prompt listlengths 1 1 | reward_model dict | data_source stringclasses 1
value | ability stringclasses 1
value | extra_info dict |
|---|---|---|---|---|
[
{
"content": "Find the smallest integer \\( n \\) such that \\( n! > 10^{100} \\).",
"role": "user"
}
] | {
"ground_truth": "70",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "200",
"problem": "Find the smallest integer \\( n \\) such that \\( n! > 10^{100} \\).",
"solution": "Okay, so I need to find the smallest integer n such that n! is greater than 10^100. Hmm, factorials grow really fast, right? But I need to figure out exactly where that happens. Let me start by recall... |
[
{
"content": "Determine the number of non-negative integer solutions to the equation \\(x_{1} + x_{2} + 3x_{3} + x_{4} = k\\), given the constraints: \\(x_{1} \\geq 1\\), \\(x_{2} \\leq 2\\), \\(x_{3} \\leq 1\\), and \\(x_{4}\\) is a multiple of 6.",
"role": "user"
}
] | {
"ground_truth": "k",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "201",
"problem": "Determine the number of non-negative integer solutions to the equation \\(x_{1} + x_{2} + 3x_{3} + x_{4} = k\\), given the constraints: \\(x_{1} \\geq 1\\), \\(x_{2} \\leq 2\\), \\(x_{3} \\leq 1\\), and \\(x_{4}\\) is a multiple of 6.",
"solution": "Okay, so I need to find the number... |
[
{
"content": "Given the differential equation \\( \\frac{{d^2x}}{{dy^2}} \\), determine the correct expression from the following options:\n\n(A) \\( {\\left( \\frac{{d^2y}}{{dx^2}} \\right)^{-1}} \\)\n\n(B) \\( -{\\left( \\frac{{d^2y}}{{dx^2}} \\right)^{-1}}{\\left( \\frac{{dy}}{{dx}} \\right)^{-3}} \\)\n\n(C)... | {
"ground_truth": "D",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "202",
"problem": "Given the differential equation \\( \\frac{{d^2x}}{{dy^2}} \\), determine the correct expression from the following options:\n\n(A) \\( {\\left( \\frac{{d^2y}}{{dx^2}} \\right)^{-1}} \\)\n\n(B) \\( -{\\left( \\frac{{d^2y}}{{dx^2}} \\right)^{-1}}{\\left( \\frac{{dy}}{{dx}} \\right)^{-3}... |
[
{
"content": "Evaluate the integral \\( \\int_{0}^{2\\pi} \\left| \\sin\\theta + \\cos\\theta \\right| \\mathrm{d}\\theta \\).",
"role": "user"
}
] | {
"ground_truth": "4\\sqrt{2}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "203",
"problem": "Evaluate the integral \\( \\int_{0}^{2\\pi} \\left| \\sin\\theta + \\cos\\theta \\right| \\mathrm{d}\\theta \\).",
"solution": "Okay, so I need to evaluate the integral of |sinθ + cosθ| from 0 to 2π. Hmm, integrating absolute values can be tricky because you have to consider where th... |
[
{
"content": "For any arbitrary set $A$, does there always exist a set $B$ such that $A$ is a proper subset of $B$?",
"role": "user"
}
] | {
"ground_truth": "Yes",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "204",
"problem": "For any arbitrary set $A$, does there always exist a set $B$ such that $A$ is a proper subset of $B$?",
"solution": "Okay, so I need to figure out if for any set A, there's always a set B where A is a proper subset of B. Hmm. Let's start by recalling some basic set theory concepts. A... |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to \\infty} x(2\\pi - 4\\arctan(x)) \\]",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "205",
"problem": "Evaluate the limit: \\[ \\lim_{x \\to \\infty} x(2\\pi - 4\\arctan(x)) \\]",
"solution": "Okay, so I need to find the limit as x approaches infinity of x times (2π minus 4 arctan(x)). Let me write that down again to make sure I got it right: \n\n\\[\n\\lim_{x \\to \\infty} x(2\\pi - ... |
[
{
"content": "Evaluate the improper integral \\( \\int_e^{\\infty}\\frac{\\ln x}{x}\\ dx \\). Choose the correct answer from the following options:\n\nA. \\( \\dfrac{1}{2} \\)\n\nB. \\( \\dfrac{e^2}{2} \\)\n\nC. \\( \\dfrac{\\ln(2e)}{2} \\)\n\nD. DNE (Does not exist)",
"role": "user"
}
] | {
"ground_truth": "D",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "206",
"problem": "Evaluate the improper integral \\( \\int_e^{\\infty}\\frac{\\ln x}{x}\\ dx \\). Choose the correct answer from the following options:\n\nA. \\( \\dfrac{1}{2} \\)\n\nB. \\( \\dfrac{e^2}{2} \\)\n\nC. \\( \\dfrac{\\ln(2e)}{2} \\)\n\nD. DNE (Does not exist)",
"solution": "Okay, let's see... |
[
{
"content": "Given that \\(\\log_{4n} 40\\sqrt{3} = \\log_{3n} 45\\), find the value of \\(n^3\\).",
"role": "user"
}
] | {
"ground_truth": "75",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "207",
"problem": "Given that \\(\\log_{4n} 40\\sqrt{3} = \\log_{3n} 45\\), find the value of \\(n^3\\).",
"solution": "Okay, let's try to solve this problem. We have the equation \\(\\log_{4n} 40\\sqrt{3} = \\log_{3n} 45\\), and we need to find \\(n^3\\). Hmm, logarithms with different bases. I rememb... |
[
{
"content": "Determine the number of connected components of the set \\(\\{(x,y) \\in \\mathbb{R}^2 : (xy)^2 = xy\\}\\) in \\(\\mathbb{R} \\times \\mathbb{R}\\).",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "208",
"problem": "Determine the number of connected components of the set \\(\\{(x,y) \\in \\mathbb{R}^2 : (xy)^2 = xy\\}\\) in \\(\\mathbb{R} \\times \\mathbb{R}\\).",
"solution": "Okay, so I need to figure out the number of connected components for the set defined by the equation \\((xy)^2 = xy\\) i... |
[
{
"content": "Let \\( X \\) be a single observation from the uniform distribution with probability density function \\( f(x) = \\frac{1}{\\theta} \\) for \\( 0 < x < \\theta \\) and \\( f(x) = 0 \\) otherwise. Suppose we use \\((X, 1.5X)\\) as a confidence interval for \\( \\theta \\). Determine the confidence ... | {
"ground_truth": "\\dfrac{1}{3}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "209",
"problem": "Let \\( X \\) be a single observation from the uniform distribution with probability density function \\( f(x) = \\frac{1}{\\theta} \\) for \\( 0 < x < \\theta \\) and \\( f(x) = 0 \\) otherwise. Suppose we use \\((X, 1.5X)\\) as a confidence interval for \\( \\theta \\). Determine the... |
[
{
"content": "Evaluate the limit of the expression \\(\\lim\\limits_{n\\to \\infty} \\left(\\frac{1}{e}\\right)^{\\frac{3n+2+\\frac{1}{(n-1)!}}{2}}\\). Specifically, determine the behavior of the term \\(\\frac{1}{(n-1)!}\\) as \\(n\\) approaches infinity.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "210",
"problem": "Evaluate the limit of the expression \\(\\lim\\limits_{n\\to \\infty} \\left(\\frac{1}{e}\\right)^{\\frac{3n+2+\\frac{1}{(n-1)!}}{2}}\\). Specifically, determine the behavior of the term \\(\\frac{1}{(n-1)!}\\) as \\(n\\) approaches infinity.",
"solution": "Okay, so I need to evaluat... |
[
{
"content": "Suppose 28 objects are placed around a circle at equal distances. In how many ways can you choose 3 objects from among them so that no two of the 3 chosen are adjacent nor diametrically opposite?",
"role": "user"
}
] | {
"ground_truth": "2268",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "211",
"problem": "Suppose 28 objects are placed around a circle at equal distances. In how many ways can you choose 3 objects from among them so that no two of the 3 chosen are adjacent nor diametrically opposite?",
"solution": "Okay, so I need to figure out how many ways there are to choose 3 objects... |
[
{
"content": "Given a connected graph with up to 105 nodes, determine the minimum number of nodes that need to be removed to make the graph disconnected, such that there exists at least one node that cannot reach all other nodes. What is this minimum number?",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "212",
