N \\) that completes the given arithme(...TRUNCATED)	" Let's denote the three distinct arithmetic sequences as \\(a_1, a_2, a_3\\) in the row of squares (...TRUNCATED)	" The given problem presents a grid of dots with the numbers $21$, $14$, and $18$ at the respective (...TRUNCATED)	" \nThe problem asks us to find the value of \\(N\\) in the given arithmetic sequences. We will use (...TRUNCATED)	" To solve the problem, we need to identify the three distinct arithmetic sequences and then find th(...TRUNCATED)	" To solve this problem, we need to identify the arithmetic sequences and how they relate to the giv(...TRUNCATED)	" Let's approach this problem systematically by unpacking the information provided and solving it st(...TRUNCATED)		0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	500	466	14	1,300	477	697	864	1,443	2,557	503	1,499	466	687	914	3,245	337	4,205	495	389	891	2,322	283	1,241	1,540	23	538	0	1,473	526	1,106	762	579	374	1,164	414	683	887	981	919	443	199	579	2,756	628	1,648	976	774	345	765	415	1,148	677	571	984	976	392	761	789	786	2,008	1,073	766	648	1,807	625	1,211	180	1,313	1,013	0	1,298	1,248	152	910	177	612	757	874	465	0	495	661	641	1,166	676	1,276	672	1,558	199	1,382	259	935	502	749	520	409	800	779	510	311	4,416	0	495	932	1,134	0	460	405	934	538	920	853	897	737	605	530	472	680	239	1,885	445	507	657	514	3,303	632	694	0
Level 5	Algebra	DigitalLearningGmbH/MATH-lighteval	[{"content":"Tim wants to invest some money in a bank which compounds quarterly with an annual inter(...TRUNCATED)	math	{ "ground_truth": "\\$42409", "style": "rule" }	{ "index": 9, "split": "train" }	" To solve this problem, we need to determine the initial investment T such that the future value of(...TRUNCATED)	" First, we need to determine the quarterly interest rate. The annual interest rate is 7%, so the qu(...TRUNCATED)	" To determine how much money Tim should invest to have $60,000 at the end of 5 years with an annual(...TRUNCATED)	" To determine how much money Tim should invest now to grow to $60,000 in 5 years with a 7% annual i(...TRUNCATED)	" To determine how much Tim needs to invest in a bank that compounds quarterly with an annual intere(...TRUNCATED)	" To solve this problem, we need to use the formula for compound interest:\n\n\\[ A = P \\left(1 + \(...TRUNCATED)	" To determine the initial investment required, we can use the formula for the future value of a pri(...TRUNCATED)	" To determine how much money Tim should invest now to have $60,000 at the end of 5 years with an an(...TRUNCATED)	" To determine how much Tim should invest for a future value of $60,000 over 5 years with an annual (...TRUNCATED)	" Here is the process to solve the problem:\n\n1. Understanding Compound Interest: The formula f(...TRUNCATED)	" We can use the formula for compound interest:\n\n\\[A = P \\left(1 + \\frac{r}{n}\\right)^{nt}\\]\(...TRUNCATED)	" To determine how much money Tim needs to invest today to have \\$60,000 at the end of 5 years with(...TRUNCATED)	" To solve the problem of determining the initial investment amount required to achieve a future val(...TRUNCATED)	" To solve the problem, we will use the formula for compound interest:\n\n\\[ A = P \\left(1 + \\fra(...TRUNCATED)	" To solve this problem, we need to determine the initial investment (principal) $P$ that Tim should(...TRUNCATED)	" To solve the problem, we need to find the principal amount (\\(P\\)) that Tim needs to invest in o(...TRUNCATED)	" To determine how much money Tim should invest to have $60,000 at the end of 5 years, we need to ca(...TRUNCATED)	" To determine the initial amount Tim needs to invest, given an annual interest rate of 7%, quarterl(...TRUNCATED)	" To solve this problem, we need to use the formula for compound interest:\n\n\\[ A = P \\left(1 + \(...TRUNCATED)	" First, let's denote the initial investment by \\( P \\).