Dataset Viewer
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|---|---|---|---|---|---|---|---|
comp_polynomial_gcd_test_000
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 14*x**8 + 6*x**7 - 46*x**6 - 18*x**5 - 26*x**4 - 14*x**3 - 50*x**2 - 40*x - 24 and 14*x**5 + 4*x**4 - 46*x**3 - 2*x**2 - 40*x - 56\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
12
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_001
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 2*x**5 + 11*x**4 + 23*x**3 + 28*x**2 + 21*x + 7 and 3*x**7 + 8*x**6 + 14*x**5 + 18*x**4 + 13*x**3 + 11*x**2 + 12*x + 5\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
7
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_002
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 7*x**7 + 21*x**6 + 28*x**5 + 28*x**4 + 28*x**3 + 28*x**2 + 21*x + 7 and 7*x**8 + 16*x**7 + 13*x**6 + 13*x**5 + 23*x**4 + 24*x**3 + 16*x**2 + 9*x + 3\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
7
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_003
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 2*x**8 + 6*x**7 + 13*x**6 + 23*x**5 + 28*x**4 + 24*x**3 + 15*x**2 + 7*x + 2 and 7*x**4 + 21*x**3 + 28*x**2 + 21*x + 7\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
7
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_004
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 7*x**6 - 5*x**5 + 5*x**4 - 5*x**3 + 5*x**2 - 2*x - 5 and 7*x**6 - 4*x**2 - 3\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_005
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 14*x**7 + 10*x**6 - 52*x**5 - 36*x**4 - 12*x**3 - 12*x**2 - 16*x - 16 and 14*x**3 + 14*x**2 - 56*x - 56\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
12
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_006
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 7*x**4 + 7*x**3 - 7*x*y**3 - 7*y**3 and 5*x**6 + 7*x**5 + 2*x**4 - 5*x**3*y**3 + 3*x**3 - 7*x**2*y**3 - 2*x*y**3 - 3*y**3\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
8
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_007
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 7*x**2 - 7 and 2*x**3 + 5*x - 7\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_008
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 7*x**6 + 3*x**5 + 7*x**4 - 7*x**3*y**3 + 2*x**3 - 3*x**2*y**3 - 7*x*y**3 - 2*y**3 and 2*x**7 + 7*x**6 + 7*x**5 - 2*x**4*y**3 + 5*x**4 - 7*x**3*y**3 + 3*x**3 - 7*x**2*y**3 - 5*x*y**3 - 3*y**3\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
8
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_009
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 4*x**8 + 14*x**7 - 6*x**6 - 42*x**5 - 26*x**4 - 52*x**3 - 42*x**2 - 16*x - 56 and 14*x**4 + 14*x**3 - 52*x**2 - 56*x - 16\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
12
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_010
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 2*x**7 + 7*x**6 + 15*x**5 + 24*x**4 + 23*x**3 + 18*x**2 + 16*x + 7 and 2*x**8 + 11*x**7 + 18*x**6 + 13*x**5 + 11*x**4 + 19*x**3 + 26*x**2 + 21*x + 7\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
7
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_011
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 7*x**7 - 2*x**6 + 2*x**5 - 4*x**4 + 2*x**3 - 3*x**2 + 5*x - 7 and 7*x**2 - 2*x - 5\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_012
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 2*x**3 + 11*x**2 + 16*x + 7 and 7*x**7 + 19*x**6 + 24*x**5 + 26*x**4 + 28*x**3 + 28*x**2 + 21*x + 7\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
7
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_013
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 7*x**10 + 2*x**9 + 7*x**8 - 7*x**7*y**3 + 3*x**7 - 2*x**6*y**3 + 7*x**6 - 7*x**5*y**3 + 3*x**5 - 3*x**4*y**3 + 7*x**4 - 7*x**3*y**3 + 2*x**3 - 3*x**2*y**3 - 7*x*y**3 + 2*x - 2*y**3 + 7 and 7*x**6 + 7*x**5 + 3*x**4 - 7*x**3*y**3 + 7*x**3 - 7*x**2*y**3 - 3*x*y**3 + 7*x - 7*y**3 + 7\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
1
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_014
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 14*x**4 + 10*x**3 - 46*x**2 - 40*x - 40 and 10*x**6 + 14*x**5 - 26*x**4 - 42*x**3 - 46*x**2 - 56*x - 40\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
12
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_015
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 5*x**5 + 7*x**4 + 5*x**3 - 5*x**2*y**3 - 7*x*y**3 - 5*y**3 and 7*x**7 + 7*x**6 + 7*x**5 - 7*x**4*y**3 + 7*x**4 - 7*x**3*y**3 + 7*x**3 - 7*x**2*y**3 - 7*x*y**3 - 7*y**3\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
8
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_016
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 7*x**6 + 5*x**5 + 7*x**4 - 7*x**3*y**3 + 7*x**3 - 5*x**2*y**3 - 7*x*y**3 - 7*y**3 and 5*x**6 + 5*x**5 + 7*x**4 - 5*x**3*y**3 + 7*x**3 - 5*x**2*y**3 - 7*x*y**3 - 7*y**3\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
8
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_017
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 14*x**7 + 14*x**6 - 46*x**5 - 42*x**4 - 26*x**3 - 42*x**2 - 56*x - 56 and 14*x**7 + 14*x**6 - 42*x**5 - 50*x**4 - 52*x**3 - 10*x**2 - 16*x - 56\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
12