"problem": "Given a connected graph with up to 105 nodes, determine the minimum number of nodes that need to be removed to make the graph disconnected, such that there exists at least one node that cannot reach all other nodes. What is this minimum number?",
"solution": "Alright, so I have this ... |
[
{
"content": "The error for the production of a machine is uniformly distributed over the interval \\([-0.7, 0.5]\\) units. If there are 10 machines working simultaneously, approximate the probability that the total production error exceeds 1 unit in magnitude.",
"role": "user"
}
] | {
"ground_truth": "0.534",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "213",
"problem": "The error for the production of a machine is uniformly distributed over the interval \\([-0.7, 0.5]\\) units. If there are 10 machines working simultaneously, approximate the probability that the total production error exceeds 1 unit in magnitude.",
"solution": "Okay, so I need to fi... |
[
{
"content": "In a dice game, each player rolls \\( k \\) dice on their turn. The first player to accumulate a total of \\( k \\) sixes over all their turns wins the game. What is the expected number of turns needed for a player to win the game?",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "214",
"problem": "In a dice game, each player rolls \\( k \\) dice on their turn. The first player to accumulate a total of \\( k \\) sixes over all their turns wins the game. What is the expected number of turns needed for a player to win the game?",
"solution": "Okay, so I need to figure out the exp... |
[
{
"content": "In a right triangle $\\triangle ABC$ with $\\angle B = 90^\\circ$, an interior bisector $BD$ is drawn, where $I$ is the incenter and $3BI = 4ID$. Find the relationship between the circumradius $R$ and the inradius $r$ of $\\triangle ABC$.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "215",
"problem": "In a right triangle $\\triangle ABC$ with $\\angle B = 90^\\circ$, an interior bisector $BD$ is drawn, where $I$ is the incenter and $3BI = 4ID$. Find the relationship between the circumradius $R$ and the inradius $r$ of $\\triangle ABC$.",
"solution": "Okay, so I have this problem h... |
[
{
"content": "A Poisson process has a parameter \\( \\lambda \\). Each time an event occurs, you receive \\( K \\) dollars. What is the expected total amount you will receive?",
"role": "user"
}
] | {
"ground_truth": "K \\lambda",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "216",
"problem": "A Poisson process has a parameter \\( \\lambda \\). Each time an event occurs, you receive \\( K \\) dollars. What is the expected total amount you will receive?",
"solution": "Okay, so I need to figure out the expected total amount I'll receive from a Poisson process with parameter ... |
[
{
"content": "Given a point P = (0,0,0) and a line L in 3D space defined by L = {P + k*w} where w = (-1, 0, 0), find the scalar q such that the point P' = P + q*w satisfies the condition that the dot product of the unit vector from P' to Q = (0, 1, 0) with the vector v = (0, 1, 0) is as close as possible to the... | {
"ground_truth": "\\pm \\sqrt{3}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "217",
"problem": "Given a point P = (0,0,0) and a line L in 3D space defined by L = {P + k*w} where w = (-1, 0, 0), find the scalar q such that the point P' = P + q*w satisfies the condition that the dot product of the unit vector from P' to Q = (0, 1, 0) with the vector v = (0, 1, 0) is as close as pos... |
[
{
"content": "In a right triangle \\(\\triangle ABC\\), with \\(\\angle ABC = 90^\\circ\\), the altitude \\(BH\\) is drawn to the hypotenuse \\(AC\\). In triangle \\(BHC\\), a cevian \\(HM\\) is drawn such that \\(MC = AB\\). Given that \\(HC = BH + 2AH\\), find the measure of the angle \\(\\angle MHC\\). Choos... | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "218",
"problem": "In a right triangle \\(\\triangle ABC\\), with \\(\\angle ABC = 90^\\circ\\), the altitude \\(BH\\) is drawn to the hypotenuse \\(AC\\). In triangle \\(BHC\\), a cevian \\(HM\\) is drawn such that \\(MC = AB\\). Given that \\(HC = BH + 2AH\\), find the measure of the angle \\(\\angle M... |
[
{
"content": "Find the 24th prime number given that \\( \\pi(x) = 25 \\).",
"role": "user"
}
] | {
"ground_truth": "89",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "219",
"problem": "Find the 24th prime number given that \\( \\pi(x) = 25 \\).",
"solution": "Okay, let's see. I need to find the 24th prime number, and they gave me that π(x) = 25. Hmm, π(x) usually represents the prime-counting function, right? So π(x) is the number of prime numbers less than or equa... |
[
{
"content": "Given the equation \\(|mx+\\alpha|+|mx+\\beta|=c\\), where \\(x=2\\) and \\(x=4\\) are solutions, and \\(m>0\\) with \\(\\alpha, \\beta, c\\) as non-zero constants, find the value of \\(\\left|\\frac{\\alpha+\\beta}{m}\\right|\\).",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "220",
"problem": "Given the equation \\(|mx+\\alpha|+|mx+\\beta|=c\\), where \\(x=2\\) and \\(x=4\\) are solutions, and \\(m>0\\) with \\(\\alpha, \\beta, c\\) as non-zero constants, find the value of \\(\\left|\\frac{\\alpha+\\beta}{m}\\right|\\).",
"solution": "Okay, so I have this equation: |mx + α... |
[
{
"content": "Find the number of solutions in positive integers to the equation \\(x_1 + x_2 + x_3 + x_4 = 19\\) subject to the conditions \\(x_2 \\neq 2x_3\\) and \\(x_1 \\neq x_2\\).",
"role": "user"
}
] | {
"ground_truth": "702",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "221",
"problem": "Find the number of solutions in positive integers to the equation \\(x_1 + x_2 + x_3 + x_4 = 19\\) subject to the conditions \\(x_2 \\neq 2x_3\\) and \\(x_1 \\neq x_2\\).",
"solution": "Okay, so I need to find the number of solutions in positive integers to the equation \\(x_1 + x_2 ... |
[
{
"content": "In a hypothesis test where the null hypothesis is \\( H_0: \\theta > 0.0625 \\) and the alternative hypothesis is \\( H_A: \\theta \\leq 0.0625 \\), what value of \\( \\theta \\) should be used to calculate the distribution of the test statistic under the null hypothesis?",
"role": "user"
}
... | {
"ground_truth": "0.0625",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "222",
"problem": "In a hypothesis test where the null hypothesis is \\( H_0: \\theta > 0.0625 \\) and the alternative hypothesis is \\( H_A: \\theta \\leq 0.0625 \\), what value of \\( \\theta \\) should be used to calculate the distribution of the test statistic under the null hypothesis?",
"solution... |
[
{
"content": "Given the linear regression model with 100 observations:\n\\[ y_i = b_0 + b_1 x_i + \\epsilon_i \\]\nThe estimated coefficients are:\n\\[ \\hat{\\beta} = \\begin{pmatrix} 9 \\\\ 1 \\end{pmatrix} \\]\nAnd the variance-covariance matrix of the estimated coefficients is:\n\\[ V(\\hat{\\beta}) = \\hat... | {
"ground_truth": "27195",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "223",
"problem": "Given the linear regression model with 100 observations:\n\\[ y_i = b_0 + b_1 x_i + \\epsilon_i \\]\nThe estimated coefficients are:\n\\[ \\hat{\\beta} = \\begin{pmatrix} 9 \\\\ 1 \\end{pmatrix} \\]\nAnd the variance-covariance matrix of the estimated coefficients is:\n\\[ V(\\hat{\\be... |
[
{
"content": "A clock has 12 positions, numbered 1 through 12. Each step, the hour hand moves either clockwise or counterclockwise with equal probability (1/2), independently of previous steps. Starting at position 1, what is the expected number of steps required for the hour hand to reach position 12?",
"r... | {
"ground_truth": "11",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "224",
"problem": "A clock has 12 positions, numbered 1 through 12. Each step, the hour hand moves either clockwise or counterclockwise with equal probability (1/2), independently of previous steps. Starting at position 1, what is the expected number of steps required for the hour hand to reach position ... |
[
{
"content": "A white Rubik's cube has 6 sides, each with 9 spaces that can be filled with numbers from 1 to 9. A 3x3 magic square is one in which the sum of the numbers in each row, column, and both main diagonals is the same, and there are 8 possible combinations of such a square. How many ways can you fill e... | {
"ground_truth": "262144",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "225",
"problem": "A white Rubik's cube has 6 sides, each with 9 spaces that can be filled with numbers from 1 to 9. A 3x3 magic square is one in which the sum of the numbers in each row, column, and both main diagonals is the same, and there are 8 possible combinations of such a square. How many ways ca... |
[
{
"content": "Find the volume of the region bounded by the planes $6x+5y+6z = 6$, $y=x$, $x=0$, and $z=0$.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{6}{55}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "226",
"problem": "Find the volume of the region bounded by the planes $6x+5y+6z = 6$, $y=x$, $x=0$, and $z=0$.",
"solution": "Okay, so I need to find the volume of the region bounded by those four planes: 6x + 5y + 6z = 6, y = x, x = 0, and z = 0. Hmm, volume problems can be tricky, but let me try to ... |
[
{
"content": "Find the value of \\(x\\) such that \\(\\sqrt{(2+\\sqrt{3})^x} + \\sqrt{(2-\\sqrt{3})^x} = 2^x\\).",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "227",
"problem": "Find the value of \\(x\\) such that \\(\\sqrt{(2+\\sqrt{3})^x} + \\sqrt{(2-\\sqrt{3})^x} = 2^x\\).",
"solution": "Alright, so I need to solve the equation \\(\\sqrt{(2+\\sqrt{3})^x} + \\sqrt{(2-\\sqrt{3})^x} = 2^x\\). Hmm, let's break this down step by step. First, let me make sure I... |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0^-}{\\frac{-e^{1/x}}{x}} \\]",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "228",
"problem": "Evaluate the limit: \\[ \\lim_{x \\to 0^-}{\\frac{-e^{1/x}}{x}} \\]",
"solution": "Okay, so I need to find the limit as x approaches 0 from the negative side of (-e^(1/x))/x. Hmm, let me start by understanding what's happening here. \n\nFirst, when x approaches 0 from the negative si... |
[
{
"content": "Losses relating to a type of insurance policy follow a gamma distribution with mean $30,000$ and shape parameter $\\alpha = 2$. For a sample of 100 policy claims, calculate the approximate probability that the mean loss size is less than $27,000$. Provide your answer to four decimal places.",
... | {
"ground_truth": "0.0786",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "229",
"problem": "Losses relating to a type of insurance policy follow a gamma distribution with mean $30,000$ and shape parameter $\\alpha = 2$. For a sample of 100 policy claims, calculate the approximate probability that the mean loss size is less than $27,000$. Provide your answer to four decimal pl... |
[
{
"content": "Compute the sum \\( S = 1 \\times 2 \\times 3 \\times 4 + 3 \\times 4 \\times 5 \\times 6 + \\ldots + 97 \\times 98 \\times 99 \\times 100 \\).",
"role": "user"
}
] | {
"ground_truth": "974510040",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "230",
"problem": "Compute the sum \\( S = 1 \\times 2 \\times 3 \\times 4 + 3 \\times 4 \\times 5 \\times 6 + \\ldots + 97 \\times 98 \\times 99 \\times 100 \\).",
"solution": "Okay, so I need to compute the sum S = 1×2×3×4 + 3×4×5×6 + … + 97×98×99×100. Hmm, let's see. Each term in the sum is a produc... |
[
{
"content": "Evaluate the expression: \\[ \\csc^2\\left(\\frac{\\pi}{9}\\right) + \\csc^2\\left(\\frac{2\\pi}{9}\\right) + \\csc^2\\left(\\frac{4\\pi}{9}\\right) \\]",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "231",
"problem": "Evaluate the expression: \\[ \\csc^2\\left(\\frac{\\pi}{9}\\right) + \\csc^2\\left(\\frac{2\\pi}{9}\\right) + \\csc^2\\left(\\frac{4\\pi}{9}\\right) \\]",
"solution": "Okay, so I need to evaluate the expression \\(\\csc^2\\left(\\frac{\\pi}{9}\\right) + \\csc^2\\left(\\frac{2\\pi}{9}... |
[
{
"content": "Twelve politicians are seated at a round table. A committee of five is to be chosen such that no two committee members are immediate neighbors. In how many ways can this committee be selected?",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "232",
"problem": "Twelve politicians are seated at a round table. A committee of five is to be chosen such that no two committee members are immediate neighbors. In how many ways can this committee be selected?",
"solution": "Okay, so I need to figure out how many ways to select a committee of five fr... |
[
{
"content": "Determine the number of terms required in the Taylor series for \\( \\ln(1+x) \\) centered at \\( x = 0 \\) to estimate \\( \\ln(1.4) \\) to within an error of 0.0005.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "233",
"problem": "Determine the number of terms required in the Taylor series for \\( \\ln(1+x) \\) centered at \\( x = 0 \\) to estimate \\( \\ln(1.4) \\) to within an error of 0.0005.",
"solution": "Okay, so I need to figure out how many terms of the Taylor series for ln(1+x) centered at 0 are requi... |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to \\infty} x^2\\left(4^{\\frac{1}{x}} - 4^{\\frac{1}{1+x}}\\right) \\]",
"role": "user"
}
] | {
"ground_truth": "\\ln 4",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "234",
"problem": "Evaluate the limit: \\[ \\lim_{x \\to \\infty} x^2\\left(4^{\\frac{1}{x}} - 4^{\\frac{1}{1+x}}\\right) \\]",
"solution": "Alright, so I need to evaluate the limit as x approaches infinity of x squared times (4^(1/x) - 4^(1/(1+x))). Hmm, okay. Let me try to break this down step by ste... |
[
{
"content": "Determine the positive integer \\( n \\leq 5 \\) such that \\( \\int_{0}^{1} e^x (x-1)^n \\, dx = 16 - 6e \\).",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "235",
"problem": "Determine the positive integer \\( n \\leq 5 \\) such that \\( \\int_{0}^{1} e^x (x-1)^n \\, dx = 16 - 6e \\).",
"solution": "Okay, so I need to find the positive integer n that's less than or equal to 5 such that the integral from 0 to 1 of e^x times (x - 1)^n dx equals 16 - 6e. Hmm... |
[
{
"content": "Calculate the limit: \\[ \\lim_{x\\to\\infty}\\dfrac{\\sqrt{x+1}-2\\sqrt{x+2}+\\sqrt{x}}{\\sqrt{x+2}-2\\sqrt{x}+\\sqrt{x-4}} \\]",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{3}{2}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "236",
"problem": "Calculate the limit: \\[ \\lim_{x\\to\\infty}\\dfrac{\\sqrt{x+1}-2\\sqrt{x+2}+\\sqrt{x}}{\\sqrt{x+2}-2\\sqrt{x}+\\sqrt{x-4}} \\]",
"solution": "Okay, so I have this limit problem to solve: \\[ \\lim_{x\\to\\infty}\\dfrac{\\sqrt{x+1}-2\\sqrt{x+2}+\\sqrt{x}}{\\sqrt{x+2}-2\\sqrt{x}+\\sq... |
[
{
"content": "Can any set that forms a basis for a subspace of a vector space V be extended to form a basis for V? Provide a justification for your answer.",
"role": "user"
}
] | {
"ground_truth": "Yes",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "237",
"problem": "Can any set that forms a basis for a subspace of a vector space V be extended to form a basis for V? Provide a justification for your answer.",
"solution": "Okay, so the question is: Can any set that forms a basis for a subspace of a vector space V be extended to form a basis for V? ... |
[
{
"content": "An insurance policy covers a loss X with a density function given by \\( f(x) = \\frac{3}{8}x^2 \\) for \\( 0 \\le x \\le 2 \\). The time (in hours) to process a claim of size x, where \\( 0 \\le x \\le 2 \\), is uniformly distributed on the interval from x to 2x. Calculate the probability that a ... | {
"ground_truth": "\\dfrac{11}{64}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "238",
"problem": "An insurance policy covers a loss X with a density function given by \\( f(x) = \\frac{3}{8}x^2 \\) for \\( 0 \\le x \\le 2 \\). The time (in hours) to process a claim of size x, where \\( 0 \\le x \\le 2 \\), is uniformly distributed on the interval from x to 2x. Calculate the probabi... |
[
{
"content": "Given the polynomial relationship between two variables, \\(b = p_1a^3 + p_2a^2 + p_3a\\), find the second derivative of \\(b\\) with respect to \\(a\\), \\(\\frac{d^2b}{da^2}\\). Verify that the expression for the second derivative is \\(15p_1a^2 + 6p_2a + p_3\\).",
"role": "user"
}
] | {
"ground_truth": "6p_1a + 2p_2",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "239",
"problem": "Given the polynomial relationship between two variables, \\(b = p_1a^3 + p_2a^2 + p_3a\\), find the second derivative of \\(b\\) with respect to \\(a\\), \\(\\frac{d^2b}{da^2}\\). Verify that the expression for the second derivative is \\(15p_1a^2 + 6p_2a + p_3\\).",
"solution": "Oka... |
[
{
"content": "Given the transition matrix \\( M = \\begin{pmatrix} 0.25 & 0.5 & 0.25 \\\\ 0.5 & 0.25 & 0.25 \\\\ 0.5 & 0.25 & 0.25 \\end{pmatrix} \\) of a Markov chain with states \\( S = \\{0, 1, 2\\} \\), calculate the expected number of steps to reach state 1 starting from state 2.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{20}{7}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "240",
"problem": "Given the transition matrix \\( M = \\begin{pmatrix} 0.25 & 0.5 & 0.25 \\\\ 0.5 & 0.25 & 0.25 \\\\ 0.5 & 0.25 & 0.25 \\end{pmatrix} \\) of a Markov chain with states \\( S = \\{0, 1, 2\\} \\), calculate the expected number of steps to reach state 1 starting from state 2.",
"solution"... |
[
{
"content": "Given the expression \\( y = w^T X + b \\), where \\( w \\) and \\( X \\) are both \\([13, 1]\\) matrices, find the derivative \\( \\frac{dy}{dw} \\).",
"role": "user"
}
] | {
"ground_truth": "X",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "241",
"problem": "Given the expression \\( y = w^T X + b \\), where \\( w \\) and \\( X \\) are both \\([13, 1]\\) matrices, find the derivative \\( \\frac{dy}{dw} \\).",
"solution": "Okay, so I need to find the derivative of y with respect to w, where y is given by the equation y = w^T X + b. Both w ... |
[
{
"content": "For $t > 0$, how many times does the equation $S(t) = C(t)$ hold true, where $S(t) = -1200\\cos\\left(\\frac{\\pi}{6}t\\right) + 1500$ and $C(t) = 1200(1.025)^{t}$? Here, $t$ represents the number of months since January 1, 2004.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "242",
"problem": "For $t > 0$, how many times does the equation $S(t) = C(t)$ hold true, where $S(t) = -1200\\cos\\left(\\frac{\\pi}{6}t\\right) + 1500$ and $C(t) = 1200(1.025)^{t}$? Here, $t$ represents the number of months since January 1, 2004.",
"solution": "Okay, so I need to figure out how many ... |
[
{
"content": "Identify which of the following metrics is not complete on \\(\\mathbb{R}\\):\n\n1. \\(|x-y|\\)\n2. \\(|\\arctan(x) - \\arctan(y)|\\)\n3. \\(|x^3 - y^3|\\)\n4. \\(|x^{1/3} - y^{1/3}|\\)",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "243",
"problem": "Identify which of the following metrics is not complete on \\(\\mathbb{R}\\):\n\n1. \\(|x-y|\\)\n2. \\(|\\arctan(x) - \\arctan(y)|\\)\n3. \\(|x^3 - y^3|\\)\n4. \\(|x^{1/3} - y^{1/3}|\\)",
"solution": "Okay, so I need to figure out which of these four metrics on the real numbers ℝ is ... |
[
{
"content": "Determine the number of linear Boolean functions from \\({0, 1}^n\\) to \\({0, 1}\\). A function \\(F\\) is linear if for all \\(x, y \\in {0, 1}^n\\), \\(F(x \\oplus y) = F(x) \\oplus F(y)\\).",
"role": "user"
}
] | {
"ground_truth": "2^n",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "244",
"problem": "Determine the number of linear Boolean functions from \\({0, 1}^n\\) to \\({0, 1}\\). A function \\(F\\) is linear if for all \\(x, y \\in {0, 1}^n\\), \\(F(x \\oplus y) = F(x) \\oplus F(y)\\).",
"solution": "Okay, so I need to figure out how many linear Boolean functions there are f... |
[
{
"content": "In a dice game, a standard six-sided die is rolled repeatedly until the sum of the numbers on the top face equals or exceeds 100. What is the most likely sum that will be reached or exceeded?",
"role": "user"
}
] | {
"ground_truth": "100",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "245",
"problem": "In a dice game, a standard six-sided die is rolled repeatedly until the sum of the numbers on the top face equals or exceeds 100. What is the most likely sum that will be reached or exceeded?",
"solution": "Alright, so I came across this dice game problem where you roll a standard si... |
[
{
"content": "Calculate $14^{2017^{2017}} \\mod 3$. What is the result?",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "246",
"problem": "Calculate $14^{2017^{2017}} \\mod 3$. What is the result?",
"solution": "Okay, so I need to calculate \\(14^{2017^{2017}} \\mod 3\\). Hmm, modular arithmetic can be tricky, but maybe I can break it down step by step. Let me think... First, I remember that when dealing with exponents ... |
[
{
"content": "Find a number in the ring \\(\\mathbb{Z}[\\sqrt{14}, \\frac{1}{2}]\\) that is not in \\(\\mathbb{Z}[\\sqrt{14}]\\). Provide an example of such a number.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{1 + \\sqrt{14}}{2}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "247",
"problem": "Find a number in the ring \\(\\mathbb{Z}[\\sqrt{14}, \\frac{1}{2}]\\) that is not in \\(\\mathbb{Z}[\\sqrt{14}]\\). Provide an example of such a number.",
"solution": "Okay, so I need to find a number that's in the ring ℤ[√14, 1/2] but not in ℤ[√14]. Hmm, let's start by understanding... |
[
{
"content": "Find the limit: \\[ \\lim_{x\\rightarrow\\infty}\\left(\\frac{x+1}{x-2}\\right)^{2x-1} \\]",
"role": "user"
}
] | {
"ground_truth": "e^6",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "248",
"problem": "Find the limit: \\[ \\lim_{x\\rightarrow\\infty}\\left(\\frac{x+1}{x-2}\\right)^{2x-1} \\]",
"solution": "Okay, so I need to find the limit as x approaches infinity of [(x + 1)/(x - 2)]^(2x - 1). Hmm, this looks like an indeterminate form of type 1^∞, right? Because as x goes to infi... |
[
{
"content": "Given a triangle ABC with sides a, b, and c, and half-perimeter p, such that \\(\\frac{p-a}{11} = \\frac{p-b}{12} = \\frac{p-c}{13}\\), find the value of \\((\\tan\\frac{A}{2})^2\\).\n\n(A) \\(\\frac{143}{432}\\) (B) \\(\\frac{13}{33}\\) (C) \\(\\frac{11}{39}\\) (D) \\(\\frac{12}{37}\\)",
"... | {
"ground_truth": "B",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "249",
"problem": "Given a triangle ABC with sides a, b, and c, and half-perimeter p, such that \\(\\frac{p-a}{11} = \\frac{p-b}{12} = \\frac{p-c}{13}\\), find the value of \\((\\tan\\frac{A}{2})^2\\).\n\n(A) \\(\\frac{143}{432}\\) (B) \\(\\frac{13}{33}\\) (C) \\(\\frac{11}{39}\\) (D) \\(\\frac{12}{37... |
[
{
"content": "Given the cubic equation \\(x^3 - 6x^2 + 5x - 7 = 0\\), find the value of \\(\\frac{1}{r^2} + \\frac{1}{s^2} + \\frac{1}{t^2}\\), where \\(r, s, t\\) are the roots of the equation.",
"role": "user"
}
] | {
"ground_truth": "-\\dfrac{59}{49}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "250",
"problem": "Given the cubic equation \\(x^3 - 6x^2 + 5x - 7 = 0\\), find the value of \\(\\frac{1}{r^2} + \\frac{1}{s^2} + \\frac{1}{t^2}\\), where \\(r, s, t\\) are the roots of the equation.",
"solution": "Okay, so I need to find the value of \\(\\frac{1}{r^2} + \\frac{1}{s^2} + \\frac{1}{t^2}... |
[
{
"content": "For which values of \\( \\lambda \\) does the integral \\( \\int_{0}^{1} t^\\lambda \\, \\mathrm{d}t \\) exist as a Lebesgue integral?",
"role": "user"
}
] | {
"ground_truth": "\\lambda > -1",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "251",
"problem": "For which values of \\( \\lambda \\) does the integral \\( \\int_{0}^{1} t^\\lambda \\, \\mathrm{d}t \\) exist as a Lebesgue integral?",
"solution": "Okay, so I need to figure out for which values of λ the integral from 0 to 1 of t^λ dt exists as a Lebesgue integral. Hmm, let me reca... |
[
{
"content": "Find an upper bound for the recurrence relation \\( T(x) \\leq 4 T(\\left \\lfloor{\\frac{x}{2}} \\right \\rfloor) + x \\) using the master theorem, where \\( x \\in \\mathbb{N} \\).",
"role": "user"
}
] | {
"ground_truth": "O(x^2)",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "252",
"problem": "Find an upper bound for the recurrence relation \\( T(x) \\leq 4 T(\\left \\lfloor{\\frac{x}{2}} \\right \\rfloor) + x \\) using the master theorem, where \\( x \\in \\mathbb{N} \\).",
"solution": "Okay, so I need to find an upper bound for the recurrence relation \\( T(x) \\leq 4 T(... |
[
{
"content": "Count the number of 10-digit numbers using the digits from \\( \\{1, 2, 3, 4\\} \\) such that no two adjacent digits differ by 1.",
"role": "user"
}
] | {
"ground_truth": "21892",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "253",
"problem": "Count the number of 10-digit numbers using the digits from \\( \\{1, 2, 3, 4\\} \\) such that no two adjacent digits differ by 1.",
"solution": "Okay, so I need to figure out how many 10-digit numbers can be made using the digits 1, 2, 3, and 4, with the condition that no two adjacen... |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to \\infty} x \\left( \\sqrt[3]{5+8x^3} - 2x \\right) \\]",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "254",
"problem": "Evaluate the limit: \\[ \\lim_{x \\to \\infty} x \\left( \\sqrt[3]{5+8x^3} - 2x \\right) \\]",
"solution": "Okay, so I need to find the limit as x approaches infinity of x times the quantity cube root of (5 + 8x³) minus 2x. Let me write that down again to make sure I have it right:\n... |
[
{
"content": "Compute the limit: \\( \\lim_{x\\to\\infty}\\ln(x)\\cdot \\ln(1-e^{-x}) \\).",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "255",
"problem": "Compute the limit: \\( \\lim_{x\\to\\infty}\\ln(x)\\cdot \\ln(1-e^{-x}) \\).",
"solution": "Okay, so I need to compute the limit as x approaches infinity of ln(x) multiplied by ln(1 - e^(-x)). Hmm, let's see. First, let me understand the behavior of each part as x gets very large.\n\... |
[
{
"content": "Given the complex number \\( z = \\left( \\frac{\\sqrt{3} - i}{1 + i} \\right)^{12} \\), determine which of the following statements is true:\n\nA. \\( z = 2^6 \\)\nB. \\( \\arg(z) = \\pi \\)\nC. \\( |z| = 2^{12} \\)\nD. \\( z = 64i \\)\nE. \\( \\arg(z) = 2\\pi \\)",
"role": "user"
}
] | {
"ground_truth": "B",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "256",
"problem": "Given the complex number \\( z = \\left( \\frac{\\sqrt{3} - i}{1 + i} \\right)^{12} \\), determine which of the following statements is true:\n\nA. \\( z = 2^6 \\)\nB. \\( \\arg(z) = \\pi \\)\nC. \\( |z| = 2^{12} \\)\nD. \\( z = 64i \\)\nE. \\( \\arg(z) = 2\\pi \\)",
"solution": "Oka... |
[
{
"content": "Evaluate the sum \\(\\sum_{d\\mid2020}{\\sigma(d)}\\), where \\(\\sigma(n)\\) is the sum of all divisors of \\(n\\).",
"role": "user"
}
] | {
"ground_truth": "7931",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "257",
"problem": "Evaluate the sum \\(\\sum_{d\\mid2020}{\\sigma(d)}\\), where \\(\\sigma(n)\\) is the sum of all divisors of \\(n\\).",
"solution": "Alright, so I need to evaluate the sum \\(\\sum_{d\\mid2020}{\\sigma(d)}\\), where \\(\\sigma(n)\\) is the sum of all divisors of \\(n\\). Hmm, okay. Le... |
[
{
"content": "Calculate the surface integral \\( \\iint (2-y) \\, dS \\) over the surface defined by \\( y = 1 \\) and \\( 0 \\leq z \\leq 1 \\).",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "258",
"problem": "Calculate the surface integral \\( \\iint (2-y) \\, dS \\) over the surface defined by \\( y = 1 \\) and \\( 0 \\leq z \\leq 1 \\).",
"solution": "Okay, so I need to calculate this surface integral: \\( \\iint_S (2 - y) \\, dS \\) over the surface defined by \\( y = 1 \\) and \\( 0 \... |
[
{
"content": "In triangle $ABC$, where $AB = 30$, $BC = 51$, and $CA = 63$, points $P$ and $Q$ lie on line segment $BC$, point $R$ lies on line segment $CA$, and point $S$ lies on line segment $AB$. These points form a parallelogram $PQRS$ such that the center of $PQRS$ coincides with the centroid of triangle $... | {
"ground_truth": "336",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "259",
"problem": "In triangle $ABC$, where $AB = 30$, $BC = 51$, and $CA = 63$, points $P$ and $Q$ lie on line segment $BC$, point $R$ lies on line segment $CA$, and point $S$ lies on line segment $AB$. These points form a parallelogram $PQRS$ such that the center of $PQRS$ coincides with the centroid o... |
[
{
"content": "Determine if the following solution is correct:\n\nGiven that $f = 0$ almost everywhere, prove that $\\int_{\\mathbb{R}} f = 0$. The solution provided is:\n\nSuppose $\\phi$ is a simple function such that $0 \\leq \\phi \\leq f$. Since $f = 0$ almost everywhere, $\\phi = 0$ almost everywhere. Let ... | {
"ground_truth": "Yes",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "260",
"problem": "Determine if the following solution is correct:\n\nGiven that $f = 0$ almost everywhere, prove that $\\int_{\\mathbb{R}} f = 0$. The solution provided is:\n\nSuppose $\\phi$ is a simple function such that $0 \\leq \\phi \\leq f$. Since $f = 0$ almost everywhere, $\\phi = 0$ almost ever... |
[
{
"content": "Given that $x_1 < x_2 < x_3$ are real numbers and solutions to the equation $x^3 - 3x^2 + (a+2)x - a = 0$, where $a$ is a real number, find the value of the expression $4x_1 - x_1^2 + x_3^2$. The equation can be factorized as $(x-1)(x(x-2)+a)=0$. Determine the value of $4x_1 - x_1^2 + x_3^2$.",
... | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "261",
"problem": "Given that $x_1 < x_2 < x_3$ are real numbers and solutions to the equation $x^3 - 3x^2 + (a+2)x - a = 0$, where $a$ is a real number, find the value of the expression $4x_1 - x_1^2 + x_3^2$. The equation can be factorized as $(x-1)(x(x-2)+a)=0$. Determine the value of $4x_1 - x_1^2 + ... |
[
{
"content": "Given two reinforced concrete buildings A and B in a seismic region, an impending earthquake might be strong (S), moderate (M), or weak (W) with probabilities P(S) = 0.02, P(M) = 0.2, and P(W) = 0.78. The probabilities of failure for each building if these earthquakes occur are 0.20, 0.05, and 0.0... | {
"ground_truth": "0.890",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "262",
"problem": "Given two reinforced concrete buildings A and B in a seismic region, an impending earthquake might be strong (S), moderate (M), or weak (W) with probabilities P(S) = 0.02, P(M) = 0.2, and P(W) = 0.78. The probabilities of failure for each building if these earthquakes occur are 0.20, 0... |
[
{
"content": "Find the integer value of $b$ such that $44 \\equiv 7^b \\pmod{71}$.",
"role": "user"
}
] | {
"ground_truth": "43",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "263",
"problem": "Find the integer value of $b$ such that $44 \\equiv 7^b \\pmod{71}$.",
"solution": "Okay, so I need to find an integer value of \\( b \\) such that \\( 44 \\equiv 7^b \\pmod{71} \\). Hmm, that sounds like a discrete logarithm problem. From what I remember, the discrete logarithm prob... |
[
{
"content": "Evaluate the limit: \\[ \\lim_{z \\to 0}{(1+z)^{\\frac{1}{z}}} \\]",
"role": "user"
}
] | {
"ground_truth": "e",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "264",
"problem": "Evaluate the limit: \\[ \\lim_{z \\to 0}{(1+z)^{\\frac{1}{z}}} \\]",
"solution": "Okay, so I need to find the limit as z approaches 0 of (1 + z) raised to the power of 1/z. Hmm, let me think. I remember that when dealing with limits involving exponents, sometimes taking the natural l... |
[
{
"content": "Find the number of ordered pairs of integral solutions \\((x, y)\\) to the equation:\n\\[(1 + 3x)(1 - 3y) = -10xy\\]",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "265",
"problem": "Find the number of ordered pairs of integral solutions \\((x, y)\\) to the equation:\n\\[(1 + 3x)(1 - 3y) = -10xy\\]",
"solution": "Okay, so I need to find all integer solutions (x, y) to the equation (1 + 3x)(1 - 3y) = -10xy. Hmm, let's start by expanding the left side to see if I c... |
[
{
"content": "Find the smallest integer \\( n \\geq 2 \\) such that the equation \\( x^2 - x + \\hat{2} = \\hat{0} \\) has a unique solution in \\( \\mathbb{Z}_n \\).",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "266",
"problem": "Find the smallest integer \\( n \\geq 2 \\) such that the equation \\( x^2 - x + \\hat{2} = \\hat{0} \\) has a unique solution in \\( \\mathbb{Z}_n \\).",
"solution": "Okay, so I need to find the smallest integer \\( n \\geq 2 \\) such that the quadratic equation \\( x^2 - x + \\hat{... |
[
{
"content": "In triangle \\(ABC\\), \\(\\measuredangle BAC = 45^\\circ\\) and \\(\\measuredangle ABC = 30^\\circ\\). A point \\(M\\) lies inside the triangle such that \\(\\measuredangle MAB = \\measuredangle MBA = 15^\\circ\\). Determine \\(\\measuredangle BMC\\).",
"role": "user"
}
] | {
"ground_truth": "135^\\circ",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "267",
"problem": "In triangle \\(ABC\\), \\(\\measuredangle BAC = 45^\\circ\\) and \\(\\measuredangle ABC = 30^\\circ\\). A point \\(M\\) lies inside the triangle such that \\(\\measuredangle MAB = \\measuredangle MBA = 15^\\circ\\). Determine \\(\\measuredangle BMC\\).",
"solution": "Alright, let me ... |
[
{
"content": "A plane has 144 seats, and it is known that 15% of ticket holders do not show up for the flight, with each person's decision being independent. How many tickets can you sell to be 99% sure that not more than 144 people will show up for the flight?",
"role": "user"
}
] | {
"ground_truth": "157",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "268",
"problem": "A plane has 144 seats, and it is known that 15% of ticket holders do not show up for the flight, with each person's decision being independent. How many tickets can you sell to be 99% sure that not more than 144 people will show up for the flight?",
"solution": "Alright, so the probl... |
[
{
"content": "Calculate the value of the expression \\(\\arccos(1/\\sqrt{2}) + \\arcsin(1/3) + 2 \\arctan(1/\\sqrt{2})\\). Express your answer as a multiple of \\(\\pi\\).",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{3\\pi}{4}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "269",
"problem": "Calculate the value of the expression \\(\\arccos(1/\\sqrt{2}) + \\arcsin(1/3) + 2 \\arctan(1/\\sqrt{2})\\). Express your answer as a multiple of \\(\\pi\\).",
"solution": "Okay, so I need to calculate the value of this expression: \\(\\arccos(1/\\sqrt{2}) + \\arcsin(1/3) + 2 \\arcta... |
[
{
"content": "Is there a field with a characteristic that is a composite number, such as 4?",
"role": "user"
}
] | {
"ground_truth": "No",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "270",
"problem": "Is there a field with a characteristic that is a composite number, such as 4?",
"solution": "Okay, so I need to figure out if there's a field with characteristic 4. Hmm, I remember that the characteristic of a field is the smallest positive integer n such that n times the multiplicat... |
[
{
"content": "Find \\( \\lim_{x \\to 0^-} \\frac{\\sqrt{x+1}}{x} \\).",
"role": "user"
}
] | {
"ground_truth": "-\\infty",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "271",
"problem": "Find \\( \\lim_{x \\to 0^-} \\frac{\\sqrt{x+1}}{x} \\).",
"solution": "Okay, so I need to find the limit as x approaches 0 from the negative side of the function sqrt(x + 1)/x. Hmm, let me think about how to approach this. First, let me recall some basic limit principles. When dealin... |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\left| \\cos \\left( \\frac{\\pi}{4(n-1)} \\right) \\right|^{2n-1} \\]",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "272",
"problem": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\left| \\cos \\left( \\frac{\\pi}{4(n-1)} \\right) \\right|^{2n-1} \\]",
"solution": "Okay, so I need to evaluate the limit as n approaches infinity of the absolute value of cosine of (π/(4(n-1))) raised to the power of 2n-1. Let me wri... |
[
{
"content": "Mary is twice as old as Ann was when Mary was half as old as Ann will be when Ann is three times as old as Mary was when Mary was three times as old as Ann was. The combined ages of Mary and Ann is 44 years. How old is Mary?",
"role": "user"
}
] | {
"ground_truth": "27.5",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "273",
"problem": "Mary is twice as old as Ann was when Mary was half as old as Ann will be when Ann is three times as old as Mary was when Mary was three times as old as Ann was. The combined ages of Mary and Ann is 44 years. How old is Mary?",
"solution": "Okay, let's try to tackle this age problem. ... |
[
{
"content": "Consider a set of balls numbered from 1 to \\( m \\). Balls are drawn with replacement, and each draw is equally likely to be any of the \\( m \\) balls. What is the expected number of draws required to first draw the ball numbered 1 twice in a row?",
"role": "user"
}
] | {
"ground_truth": "m(m + 1)",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "274",
"problem": "Consider a set of balls numbered from 1 to \\( m \\). Balls are drawn with replacement, and each draw is equally likely to be any of the \\( m \\) balls. What is the expected number of draws required to first draw the ball numbered 1 twice in a row?",
"solution": "Okay, so I need to ... |
[
{
"content": "A casino patron will continue to make $5 bets on red in roulette until she has won 4 of these bets. Calculate her expected winnings when she stops.",
"role": "user"
}
] | {
"ground_truth": "-\\dfrac{20}{9}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "275",
"problem": "A casino patron will continue to make $5 bets on red in roulette until she has won 4 of these bets. Calculate her expected winnings when she stops.",
"solution": "Okay, so I need to calculate the expected winnings for this casino patron who keeps making $5 bets on red in roulette unt... |
[
{
"content": "Find the maximum value of \\( P(X = Y) \\) for random variables \\( X \\) and \\( Y \\) with Bernoulli distributions having parameters \\( p \\) and \\( q \\), respectively. The random variables \\( X \\) and \\( Y \\) are not necessarily independent.",
"role": "user"
}
] | {
"ground_truth": "1 - |p - q|",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "276",
"problem": "Find the maximum value of \\( P(X = Y) \\) for random variables \\( X \\) and \\( Y \\) with Bernoulli distributions having parameters \\( p \\) and \\( q \\), respectively. The random variables \\( X \\) and \\( Y \\) are not necessarily independent.",
"solution": "Okay, so I need t... |
[
{
"content": "Let \\( f: \\mathbb{R} \\to \\mathbb{R} \\) be a continuous function such that \\( f(i) = 0 \\) for all \\( i \\in \\mathbb{Z} \\). Which of the following statements is always true?\n\nA. The image of \\( f \\) is closed in \\( \\mathbb{R} \\).\nB. The image of \\( f \\) is open in \\( \\mathbb{R}... | {
"ground_truth": "D",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "277",
"problem": "Let \\( f: \\mathbb{R} \\to \\mathbb{R} \\) be a continuous function such that \\( f(i) = 0 \\) for all \\( i \\in \\mathbb{Z} \\). Which of the following statements is always true?\n\nA. The image of \\( f \\) is closed in \\( \\mathbb{R} \\).\nB. The image of \\( f \\) is open in \\(... |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\rightarrow\\pi/4}(2\\sin(x)-\\sqrt{2})\\tan(x-3\\pi/4) \\]",
"role": "user"
}
] | {
"ground_truth": "-\\sqrt{2}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "278",
"problem": "Evaluate the limit: \\[ \\lim_{x\\rightarrow\\pi/4}(2\\sin(x)-\\sqrt{2})\\tan(x-3\\pi/4) \\]",
"solution": "Okay, so I need to evaluate the limit: \\[\\lim_{x\\rightarrow\\pi/4}(2\\sin(x)-\\sqrt{2})\\tan(x-3\\pi/4)\\]. Hmm, let's see. First, I should probably check what happens when ... |
[
{
"content": "A miner is trapped in a mine with three identical doors. The first door leads to safety in 2 hours. The second door leads back to the starting point in 3 hours. The third door leads to a maze, returning to the starting point in 5 hours on the first entry, and 1 hour on subsequent entries. The mine... | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "279",
"problem": "A miner is trapped in a mine with three identical doors. The first door leads to safety in 2 hours. The second door leads back to the starting point in 3 hours. The third door leads to a maze, returning to the starting point in 5 hours on the first entry, and 1 hour on subsequent entri... |
[
{
"content": "Consider a non-zero vector \\( a \\) such that \\( a \\in \\{0,1\\}^n \\), and a vector \\( b \\) chosen uniformly and randomly from \\( \\{0,1\\}^n \\). What is the probability that the dot product \\( \\sum_{i=1}^{n} a_i b_i \\) is odd?",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{1}{2}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "280",
"problem": "Consider a non-zero vector \\( a \\) such that \\( a \\in \\{0,1\\}^n \\), and a vector \\( b \\) chosen uniformly and randomly from \\( \\{0,1\\}^n \\). What is the probability that the dot product \\( \\sum_{i=1}^{n} a_i b_i \\) is odd?",
"solution": "Okay, let's see. I need to fin... |
[
{
"content": "What is the minimum number of times you need to roll a six-sided die so that the probability of rolling each number at least once is greater than 0.5?",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "281",
"problem": "What is the minimum number of times you need to roll a six-sided die so that the probability of rolling each number at least once is greater than 0.5?",
"solution": "Okay, so I need to figure out the minimum number of times I should roll a six-sided die to have a probability greater ... |
[
{
"content": "Calculate the value of \\( z = (1+i)^8(1-i\\sqrt{3})^{-6} \\) using the trigonometric form of complex numbers.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{1}{4}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "282",
"problem": "Calculate the value of \\( z = (1+i)^8(1-i\\sqrt{3})^{-6} \\) using the trigonometric form of complex numbers.",
"solution": "Alright, so I need to calculate the value of \\( z = (1+i)^8(1-i\\sqrt{3})^{-6} \\) using the trigonometric form of complex numbers. Hmm, okay, let's start by... |
[
{
"content": "Determine whether the Laurent series of \\( \\frac{1}{(1+x)^3} \\) is the same as its Maclaurin series.",
"role": "user"
}
] | {
"ground_truth": "Yes",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "283",
"problem": "Determine whether the Laurent series of \\( \\frac{1}{(1+x)^3} \\) is the same as its Maclaurin series.",
"solution": "Okay, so I need to figure out if the Laurent series of \\( \\frac{1}{(1+x)^3} \\) is the same as its Maclaurin series. Hmm, let me recall what these terms mean. A Ma... |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to +\\infty}\\left[x - x^2 \\cdot \\ln\\left(1+ \\frac{1}{x}\\right)\\right] \\]",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{1}{2}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "284",
"problem": "Evaluate the limit: \\[ \\lim_{x \\to +\\infty}\\left[x - x^2 \\cdot \\ln\\left(1+ \\frac{1}{x}\\right)\\right] \\]",
"solution": "Alright, so I need to find the limit as x approaches positive infinity of this expression: x minus x squared times the natural logarithm of (1 plus 1 ove... |
[
{
"content": "Calculate the expected number of rolls needed to obtain two consecutive sixes when rolling a die, given that all rolls must be even numbers.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "285",
"problem": "Calculate the expected number of rolls needed to obtain two consecutive sixes when rolling a die, given that all rolls must be even numbers.",
"solution": "Alright, so I need to figure out the expected number of rolls required to get two consecutive sixes, but with the catch that all... |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0} \\left( \\frac{\\cot(x)(1-\\cos^2(3x))}{x^2 + 5x} \\right) \\]",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{9}{5}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "286",
"problem": "Evaluate the limit: \\[ \\lim_{x \\to 0} \\left( \\frac{\\cot(x)(1-\\cos^2(3x))}{x^2 + 5x} \\right) \\]",
"solution": "Okay, so I need to evaluate the limit as x approaches 0 of [cot(x)(1 - cos²(3x))] divided by (x² + 5x). Let me start by breaking this down step by step. First, maybe... |
[
{
"content": "Given that a vowel (v) is followed by a consonant (c) 75% of the time and a consonant is followed by a vowel 65% of the time, determine the most likely sequence of vowels and consonants when three successive letters are chosen.",
"role": "user"
}
] | {
"ground_truth": "cvc",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "287",