\n\nThe formula for the future value \\( A(...TRUNCATED)	" To determine how much Tim should invest to have $60,000 after 5 years with quarterly compounding a(...TRUNCATED)	" To solve this problem, we need to determine how much money Tim should invest today to have a total(...TRUNCATED)	" To solve the problem of finding the initial amount of money (principal) that needs to be invested (...TRUNCATED)	" \n\nIf Tim wants to have $60,000 after 5 years, and if our investment had simple interest, it woul(...TRUNCATED)	" To find the initial investment required for Tim to have $60,000 at the end of 5 years with a 7% an(...TRUNCATED)	" ## Step-by-Step Solution:\n\n1. Determine the effective quarterly interest rate:\n\n The ann(...TRUNCATED)	" To solve this problem, we can use the formula for compound interest: \n\n\\[ A = P \\left(1 + \\fr(...TRUNCATED)	" \n\nFirst, we need to find the formula for the future value of an investment which compounds quart(...TRUNCATED)	" To solve the problem, we need to determine the initial investment \\(P\\) required to achieve a to(...TRUNCATED)	" To determine the amount of money Tim should invest, we'll use the formula for compound interest:\n(...TRUNCATED)	" To solve this problem, we need to determine the initial investment (principal, $P$), which will gr(...TRUNCATED)	" To determine how much money Tim should invest in a bank that compounds quarterly with an annual in(...TRUNCATED)	" To determine how much Tim should invest to achieve a total of $60,000 in 5 years with an interest (...TRUNCATED)	" To determine how much money Tim should invest today to reach $60,000 in 5 years with an annual int(...TRUNCATED)	" We'll use the formula for compound interest to determine the initial investment.\n\nGiven:\n- The (...TRUNCATED)	" To solve this problem, we need to determine the present value of $60,000 that will grow to $60,000(...TRUNCATED)	" To solve this problem, we can use the formula for compound interest:\n\n\\[A = P \\left(1 + \\frac(...TRUNCATED)	" To solve this problem, we need to use the formula for compound interest:\n\n\\[ A = P \\left(1 + \(...TRUNCATED)	" ```python\n# Given data\nfuture_value = 60000\nannual_interest_rate = 0.07\nyears = 5\ncompounding(...TRUNCATED)	" To determine how much Tim should invest initially to achieve a total of $60,000 at the end of 5 ye(...TRUNCATED)	" We will use the formula for compound interest to find the initial principal amount. The formula is(...TRUNCATED)	" \n\n1. Identify the given values:\n - The annual interest rate is \\(7\\%\\) per year.\n - The(...TRUNCATED)	" To solve this problem, we need to use the formula for compound interest. The formula is:\n\n\\[\nA(...TRUNCATED)	" To solve this problem, we need to use the compound interest formula:\n\n\\[ A = P \\left(1 + \\fra(...TRUNCATED)	" To determine how much Tim should initially invest to end up with $60,000 in 5 years with an annual(...TRUNCATED)	" To determine how much Tim should invest, we need to use the formula for compound interest. The for(...TRUNCATED)	" We need to find the initial amount Tim needs to invest so that he will have $60,000 after 5 years.(...TRUNCATED)	" Given:\n- Annual interest rate, \\( r = 7\\% = 0.07 \\)\n- Number of years, \\( t = 5 \\)\n- Futur(...TRUNCATED)	" We can use the formula for compound interest: A = P(1 + r/n)^(nt), where:\n\n- A is the future val(...TRUNCATED)	" Given:\n- Interest rate per year compounded quarterly\n- Principal amount to be invested\n- The de(...TRUNCATED)	" To calculate how much money Tim should invest initially to reach \\$60,000 at the end of 5 years w(...TRUNCATED)	" To determine how much Tim should invest initially to have $60,000 after 5 years with quarterly com(...TRUNCATED)	" To determine the amount of money Tim should invest in a bank that compounds quarterly with an annu(...TRUNCATED)	" To determine the amount of money Tim should invest to have $60,000 in 5 years with quarterly compo(...TRUNCATED)	" ```python\n# Let's calculate the amount of money Tim has to invest for his desired future value.