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_018
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 7*x**8 + 5*x**7 + 3*x**6 - 7*x**5*y**3 + 2*x**5 - 5*x**4*y**3 + 7*x**4 - 3*x**3*y**3 + 7*x**3 - 2*x**2*y**3 - 7*x*y**3 - 7*y**3 and 5*x**10 + 2*x**9 + 7*x**8 - 5*x**7*y**3 + 5*x**7 - 2*x**6*y**3 + 7*x**6 - 7*x**5*y**3 + 7*x**5 - 5*x**4*y**3 + 7*x**4 - 7*x**3*y**3 + 7*x**3 - 7*x**2*y**3 - 7*x*y**3 - 7*y**3\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
8
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_019
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 10*x**8 + 14*x**7 - 30*x**6 - 52*x**5 - 26*x**4 - 2*x**3 - 42*x**2 - 56*x - 56 and 6*x**5 + 14*x**4 - 14*x**3 - 52*x**2 - 40*x - 16\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
12
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_020
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 14*x**4 + 10*x**3 - 46*x**2 - 40*x - 40 and 14*x**4 + 14*x**3 - 42*x**2 - 56*x - 56\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
12
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_021
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 4*x**3 + 14*x**2 - 16*x - 56 and 10*x**5 + 4*x**4 - 26*x**3 - 2*x**2 - 56*x - 56\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
12
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_022
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 7*x**4 + 7*x**3 - 7*x*y**3 - 7*y**3 and 2*x**5 + 7*x**4 + 2*x**3 - 2*x**2*y**3 - 7*x*y**3 - 2*y**3\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
8
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_023
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 14*x**8 + 14*x**7 - 52*x**6 - 46*x**5 - 6*x**4 - 36*x**3 - 26*x**2 - 16*x - 56 and 14*x**6 + 14*x**5 - 42*x**4 - 46*x**3 - 46*x**2 - 40*x - 40\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
12
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_024
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 7*x**9 + 7*x**8 + 7*x**7 - 7*x**6*y**3 + 2*x**6 - 7*x**5*y**3 + 5*x**5 - 7*x**4*y**3 + 7*x**4 - 2*x**3*y**3 + 5*x**3 - 5*x**2*y**3 - 7*x*y**3 - 5*y**3 and 3*x**5 + 2*x**4 + 5*x**3 - 3*x**2*y**3 - 2*x*y**3 - 5*y**3\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
8
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_025
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 2*x**6 + 5*x**5 - 7 and 3*x**4 + 4*x**3 - 2*x - 5\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_026
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 7*x**6 - 5*x**4 + 5*x**2 - 7 and 7*x**2 - 5*x - 2\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_027
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 7*x**10 + 5*x**9 + 7*x**8 - 7*x**7*y**3 + 2*x**7 - 5*x**6*y**3 + 2*x**6 - 7*x**5*y**3 + 5*x**5 - 2*x**4*y**3 + 5*x**4 - 2*x**3*y**3 + 2*x**3 - 5*x**2*y**3 - 5*x*y**3 - 2*y**3 and 7*x**7 + 2*x**6 + 2*x**5 - 7*x**4*y**3 + 7*x**4 - 2*x**3*y**3 + 7*x**3 - 2*x**2*y**3 - 7*x*y**3 - 7*y**3\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
8
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_028
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 3*x**2 - 3 and 7*x**5 - 4*x**4 + 4*x**3 - 5*x**2 + 5*x - 7\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_029
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 5*x**9 + 17*x**8 + 26*x**7 + 28*x**6 + 28*x**5 + 23*x**4 + 13*x**3 + 13*x**2 + 16*x + 7 and 7*x**9 + 21*x**8 + 23*x**7 + 13*x**6 + 13*x**5 + 23*x**4 + 26*x**3 + 19*x**2 + 9*x + 2\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
7
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_030
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 3*x**4 + 13*x**3 + 19*x**2 + 11*x + 2 and 7*x**7 + 21*x**6 + 24*x**5 + 18*x**4 + 20*x**3 + 26*x**2 + 21*x + 7\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
7
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_031
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 7*x**4 - 2*x**3 - 3*x**2 + 5*x - 7 and 7*x**4 - 2*x**3 + 2*x**2 - 7\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_032
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 3*x**2 - x - 2 and 7*x**3 - 5*x**2 + x - 3\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_033
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 7*x**9 + 19*x**8 + 24*x**7 + 24*x**6 + 24*x**5 + 24*x**4 + 24*x**3 + 21*x**2 + 11*x + 2 and 5*x**7 + 17*x**6 + 26*x**5 + 28*x**4 + 23*x**3 + 16*x**2 + 12*x + 5\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
7
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_034
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 2*x**8 + 3*x**7 + 7*x**6 - 2*x**5*y**3 + 2*x**5 - 3*x**4*y**3 + 3*x**4 - 7*x**3*y**3 + 3*x**3 - 2*x**2*y**3 - 3*x*y**3 - 3*y**3 and 7*x**4 + 7*x**3 - 7*x*y**3 - 7*y**3\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
8
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_035
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 5*x**7 + 13*x**6 + 18*x**5 + 19*x**4 + 18*x**3 + 23*x**2 + 21*x + 7 and 2*x**7 + 11*x**6 + 21*x**5 + 24*x**4 + 22*x**3 + 20*x**2 + 17*x + 7\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
7
|
math
|
comp_polynomial_gcd
|
test
|
comp_polynomial_gcd_test_036
|
comp_polynomial_gcd
|
Algebra (Polynomial Roots) + Arithmetic (GCD)
|
[
{
"content": "Find the absolute sum of the coefficients and exponents in the polynomial that is the Greatest Common Divisor of: 7*x**3 - 2*x**2 - 2*x - 3 and 3*x**7 + 4*x**6 - 2*x**5 - 3*x**4 + 5*x**2 - 7\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3
|
math
|
comp_polynomial_gcd
|
test
|
comp_n_gon_test_000
|
comp_n_gon
|
Geometry (Rotation) + Combinatorics (Pattern Matching)
|
[
{
"content": "Find the number of rectangles that can be formed inside a fixed regular dodecagon (12-gon) where each side of the rectangle lies on either a side or a diagonal of the dodecagon. Note that it is possible for a rectangle to be contained within another rectangle, and that the rectangles may not extend beyond the boundaries of the 12-gon.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
315
|
math
|
comp_n_gon
|
test
|
comp_n_gon_test_001
|
comp_n_gon
|
Geometry (Rotation) + Combinatorics (Pattern Matching)
|
[
{
"content": "Find the number of squares that can be formed inside a fixed regular dodecagon (12-gon) where each side of the square lies on either a side or a diagonal of the 5odecagon. Note that it is possible for a square to be contained within another square, and that the squares may not extend beyond the boundaries of the 12-gon.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
45
|
math
|
comp_n_gon
|
test
|
comp_n_gon_test_002
|
comp_n_gon
|
Geometry (Rotation) + Combinatorics (Pattern Matching)
|
[
{
"content": "Find the number of rectangles that can be formed inside a fixed regular 19-gon where each side of the rectangle lies on either a side or a diagonal of the 19-gon. Note that it is possible for a rectangle to be contained within another rectangle, and that the rectangles may not extend beyond the boundaries of the 19-gon.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
math
|
comp_n_gon
|
test
|
comp_n_gon_test_003
|
comp_n_gon
|
Geometry (Rotation) + Combinatorics (Pattern Matching)
|
[
{
"content": "Find the number of squares that can be formed inside a fixed regular hexagon where each side of the square lies on either a side or a diagonal of the hexagon. Note that it is possible for a square to be contained within another square, and that the squares may not extend beyond the boundaries of the hexagon.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
math
|
comp_n_gon
|
test
|
comp_n_gon_test_004
|
comp_n_gon
|
Geometry (Rotation) + Combinatorics (Pattern Matching)
|
[
{
"content": "Find the number of rectangles that can be formed inside a fixed regular hexagon where each side of the rectangle lies on either a side or a diagonal of the hexagon. Note that it is possible for a rectangle to be contained within another rectangle, and that the rectangles may not extend beyond the boundaries of the hexagon.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
9
|
math
|
comp_n_gon
|
test
|
comp_n_gon_test_005
|
comp_n_gon
|
Geometry (Rotation) + Combinatorics (Pattern Matching)
|
[
{
"content": "Find the number of rectangles that can be formed inside a fixed regular octagon where each side of the rectangle lies on either a side or a diagonal of the octagon. Note that it is possible for a rectangle to be contained within another rectangle, and that the rectangles may not extend beyond the boundaries of the octagon.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
40
|
math
|
comp_n_gon
|
test
|
comp_n_gon_test_006
|
comp_n_gon
|
Geometry (Rotation) + Combinatorics (Pattern Matching)
|
[
{
"content": "Find the number of squares that can be formed inside a fixed regular octagon where each side of the square lies on either a side or a diagonal of the octagon. Note that it is possible for a square to be contained within another square, and that the squares may not extend beyond the boundaries of the octagon.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
20
|
math
|
comp_n_gon
|
test
|
comp_n_gon_test_007
|
comp_n_gon
|
Geometry (Rotation) + Combinatorics (Pattern Matching)
|
[
{
"content": "Find the number of rectangles that can be formed inside a fixed regular 10-gon where each side of the rectangle lies on either a side or a diagonal of the 10-gon. Note that it is possible for a rectangle to be contained within another rectangle, and that the rectangles may not extend beyond the boundaries of the 10-gon.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
125
|
math
|
comp_n_gon
|
test
|
comp_n_gon_test_008
|
comp_n_gon
|
Geometry (Rotation) + Combinatorics (Pattern Matching)
|
[
{
"content": "A regular hexagon is drawn, and all of the diagonals between vertices two-apart are drawn (i.e, none of the longest diagonals are drawn). Find the number of triangles that can be formed inside the hexagon, where each side of the triangle lies on a side or one of the diagonals drawn. Note that some triangles may be contained within other triangles.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
32
|
math
|
comp_n_gon
|
test
|
comp_n_gon_test_009
|
comp_n_gon
|
Geometry (Rotation) + Combinatorics (Pattern Matching)
|
[
{
"content": "A regular hexagon is drawn, and all of the diagonals between vertices two-apart are drawn (i.e, none of the longest diagonals are drawn). Find the number of non-equilateral isoceles triangles that can be formed inside the hexagon, where each side of the triangle lies on a side or one of the diagonals drawn. Note that some triangles may be contained within other triangles.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
12
|
math
|
comp_n_gon
|
test
|
comp_n_gon_test_010
|
comp_n_gon
|
Geometry (Rotation) + Combinatorics (Pattern Matching)
|
[
{
"content": "A regular hexagon is drawn, and all of the diagonals between vertices three-apart are drawn (i.e, the longest diagonals are drawn). Find the number of convex polygons that can be formed inside the hexagon, where each side of the polygon lies on a side or one of the diagonals drawn, not including the hexagon itself. Note that some polygons may be contained within other polygons.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
18
|
math
|
comp_n_gon
|
test
|
comp_n_gon_test_011
|
comp_n_gon
|
Geometry (Rotation) + Combinatorics (Pattern Matching)
|
[
{
"content": "A regular hexagon is drawn, and all of the diagonals between vertices three-apart are drawn (i.e, the longest diagonals are drawn). Find the number of triangles that can be formed inside the hexagon, where each side of the polygon lies on a side or one of the diagonals drawn, not including the hexagon itself. Note that some polygons may be contained within other polygons.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
6
|
math
|
comp_n_gon
|
test
|
comp_n_gon_test_012
|
comp_n_gon
|
Geometry (Rotation) + Combinatorics (Pattern Matching)
|
[
{
"content": "Find the number of triangles that can be formed inside a fixed regular pentagon where each side of the triangle lies on either a side or a diagonal of the pentagon. Note that it is possible for a triangle to be contained within another triangle, and that the triangles may not extend beyond the boundaries of the pentagon.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
35
|
math
|
comp_n_gon
|
test
|
comp_n_gon_test_013
|
comp_n_gon
|
Geometry (Rotation) + Combinatorics (Pattern Matching)
|
[
{
"content": "Find the number of equilateral triangles that can be formed inside a fixed regular hexagon where each side of the triangle lies on either a side or a diagonal of the hexagon. Note that it is possible for a triangle to be contained within another triangle, and that the triangles may not extend beyond the boundaries of the hexagon.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
8
|
math
|
comp_n_gon
|
test
|
comp_n_gon_test_014
|
comp_n_gon
|
Geometry (Rotation) + Combinatorics (Pattern Matching)
|
[
{
"content": "Find the number of non-equilateral isoceles triangles that can be formed inside a fixed regular hexagon where each side of the triangle lies on either a side or a diagonal of the hexagon. Note that it is possible for a triangle to be contained within another triangle, and that the triangles may not extend beyond the boundaries of the hexagon.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
18
|
math
|
comp_n_gon
|
test
|
comp_n_gon_test_015
|
comp_n_gon
|
Geometry (Rotation) + Combinatorics (Pattern Matching)
|
[
{
"content": "A regular hexagon is drawn, and all of its diagonals are drawn. Find the number of regular hexagons that can be formed, where each side of the formed hexagons lies on an edge of the original hexagon, or one of the drawn diagonals. Do not include the original hexagon.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
1
|
math
|
comp_n_gon
|
test
|
comp_n_gon_test_016
|
comp_n_gon
|
Geometry (Rotation) + Combinatorics (Pattern Matching)
|
[
{
"content": "A square is drawn, and its diagonals are drawn. Find the number of triangles that can be formed, where each side of the triangle lies on a side of the square, or one of the drawn diagonals. Note that it is possible for a triangle to be contained within another triangle\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
8
|
math
|
comp_n_gon
|
test
|
comp_n_gon_test_017
|
comp_n_gon
|
Geometry (Rotation) + Combinatorics (Pattern Matching)
|
[
{
"content": "A square is drawn, and its midpoints are marked. Lines are drawn connecting each pair of midpoints. Find the number of squares that can be formed, where each side of the square lies on a side of the original square, or one of the lines drawn. Note that it is possible for a square to be contained within another square\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
6
|
math
|
comp_n_gon
|
test
|
comp_n_gon_test_018
|
comp_n_gon
|
Geometry (Rotation) + Combinatorics (Pattern Matching)
|
[
{
"content": "A square is drawn, and its midpoints are marked. Lines are drawn connecting each pair of midpoints. Find the number of triangles that can be formed, where each side of the triangle lies on a side of the original square, or one of the lines drawn. Note that it is possible for a triangle to be contained within another triangle\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
12
|
math
|
comp_n_gon
|
test
|
comp_n_gon_test_019
|
comp_n_gon
|
Geometry (Rotation) + Combinatorics (Pattern Matching)
|
[
{
"content": "A square is drawn, and its midpoints are marked. Lines are drawn connecting each pair of opposing midpoints. Find the number of squares that can be formed, where each side of the square lies on a side of the original square, or one of the lines drawn. Note that it is possible for a square to be contained within another square\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
5
|
math
|
comp_n_gon
|
test
|
comp_n_gon_test_020
|
comp_n_gon
|
Geometry (Rotation) + Combinatorics (Pattern Matching)
|
[
{
"content": "A square is drawn, and its midpoints are marked. Lines are drawn connecting each pair of opposing midpoints. Find the number of rectangles that can be formed, where each side of the rectangle lies on a side of the original square, or one of the lines drawn. Note that it is possible for a rectangle to be contained within another rectangle\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
9
|
math
|
comp_n_gon
|
test
|
comp_circles_algebra_test_000
|
comp_circles_algebra
|
Geometry (Circle) + Algebra (Function Intersection)
|
[
{
"content": "A circle with radius 4 is moving on the coordinate plane such that its center moves along the curve P(t) = \\langle t, t^2 \\rangle starting at t=0. Find the first value of t for which the circle lies tangent to the x-axis Provide your answer as a decimal number, rounded to the nearest two decimal points.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2
|
math
|
comp_circles_algebra
|
test
|
comp_circles_algebra_test_001
|
comp_circles_algebra
|
Geometry (Circle) + Algebra (Function Intersection)
|
[
{
"content": "A circle with radius 3 is moving on the coordinate plane such that its center moves along the curve P(t) = \\langle t, t \\rangle starting at t=0. Find the first value of t for which the circle lies tangent to the y-axis. Provide your answer as a decimal number, rounded to the nearest two decimal points.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3
|
math
|
comp_circles_algebra
|
test
|
comp_circles_algebra_test_002
|
comp_circles_algebra
|
Geometry (Circle) + Algebra (Function Intersection)
|
[
{
"content": "A circle with radius 1 is moving on the coordinate plane such that its center moves along the curve P(t) = \\langle t, sin(t) \\rangle starting at t=0. Find the first value of t for which the circle lies tangent to the x-axis. Provide your answer as a decimal number, rounded to the nearest two decimal points.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0.79
|
math
|
comp_circles_algebra
|
test
|
comp_circles_algebra_test_003
|
comp_circles_algebra
|
Geometry (Circle) + Algebra (Function Intersection)
|
[
{
"content": "A circle with radius r is moving on the coordinate plane such that its center moves along the curve P(t) = \\langle t, sin(t) \\rangle starting at t=0. Find the least value of r for which the circle at some point lies tangent to both the x and y axes. Provide your answer as a decimal number, rounded to the nearest two decimal points.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0.71
|
math
|
comp_circles_algebra
|
test
|
comp_circles_algebra_test_004
|
comp_circles_algebra
|
Geometry (Circle) + Algebra (Function Intersection)
|
[
{
"content": "A circle with radius 1 is moving on the coordinate plane such that its center is at the point (t, 0) starting at t=0. Find the least value of t for which the circle lies tangent to the curve y = sin(x). Provide your answer as a decimal number, rounded to the nearest two decimal points.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0.71
|
math
|
comp_circles_algebra
|
test
|
comp_circles_algebra_test_005
|
comp_circles_algebra
|
Geometry (Circle) + Algebra (Function Intersection)
|
[
{
"content": "A circle with radius 1 is moving on the coordinate plane such that its center is at the point (t, 0) starting at t=0. Another circle with radius 1 lies tangent to the curve y = cos(x) at x = \\pi. Find the least value of t for which the two circles are tangent. Provide your answer as a decimal number, rounded to the nearest two decimal points.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
1.14
|
math
|
comp_circles_algebra
|
test
|
comp_circles_algebra_test_006
|
comp_circles_algebra
|
Geometry (Circle) + Algebra (Function Intersection)
|
[
{
"content": "A circle with radius 5 is centered at the origin. A chord is constructed on the circle parallel to the x-axis whose length is 8. A line with the equation y = mx is constructed. Find the largest value of m for which the line intersects the chord. Provide your answer as a decimal number, rounded to the nearest two decimal points.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0.75
|
math
|
comp_circles_algebra
|
test
|
comp_circles_algebra_test_007
|
comp_circles_algebra
|
Geometry (Circle) + Algebra (Function Intersection)
|
[
{
"content": "A circle with radius 5 is centered at the origin. A chord is constructed on the circle above the x-axis, parallel to the x-axis, whose length is 8. A parabola with the equation y = ax^2 - 1 is constructed. Find the smallest value of a for which the parabola intersects the chord. Provide your answer as a decimal number, rounded to the nearest two decimal points.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0.25
|
math
|
comp_circles_algebra
|
test
|
comp_circles_algebra_test_008
|
comp_circles_algebra
|
Geometry (Circle) + Algebra (Function Intersection)
|
[
{
"content": "A circle with radius 5 is constructed in the coordinate plane. Two chords are constructed on the circle parallel to the x-axis, whose lengths are 8. Find the slope of the steepest line that intersects both chords. Provide your answer as a decimal number, rounded to the nearest two decimal points.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0.75
|
math
|
comp_circles_algebra
|
test
|
comp_circles_algebra_test_009
|
comp_circles_algebra
|
Geometry (Circle) + Algebra (Function Intersection)
|
[
{
"content": "A circle with radius 1 is moving on the coordinate plane such that its center is at the coordinate (0, t) starting at t=0. A parabola with equation x^2 + 2 is placed in the coordinate plane. Find the first value of t for which the circle and the parabola intersect at one point. Provide your answer as a decimal number, rounded to the nearest two decimal points.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2
|
math
|
comp_circles_algebra
|
test
|
comp_circles_algebra_test_010
|
comp_circles_algebra
|
Geometry (Circle) + Algebra (Function Intersection)
|
[
{
"content": "A circle with radius 3 is moving on the coordinate plane such that its center is at the coordinate (0, t) starting at t=0. A parabola with equation x^2 + 2 is placed in the coordinate plane. Find the first value of t for which the circle and the parabola intersect at three points. Provide your answer as a decimal number, rounded to the nearest two decimal points.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
5
|
math
|
comp_circles_algebra
|
test
|
comp_circles_algebra_test_011
|
comp_circles_algebra
|
Geometry (Circle) + Algebra (Function Intersection)
|
[
{
"content": "A circle with radius 1 moves upwards on the coordinate plane such that its center is at the coordinate (0, t) starting at t=0. A parabola with equation x^2 + 2 is placed in the coordinate plane. Find the largest value of t for which the circle and the parabola intersect at two points. Provide your answer as a decimal number, rounded to the nearest two decimal points.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
3.25
|
math
|
comp_circles_algebra
|
test
|
comp_circles_algebra_test_012
|
comp_circles_algebra
|
Geometry (Circle) + Algebra (Function Intersection)
|
[
{
"content": "A circle with radius r is placed so that it is centered at (0, r). A parabola with equation x^2 is also placed in the coordinate plane. Find the infimum of all values of r for which the circle and the parabola intersect at three points. Provide your answer as a decimal number, rounded to the nearest two decimal points.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
math
|
comp_circles_algebra
|
test
|
comp_circles_algebra_test_013
|
comp_circles_algebra
|
Geometry (Circle) + Algebra (Function Intersection)
|
[
{
"content": "A circle with radius 1 is centered at the origin. A second circle with radius 1 lies externally tangent to the first circle at x = 1. Find the number of intersections of the second circle with the line y = 0.5x.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2
|
math
|
comp_circles_algebra
|
test
|
comp_circles_algebra_test_014
|
comp_circles_algebra
|
Geometry (Circle) + Algebra (Function Intersection)
|
[
{
"content": "A circle with radius 1 is centered at the origin. A second circle with radius 1 lies externally tangent to the first circle at x = 1. Find the number of intersections of the second circle with the curve y = sin(2x).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2
|
math
|
comp_circles_algebra
|
test
|
comp_circles_algebra_test_015
|
comp_circles_algebra
|
Geometry (Circle) + Algebra (Function Intersection)
|
[
{
"content": "A circle with radius 1 is centered at the origin. A second circle with radius 1 lies externally tangent to the first circle at x = 1. A third circle lies tangent to the other two circles with radius 1 and center above the x-axis. Find the number of intersections of the third circle with the parabola y = x^2.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2
|
math
|
comp_circles_algebra
|
test
|
comp_circles_algebra_test_016
|
comp_circles_algebra
|
Geometry (Circle) + Algebra (Function Intersection)
|
[
{
"content": "A circle with radius 1 is centered at the origin. A second circle with radius 1 lies externally tangent to the first circle at x = 1. A third circle lies tangent to the other two circles with radius 1 and center above the x-axis. Find the least value of x for which the curve y = sin(x) intersects the third circle. Provide your answer as a decimal number, rounded to the nearest two decimal points.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0.84
|
math
|
comp_circles_algebra
|
test
|
comp_circles_algebra_test_017
|
comp_circles_algebra
|
Geometry (Circle) + Algebra (Function Intersection)
|
[
{
"content": "A circle with radius 1 is centered at the origin. A second circle with radius 1 lies externally tangent to the first circle at x = 1. A third circle lies tangent to the other two circles with radius 1 and center above the x-axis. Find the number of intersections of the third circle with the curve y = 3sin(4x).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
6
|
math
|
comp_circles_algebra
|
test
|
comp_circles_algebra_test_018
|
comp_circles_algebra
|
Geometry (Circle) + Algebra (Function Intersection)
|
[
{
"content": "A circle with radius 2 is centered at the origin. A second circle with radius 2 lies externally tangent to the first circle at x = 1. A third circle of radius 1 lies internally tangent to the second circle at its rightmost point. Find the value of a for which the curve x = ay^2 passes through the center of the third circle. Provide your answer as a decimal number, rounded to the nearest two decimal points.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0.25
|
math
|
comp_circles_algebra
|
test
|
comp_circles_algebra_test_019
|
comp_circles_algebra
|
Geometry (Circle) + Algebra (Function Intersection)
|
[
{
"content": "A circle with radius 2 is centered at the origin. A second circle with radius 2 lies externally tangent to the first circle at x = 1. A third circle of radius 1 lies internally tangent to the second circle at its rightmost point. Find the number of intersections of the third circle with the curve y = 4sin(5x).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
math
|
comp_circles_algebra
|
test
|
comp_circles_algebra_test_020
|
comp_circles_algebra
|
Geometry (Circle) + Algebra (Function Intersection)
|
[
{
"content": "A circle with radius 2 is centered at the origin. A second circle with radius 2 lies externally tangent to the first circle at x = 1. A third circle with radius 4 is centered at the origin. A fourth circle is constructed passing through all three circles at their rightmost points. Find the number of intersections of the fourth circle with the line y = 3x.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
math
|
comp_circles_algebra
|
test
|
comp_circles_algebra_test_021
|
comp_circles_algebra
|
Geometry (Circle) + Algebra (Function Intersection)
|
[
{
"content": "A circle with radius 2 is centered at the origin. A second circle with radius 2 lies externally tangent to the first circle at x = 1 with a positive y value. A third circle with radius 4 is centered at the origin. A fourth circle is constructed passing through all three circles at their rightmost points. Find the number of intersections of the fourth circle with the curve y = -(x-3)^2 + 4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
math
|
comp_circles_algebra
|
test
|
comp_circles_algebra_test_022
|
comp_circles_algebra
|
Geometry (Circle) + Algebra (Function Intersection)
|
[
{
"content": "A circle with radius 2 is centered at the origin. A second circle with radius 2 lies externally tangent to the first circle at x = 1 with a positive y value. A third circle with radius 4 is centered at the origin. A fourth circle is constructed passing through all three circles at their rightmost points. Find the number of intersections of the fourth circle with the curve y = sin(3x).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
math
|
comp_circles_algebra
|
test
|
comp_circles_algebra_test_023
|
comp_circles_algebra
|
Geometry (Circle) + Algebra (Function Intersection)
|
[
{
"content": "A circle with radius 2 is centered at the origin. A second circle with radius 2 lies externally tangent to the first circle at x = 1. A third circle with radius 4 is centered at the origin. A fourth circle is constructed passing through all three circles at their rightmost points. Find the number of intersections of the fourth circle with the curve y = sin(3x).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
4
|
math
|
comp_circles_algebra
|
test
|
comp_circles_algebra_test_024
|
comp_circles_algebra
|
Geometry (Circle) + Algebra (Function Intersection)
|
[
{
"content": "A circle with radius 2 is centered at the origin. A second circle with radius 2 lies externally tangent to the first circle at x = 1. A third circle with radius 4 passes through the rightmost points of the first two circles and has center with a positive x-value. Find the number of intersections of the third circle with the line y = 4.\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
1
|
math
|
comp_circles_algebra
|
test
|
comp_circles_algebra_test_025
|
comp_circles_algebra
|
Geometry (Circle) + Algebra (Function Intersection)
|
[
{
"content": "A circle with radius 2 is centered at the origin. A second circle with radius 2 lies externally tangent to the first circle at x = 1. A third circle with radius 4 passes through the rightmost points of the first two circles and has center with a positive x-value. Find the number of intersections of the third circle with the curve y = sin(3x).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
2
|
math
|
comp_circles_algebra
|
test
|
comp_circles_algebra_test_026
|
comp_circles_algebra
|
Geometry (Circle) + Algebra (Function Intersection)
|
[
{
"content": "A circle with radius 2 is centered at the origin. A second circle with radius 2 lies externally tangent to the first circle at x = 1. A third circle with radius 4 passes through the rightmost points of the first two circles and has center with a positive x-value. Find the number of intersections of the third circle with the curve y = 4sin(5x).\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
24
|
math
|
comp_circles_algebra
|
test
|
comp_parametric_intersection_test_000
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "Considering the functions f(x) = ax and g(x) = px^2 + qx, where a, p, q can each take integer values from 1 to 5, how many different combinations of parameter values result in at least 1 intersection points in the range [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
73
|
math
|
comp_parametric_intersection
|
test
|
comp_parametric_intersection_test_001
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "If a, b, p are integers between 1 and 5 inclusive, how many different combinations of these values make the functions f(x) = asin(bπx) + 2 and g(x) = px intersect at least 6 times in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
53
|
math
|
comp_parametric_intersection
|
test
|
comp_parametric_intersection_test_002
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "Pick values for a, b, p, q from 1 to 5, how many combinations can we have so that the function f(x) = ax^2 + bx and g(x) = px^2 + qx have at least 1 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
388
|
math
|
comp_parametric_intersection
|
test
|
comp_parametric_intersection_test_003
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "Pick values for a, b, p, q from 1 to 5, how many combinations can we have so that the function f(x) = asin(bπx) and g(x) = pcos(qπx) - 3 have at least 1 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
545
|
math
|
comp_parametric_intersection
|
test
|
comp_parametric_intersection_test_004
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "If a, b, p are integers between 1 and 5 inclusive, how many different combinations of these values make the functions f(x) = ax^2 + bx and g(x) = px + 3 intersect at least 1 times in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
62
|
math
|
comp_parametric_intersection
|
test
|
comp_parametric_intersection_test_005
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "Pick values for a, b, p from 1 to 5, how many combinations can we have so that the function f(x) = ax^2 + bx and g(x) = px have at least 2 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
60
|
math
|
comp_parametric_intersection
|
test
|
comp_parametric_intersection_test_006
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "Considering the functions f(x) = asin(bπx) and g(x) = psin(qπx), where a, b, p, q can each take integer values from 1 to 5, how many different combinations of parameter values result in at least 7 intersection points in the range [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
508
|
math
|
comp_parametric_intersection
|
test
|
comp_parametric_intersection_test_007
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "Pick values for a, b, p, q from 1 to 5, how many combinations can we have so that the function f(x) = ax^2 + bx and g(x) = px^2 + qx - 2 have at least 1 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
390
|
math
|
comp_parametric_intersection
|
test
|
comp_parametric_intersection_test_008
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "Pick values for a, b, p from 1 to 5, how many combinations can we have so that the function f(x) = ax^2 + bx and g(x) = px have at least 1 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
65
|
math
|
comp_parametric_intersection
|
test
|
comp_parametric_intersection_test_009
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "Considering the functions f(x) = ax and g(x) = px + 3, where a, p can each take integer values from 1 to 5, how many different combinations of parameter values result in at least 1 intersection points in the range [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
20
|
math
|
comp_parametric_intersection
|
test
|
comp_parametric_intersection_test_010
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "Pick values for a, p, q from 1 to 5, how many combinations can we have so that the function f(x) = ax and g(x) = px^2 + qx have at least 2 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
63
|
math
|
comp_parametric_intersection
|
test
|
comp_parametric_intersection_test_011
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "If a, b, p, q are integers between 1 and 5 inclusive, how many different combinations of these values make the functions f(x) = acos(bπx) and g(x) = psin(qπx) intersect at least 2 times in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
562
|
math
|
comp_parametric_intersection
|
test
|
comp_parametric_intersection_test_012
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "If a, b, p, q are integers between 1 and 5 inclusive, how many different combinations of these values make the functions f(x) = ax^2 + bx and g(x) = pcos(qπx) intersect at least 1 times in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
515
|
math
|
comp_parametric_intersection
|
test
|
comp_parametric_intersection_test_013
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "Pick values for a, b, p, q from 1 to 5, how many combinations can we have so that the function f(x) = asin(bπx) and g(x) = pcos(qπx) have at least 1 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
546
|
math
|
comp_parametric_intersection
|
test
|
comp_parametric_intersection_test_014
|
comp_parametric_intersection
|
Combinatorics (Probability) + Algebra (Function Intersection)
|
[
{
"content": "Pick values for a, p, q from 1 to 5, how many combinations can we have so that the function f(x) = ax and g(x) = px^2 + qx have at least 3 intersections in the interval [-10, 10]?\n\nPresent the answer in LaTex format: \\boxed{Your answer}",
"role": "user"
}
] |
0
|
math
|
comp_parametric_intersection
|
test
|
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