"problem": "Given that a vowel (v) is followed by a consonant (c) 75% of the time and a consonant is followed by a vowel 65% of the time, determine the most likely sequence of vowels and consonants when three successive letters are chosen.",
"solution": "Okay, so I need to figure out the most li... |
[
{
"content": "What is the smallest value of \\( n \\) such that the final digit of \\( 13^n \\) is one more than the digit adjacent to it?",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "288",
"problem": "What is the smallest value of \\( n \\) such that the final digit of \\( 13^n \\) is one more than the digit adjacent to it?",
"solution": "Okay, so I need to find the smallest value of \\( n \\) such that when you look at the number \\( 13^n \\), the final digit is one more than the... |
[
{
"content": "Suppose the quadratic polynomial \\(p(x) = ax^2 + bx + c\\) has positive coefficients \\(a, b, c\\) in arithmetic progression in that order. If \\(p(x) = 0\\) has integer roots \\(\\alpha\\) and \\(\\beta\\), find the value of \\(\\alpha + \\beta + \\alpha \\beta\\).",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "289",
"problem": "Suppose the quadratic polynomial \\(p(x) = ax^2 + bx + c\\) has positive coefficients \\(a, b, c\\) in arithmetic progression in that order. If \\(p(x) = 0\\) has integer roots \\(\\alpha\\) and \\(\\beta\\), find the value of \\(\\alpha + \\beta + \\alpha \\beta\\).",
"solution": "O... |
[
{
"content": "Find the number of integer solutions to the equation \\(a+b+c+d+e+f=20\\) subject to the constraints \\(2 \\leq a,b,c,d,e,f \\leq 6\\).",
"role": "user"
}
] | {
"ground_truth": "951",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "290",
"problem": "Find the number of integer solutions to the equation \\(a+b+c+d+e+f=20\\) subject to the constraints \\(2 \\leq a,b,c,d,e,f \\leq 6\\).",
"solution": "Okay, so I need to find the number of integer solutions to the equation \\(a + b + c + d + e + f = 20\\) where each variable is betwe... |
[
{
"content": "Calculate the surface area of the surface \\( M = \\{(x,y,z) \\in \\mathbb{R}^3 : x^2+y^2<1 \\text{ and } z=3-2x-2y\\} \\).",
"role": "user"
}
] | {
"ground_truth": "3\\pi",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "291",
"problem": "Calculate the surface area of the surface \\( M = \\{(x,y,z) \\in \\mathbb{R}^3 : x^2+y^2<1 \\text{ and } z=3-2x-2y\\} \\).",
"solution": "Okay, so I need to find the surface area of this surface M. Let me first understand what the surface is. The problem says it's the set of all poi... |
[
{
"content": "Find the largest eigenvalue of the matrix \\(\\begin{bmatrix} 1 & 4 & 16 \\\\ 4 & 16 & 1 \\\\ 16 & 1 & 4 \\end{bmatrix}\\).",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "292",
"problem": "Find the largest eigenvalue of the matrix \\(\\begin{bmatrix} 1 & 4 & 16 \\\\ 4 & 16 & 1 \\\\ 16 & 1 & 4 \\end{bmatrix}\\).",
"solution": "Okay, so I need to find the largest eigenvalue of this matrix:\n\n\\[\n\\begin{bmatrix}\n1 & 4 & 16 \\\\\n4 & 16 & 1 \\\\\n16 & 1 & 4\n\\end{bmat... |
[
{
"content": "Calculate the number of elements of order 24 in the group \\( C_{24} \\times C_{40} \\).",
"role": "user"
}
] | {
"ground_truth": "96",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "293",
"problem": "Calculate the number of elements of order 24 in the group \\( C_{24} \\times C_{40} \\).",
"solution": "Okay, so I need to find the number of elements of order 24 in the group \\( C_{24} \\times C_{40} \\). Hmm, let me start by recalling some group theory concepts. \n\nFirst, \\( C_n... |
[
{
"content": "Find the minimum positive integer value of \\( q \\) such that there exists a fraction \\( \\frac{p}{q} \\) with \\( p \\) and \\( q \\) as positive integers, satisfying the inequality \\( \\frac{95}{36} > \\frac{p}{q} > \\frac{96}{37} \\).",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "294",
"problem": "Find the minimum positive integer value of \\( q \\) such that there exists a fraction \\( \\frac{p}{q} \\) with \\( p \\) and \\( q \\) as positive integers, satisfying the inequality \\( \\frac{95}{36} > \\frac{p}{q} > \\frac{96}{37} \\).",
"solution": "Okay, let's see. I need to f... |
[
{
"content": "Find the angle \\( \\theta \\) in the range \\( 0 \\leq \\theta < 360 \\) degrees that minimizes the x-coordinate of the Archimedean spiral given by \\( r = 1 - \\frac{0.5}{2\\pi}\\theta \\).",
"role": "user"
}
] | {
"ground_truth": "174",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "295",
"problem": "Find the angle \\( \\theta \\) in the range \\( 0 \\leq \\theta < 360 \\) degrees that minimizes the x-coordinate of the Archimedean spiral given by \\( r = 1 - \\frac{0.5}{2\\pi}\\theta \\).",
"solution": "Alright, so I need to find the angle θ between 0 and 360 degrees that minimiz... |
[
{
"content": "Find the set of values of \\( \\lambda \\) for which the perpendicular distance from the point \\( P \\) on the line \\( l \\), given by \\( r = 3i + 8j + 2k + \\lambda(4i + 6j + 5k) \\), to the plane \\( \\Pi_1 \\) with equation \\( 2x + 3y + 6z = 14 \\) is not greater than 4.",
"role": "user... | {
"ground_truth": "[-1, 0]",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "296",
"problem": "Find the set of values of \\( \\lambda \\) for which the perpendicular distance from the point \\( P \\) on the line \\( l \\), given by \\( r = 3i + 8j + 2k + \\lambda(4i + 6j + 5k) \\), to the plane \\( \\Pi_1 \\) with equation \\( 2x + 3y + 6z = 14 \\) is not greater than 4.",
"so... |
[
{
"content": "Given that $f'(x) = \\sqrt{1 + x^3}$ and $f(1) = 0.5$, find the value of $f(4)$. Use the Fundamental Theorem of Calculus to solve the problem.",
"role": "user"
}
] | {
"ground_truth": "13.372",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "297",
"problem": "Given that $f'(x) = \\sqrt{1 + x^3}$ and $f(1) = 0.5$, find the value of $f(4)$. Use the Fundamental Theorem of Calculus to solve the problem.",
"solution": "Okay, so I need to find the value of f(4) given that the derivative f'(x) is sqrt(1 + x³) and that f(1) = 0.5. The problem say... |
[
{
"content": "Find the length of the curve \\( y = \\frac{3}{4}x^\\frac{4}{3} - \\frac{3}{8}x^\\frac{2}{3} + 7 \\) from \\( x = 1 \\) to \\( x = 8 \\).",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{99}{8}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "298",
"problem": "Find the length of the curve \\( y = \\frac{3}{4}x^\\frac{4}{3} - \\frac{3}{8}x^\\frac{2}{3} + 7 \\) from \\( x = 1 \\) to \\( x = 8 \\).",
"solution": "Okay, so I need to find the length of the curve given by \\( y = \\frac{3}{4}x^{\\frac{4}{3}} - \\frac{3}{8}x^{\\frac{2}{3}} + 7 \\... |
[
{
"content": "A teaching assistant (TA) needs to keep his computer running continuously for the next 7 days (168 hours) using batteries, as his power cable is missing. He has 100 batteries available, each with an average lifetime of 2 hours and a standard deviation of 30 minutes. What is the minimum number of b... | {
"ground_truth": "88",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "299",
"problem": "A teaching assistant (TA) needs to keep his computer running continuously for the next 7 days (168 hours) using batteries, as his power cable is missing. He has 100 batteries available, each with an average lifetime of 2 hours and a standard deviation of 30 minutes. What is the minimum... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.