\n(...TRUNCATED)	" To determine how much interest is earned in one month, you would divide the interest earned in the(...TRUNCATED)	" To solve this problem using Python and sympy, we need to use the formula for compound interest. Th(...TRUNCATED)	" To solve this problem, we need to use the compound interest formula:\n\n\\[ A = P \\left(1 + \\fra(...TRUNCATED)	" To determine how much Tim needs to invest initially, we can use the formula for compound interest (...TRUNCATED)	" To determine the initial investment $P$ required to grow to $60,000 in 5 years with an annual inte(...TRUNCATED)	" Let's break down the problem and solve it step-by-step using the formula for compound interest:\n\(...TRUNCATED)	" To determine the initial investment required to reach a certain future value with a given interest(...TRUNCATED)	" To solve this problem, we need to use the formula for compound interest:\n\n\\[ A = P \\left(1 + \(...TRUNCATED)	" To solve this problem, we need to use the compound interest formula:\n\n\\[ A = P \\left(1 + \\fra(...TRUNCATED)	" The problem states that Tim wants to invest some amount of money, let's call it \\( P \\), such t(...TRUNCATED)	" To solve this problem, we need to determine the present value of an investment that will grow to $(...TRUNCATED)	" To determine how much money Tim should invest now to have $60,000 in 5 years with an annual intere(...TRUNCATED)	" ```python\n# Constants\r\ninitial_investment = 0\r\nannual_interest_rate = 0.07\r\ncompound_period(...TRUNCATED)	" To solve the problem, we need to calculate the initial investment amount such that it grows to $60(...TRUNCATED)	" To determine how much money Tim should invest in his bank to have $60,000 at the end of 5 years wi(...TRUNCATED)	" To solve this problem, we can use the formula for compound interest:\n\n\\[ A = P \\left(1 + \\fra(...TRUNCATED)	" To solve this problem, we need to use the formula for compound interest. The formula is:\n\n\\[ A (...TRUNCATED)	" To solve this problem, we need to use the formula for compound interest:\n\n\\[ A = P \\left(1 + \(...TRUNCATED)	" \n\nTo find out how much money Tim should invest now in a bank that compounds quarterly with an an(...TRUNCATED)	" \n\n1. The initial investment amount (continuously compounded) can be calculated using the formula(...TRUNCATED)	" ### Reasoning:\n\nTo find the present value \\( P \\) of an amount after a given time period, we u(...TRUNCATED)	" To solve this problem, we need to determine the principal amount that Tim should invest in the ban(...TRUNCATED)	" To determine how much Tim should invest now to have $60,000 in 5 years with an annual interest rat(...TRUNCATED)	" To determine how much money Tim should invest to have $\\$60,\\!000$ in 5 years with an annual int(...TRUNCATED)	" To determine how much money Tim needs to invest to have $60,000 in 5 years with quarterly compound(...TRUNCATED)	" To solve this problem, we need to determine how much money Tim should invest today, given an annua(...TRUNCATED)	" To solve this problem, we need to determine how much money Tim should invest to achieve a total of(...TRUNCATED)		" Here's the solution to the problem:\n\n1. Understand the interest formula during compounding: (...TRUNCATED)	" To solve this problem, we'll use the formula for compound interest:\n\n\\[ A = P \\left(1 + \\frac(...TRUNCATED)	" \n\nWe are told that Tim will invest some money in a bank. \n\nThe interest rate is given as $7\\%(...TRUNCATED)	" To solve this problem, we can use the formula for compound interest:\n\n\\[ A = P \\left(1 + \\fra(...TRUNCATED)	" The compound interest formula is given by:\n\n\\[A = P \\left(1 + \\frac{r}{n}\\right)^{nt}\\]\n\n(...TRUNCATED)	" To go through the content of the problem step by step and find the derived answer. We need the cal(...TRUNCATED)	" To determine how much Tim should invest initially, we need to account for the compounding interest(...TRUNCATED)	" To solve this problem, we can use the formula for compound interest:\n\n\\[A = P \\left(1 + \\frac(...TRUNCATED)	" To determine how much Tim should invest in order to have $60,000 after 5 years with quarterly comp(...TRUNCATED)	" The initial investment \\( P \\) and the bank's annual interest rate \\( r = 7\\% = 0.07 \\) compo(...TRUNCATED)	" To determine how much money Tim should invest in a bank compounded quarterly with an annual intere(...TRUNCATED)	" To solve this problem, we need to use the formula for compound interest. The formula for the futur(...TRUNCATED)	" To determine how much money Tim should invest to achieve a total of $60,000 in 5 years with a 7% a(...TRUNCATED)	" To determine how much Tim should invest in a bank compounding quarterly with an annual interest ra(...TRUNCATED)	" To solve the problem, we'll use the compound interest formula:\n\n\\[ A = P \\left(1 + \\frac{r}{n(...TRUNCATED)	" To find out how much money Tim needs to invest to have $\\$60,000$ after 5 years with an annual in(...TRUNCATED)	" To determine how much money Tim needs to invest to reach $60,000 in 5 years with an 7% annual inte(...TRUNCATED)	" Let's denote the initial amount of money Tim invests as \\(P\\). The bank compounds quarterly with(...TRUNCATED)	" Suppose Tim invests a principal amount $P$ dollars in the bank. The interest is compounded quarter(...TRUNCATED)	" To solve the problem, we need to use the formula for compound interest, which is given by:\n\n\\[ (...TRUNCATED)	" ```python\n# Define the target amount after 5 years\ntarget_amount = 60000\n\n# Define the annual (...TRUNCATED)	" ```python\n# Let's calculate the present value of the investment to achieve $60,000 at the end of (...TRUNCATED)	" To solve this problem, I need to use the formula for compound interest:\n\namount = principal * (1(...TRUNCATED)	" Denote the required investment as $P$. The interest rate is $7\\%$, so the quarterly interest rate(...TRUNCATED)	" To determine how much Ann should invest so that she will have $\\$60,000$ at the end of 50 years a(...TRUNCATED)	" Define the initial amount of money Tim invests as \\( P \\). The interest rate is \\( 0.07 \\) ann(...TRUNCATED)	" To solve the problem, we need to use the formula for compound interest. The formula is:\n\n\\[ A =(...TRUNCATED)	" To solve this problem, we need to use the formula for compound interest, which is given by (A = P((...TRUNCATED)	" We need to find how much money Tim should invest initially to achieve $60,000 in 5 years with an a(...TRUNCATED)	" To find out how much money Tim should invest, we'll use the formula for compound interest. The for(...TRUNCATED)	" To determine how much money Tim should invest in a bank that compounds quarterly at an annual inte(...TRUNCATED)	" To solve this problem, we need to determine the present value of an investment that will grow to $(...TRUNCATED)	" We need to determine how much money Tim should invest in a bank that compounds quarterly with an a(...TRUNCATED)	" \n\n1. Define the principal amount (initial investment) as \\( P \\).\n2. Determine the annual int(...TRUNCATED)	" To solve this problem, we need to use the formula for compound interest. For each compounding peri(...TRUNCATED)	" \nTo determine how much money Tim should invest now to have $60,000 after 5 years with annual inte(...TRUNCATED)	" Let's denoting P as the initial investment, f as the annual interest rate, i as the number of comp(...TRUNCATED)	" Suppose we let 100 represent \\( \\$100^{1/4} \\). Therefore, the quarterly interest rate is 7%, s(...TRUNCATED)	" Let's define some variables to help solve this problem:\n\n- Let \\( P \\) be the initial amount o(...TRUNCATED)	" Keep writing in more details!\nTo solve this problem, we need to calculate the initial amount that(...TRUNCATED)	" To solve the problem, let's use the formula for compound interest:\n\n\\[ A = P \\left(1 + \\frac{(...TRUNCATED)	" The problem can be formulated as solving for the present value \\(P\\) that will grow to $60,000 a(...TRUNCATED)	" To determine how much money Tim needs to invest now to have $\\$60,000$ in 5 years with an annual (...TRUNCATED)	" ```python\nfrom sympy import symbols, Eq, solve\n\n# Define the variables\nP, r, n, t, A = symbols(...TRUNCATED)	" To determine how much money Tim should invest to have \\(\\$60,000\\) in 5 years with an annual in(...TRUNCATED)	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	616	648	471	516	588	431	468	438	388	621	450	461	405	398	441	478	4,649	518	450	442	403	491	466	170	498	584	452	397	468	401	404	494	504	434	471	442	3,409	418	281	407	415	534	411	388	551	455	416	603	319	763	1,961	421	443	519	283	345	441	444	466	511	495	442	397	413	1,795	555	568	286	460	443	1,845	427	409	439	323	453	437	571	595	482	433	399	0	543	388	458	420	487	123	420	428	394	469	426	385	4,526	515	481	405	411	390	553	434	304	323	204	239	433	451	582	370	491	432	462	499	538	727	454	406	330	752	594	699	401	492	492	641	598

End of preview. Expand in Data Studio

README.md exists but content is empty.

Downloads last month: 8

Size of the auto-converted Parquet files:

Number of rows: