prompt listlengths 1 1 | reward_model dict | data_source stringclasses 1
value | ability stringclasses 1
value | extra_info dict |
|---|---|---|---|---|
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0} \\left(\\dfrac{1}{\\tan^2 x}-\\dfrac{1}{x^2} \\right) \\]",
"role": "user"
}
] | {
"ground_truth": "-\\dfrac{2}{3}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "0",
"problem": "Evaluate the limit: \\[ \\lim_{x \\to 0} \\left(\\dfrac{1}{\\tan^2 x}-\\dfrac{1}{x^2} \\right) \\]",
"solution": "Okay, so I need to evaluate the limit as x approaches 0 of (1/tan²x - 1/x²). Hmm, let me think. First, I remember that tanx is sinx/cosx, so tan²x would be sin²x/cos²x. The... |
[
{
"content": "Determine the minimum sample size required such that the probability of at least two people being a match at all six genetic markers exceeds 0.001, given that the probability of a match at each individual marker is 1/9.",
"role": "user"
}
] | {
"ground_truth": "34",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "1",
"problem": "Determine the minimum sample size required such that the probability of at least two people being a match at all six genetic markers exceeds 0.001, given that the probability of a match at each individual marker is 1/9.",
"solution": "Okay, so I need to figure out the minimum sample si... |
[
{
"content": "Let $A$ be a proper infinite subset of a set $X$. If $x$ and $y$ are two distinct elements of $X$ that are not in $A$, and we define $B = \\{x, y\\} \\cup A$, what is the cardinality of $B$ in terms of the cardinality of $A$?",
"role": "user"
}
] | {
"ground_truth": "|A|",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "2",
"problem": "Let $A$ be a proper infinite subset of a set $X$. If $x$ and $y$ are two distinct elements of $X$ that are not in $A$, and we define $B = \\{x, y\\} \\cup A$, what is the cardinality of $B$ in terms of the cardinality of $A$?",
"solution": "Okay, let's try to figure out the cardinality... |
[
{
"content": "Find the longest geometric progression with a common ratio greater than 1 that can be formed from the set \\( \\{100, 101, 102, \\ldots, 1000\\} \\). What is the length of this progression?",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "3",
"problem": "Find the longest geometric progression with a common ratio greater than 1 that can be formed from the set \\( \\{100, 101, 102, \\ldots, 1000\\} \\). What is the length of this progression?",
"solution": "Okay, so I need to find the longest geometric progression (GP) with a common rati... |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\frac{\\sum_{k=1}^n k^p}{n^{p+1}} \\]",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{1}{p+1}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "4",
"problem": "Evaluate the limit: \\[ \\lim_{n \\to \\infty} \\frac{\\sum_{k=1}^n k^p}{n^{p+1}} \\]",
"solution": "Okay, so I need to find the limit as n approaches infinity of the sum from k=1 to n of k^p divided by n^(p+1). Hmm. Let me think. I remember that when dealing with sums involving powers... |
[
{
"content": "Evaluate the limit: \\[ \\lim_{t\\to 0}\\left(\\frac{1}{\\ln(1 + t)}+\\frac{1}{\\ln(1-t)}\\right). \\]",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "5",
"problem": "Evaluate the limit: \\[ \\lim_{t\\to 0}\\left(\\frac{1}{\\ln(1 + t)}+\\frac{1}{\\ln(1-t)}\\right). \\]",
"solution": "Okay, so I need to find the limit as t approaches 0 of the expression (1 / ln(1 + t)) + (1 / ln(1 - t)). Hmm, let me think about how to approach this. First, I remember... |
[
{
"content": "Determine the radius of convergence for the function \\( \\frac{1}{z^2 + 2z + 2} \\).",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{2}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "6",
"problem": "Determine the radius of convergence for the function \\( \\frac{1}{z^2 + 2z + 2} \\).",
"solution": "Okay, so I need to find the radius of convergence for the function \\( \\frac{1}{z^2 + 2z + 2} \\). Hmm, let me think. I remember that the radius of convergence is related to power seri... |
[
{
"content": "A cube is to be colored using 6 distinct colors such that no two adjacent faces share the same color. How many distinct ways can this be done?",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "7",
"problem": "A cube is to be colored using 6 distinct colors such that no two adjacent faces share the same color. How many distinct ways can this be done?",
"solution": "Okay, so I need to figure out how many distinct ways there are to color a cube using 6 distinct colors, with the condition that ... |
[
{
"content": "In a group of 10 chickens, 9 are real and each weighs 2kg, while 1 is a false chicken that weighs either 1kg or 3kg, each with a probability of \\(\\frac{1}{2}\\). A machine randomly selects each chicken with a probability of \\(\\frac{1}{2}\\) for weighing, and after weighing, all chickens are re... | {
"ground_truth": "\\dfrac{3}{5}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "8",
"problem": "In a group of 10 chickens, 9 are real and each weighs 2kg, while 1 is a false chicken that weighs either 1kg or 3kg, each with a probability of \\(\\frac{1}{2}\\). A machine randomly selects each chicken with a probability of \\(\\frac{1}{2}\\) for weighing, and after weighing, all chick... |
[
{
"content": "Evaluate the integral \\( \\int_{-\\infty}^0 xe^{-4x} \\, dx \\).",
"role": "user"
}
] | {
"ground_truth": "-\\infty",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "9",
"problem": "Evaluate the integral \\( \\int_{-\\infty}^0 xe^{-4x} \\, dx \\).",
"solution": "Okay, so I need to evaluate the integral from negative infinity to 0 of x times e to the power of -4x dx. Hmm, let's see. I remember that integrals involving infinity are improper integrals, so I probably ... |
[
{
"content": "If a function $f(x)$ is differentiable at $x = a$, is it necessarily differentiable in an open interval including $x = a$?",
"role": "user"
}
] | {
"ground_truth": "No",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "10",
"problem": "If a function $f(x)$ is differentiable at $x = a$, is it necessarily differentiable in an open interval including $x = a$?",
"solution": "Okay, so I need to figure out if a function f(x) being differentiable at a single point x = a necessarily means it's differentiable in some open in... |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to \\infty} x \\left( \\left(1 + \\frac{1}{x}\\right)^{1 + \\frac{1}{x}} - 1 \\right) \\]",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "11",
"problem": "Evaluate the limit: \\[ \\lim_{x \\to \\infty} x \\left( \\left(1 + \\frac{1}{x}\\right)^{1 + \\frac{1}{x}} - 1 \\right) \\]",
"solution": "Okay, so I need to evaluate the limit as x approaches infinity of x times the quantity ((1 + 1/x)^(1 + 1/x) minus 1). Hmm, that looks a bit intim... |
[
{
"content": "Determine the number of distinct terms in the expansion of \\((x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8)^4\\).",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "12",
"problem": "Determine the number of distinct terms in the expansion of \\((x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8)^4\\).",
"solution": "Okay, so I need to find the number of distinct terms in the expansion of \\((x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8)^4\\). Hmm, let's see. When you expand a po... |
[
{
"content": "Let $A$ be a $3 \\times 3$ matrix and $\\vec{x}, \\vec{y}, \\vec{z}$ be vectors in $\\mathbb{R}^3$. Given the equations:\n\\[ A\\vec{x} = \\begin{pmatrix} 1 \\\\ 0 \\\\ 1 \\end{pmatrix}, \\quad A\\vec{y} = \\begin{pmatrix} 0 \\\\ 1 \\\\ 0 \\end{pmatrix}, \\quad A\\vec{z} = \\begin{pmatrix} 1 \\\\ ... | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "13",
"problem": "Let $A$ be a $3 \\times 3$ matrix and $\\vec{x}, \\vec{y}, \\vec{z}$ be vectors in $\\mathbb{R}^3$. Given the equations:\n\\[ A\\vec{x} = \\begin{pmatrix} 1 \\\\ 0 \\\\ 1 \\end{pmatrix}, \\quad A\\vec{y} = \\begin{pmatrix} 0 \\\\ 1 \\\\ 0 \\end{pmatrix}, \\quad A\\vec{z} = \\begin{pmatr... |
[
{
"content": "If a matrix $A \\in \\mathbb{R}^{m \\times n}$ is real, does there exist a singular value decomposition (SVD) $A = U\\Sigma V^T$ where both $U$ and $V$ are real matrices?",
"role": "user"
}
] | {
"ground_truth": "Yes",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "14",
"problem": "If a matrix $A \\in \\mathbb{R}^{m \\times n}$ is real, does there exist a singular value decomposition (SVD) $A = U\\Sigma V^T$ where both $U$ and $V$ are real matrices?",
"solution": "Okay, so I need to figure out if a real matrix A has an SVD where both U and V are real matrices. H... |
[
{
"content": "Calculate the value of \\(13^{498} \\mod 997\\).",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "15",
"problem": "Calculate the value of \\(13^{498} \\mod 997\\).",
"solution": "Okay, so I need to calculate \\(13^{498} \\mod 997\\). Hmm, 997 is a prime number, right? Let me check. Yeah, 997 is indeed a prime. That might be useful because there are some theorems related to primes in modular arithm... |
[
{
"content": "Evaluate the ordinal expression $1^\\omega$ using the definition of ordinal exponentiation for limit ordinals.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "16",
"problem": "Evaluate the ordinal expression $1^\\omega$ using the definition of ordinal exponentiation for limit ordinals.",
"solution": "Okay, so I need to evaluate the ordinal expression \\(1^{\\omega}\\) using the definition of ordinal exponentiation for limit ordinals. Hmm, let me start by re... |
[
{
"content": "Find the value of \\(i^i\\), where \\(i\\) is the imaginary unit.",
"role": "user"
}
] | {
"ground_truth": "e^{-\\pi/2}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "17",
"problem": "Find the value of \\(i^i\\), where \\(i\\) is the imaginary unit.",
"solution": "Okay, so I need to find the value of \\(i^i\\). Hmm, I remember that \\(i\\) is the imaginary unit, defined by \\(i^2 = -1\\). But raising \\(i\\) to the power of \\(i\\) seems a bit tricky. Let me think ... |
[
{
"content": "Express the conjugate \\( \\bar{z} \\) in terms of \\( z \\) for the equation \\( z^2 = \\bar{z} \\).",
"role": "user"
}
] | {
"ground_truth": "\\bar{z} = z^2",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "18",
"problem": "Express the conjugate \\( \\bar{z} \\) in terms of \\( z \\) for the equation \\( z^2 = \\bar{z} \\).",
"solution": "Okay, so I need to find the conjugate of z, which is \\(\\bar{z}\\), in terms of z for the equation \\(z^2 = \\bar{z}\\). Hmm, let me think. \n\nFirst, I remember that ... |
[
{
"content": "Company A runs buses between New York and Newark, with a bus leaving New York every half hour starting from 0:00, 24 hours a day. Company B also runs buses on the same route, with their buses leaving New York on average twice per hour, but not necessarily on a fixed schedule. Assuming passengers a... | {
"ground_truth": "Yes",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "19",
"problem": "Company A runs buses between New York and Newark, with a bus leaving New York every half hour starting from 0:00, 24 hours a day. Company B also runs buses on the same route, with their buses leaving New York on average twice per hour, but not necessarily on a fixed schedule. Assuming p... |
[
{
"content": "Find the number of non-negative integral solutions for the equation $x + 2y + 3z = 33$.",
"role": "user"
}
] | {
"ground_truth": "108",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "20",
"problem": "Find the number of non-negative integral solutions for the equation $x + 2y + 3z = 33$.",
"solution": "Alright, let's see. I need to find the number of non-negative integral solutions to the equation x + 2y + 3z = 33. Hmm, okay. So, x, y, z are all non-negative integers. That means th... |
[
{
"content": "Find the minimum value of the function \\( f = \\sin\\theta_1 + \\sin\\theta_2 \\), given that \\( \\theta_1 + \\theta_2 + \\phi = \\pi \\) and \\( 0 < \\phi < \\pi \\).",
"role": "user"
}
] | {
"ground_truth": "\\sin \\phi",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "21",
"problem": "Find the minimum value of the function \\( f = \\sin\\theta_1 + \\sin\\theta_2 \\), given that \\( \\theta_1 + \\theta_2 + \\phi = \\pi \\) and \\( 0 < \\phi < \\pi \\).",
"solution": "Okay, so I need to find the minimum value of the function \\( f = \\sin\\theta_1 + \\sin\\theta_2 \\... |
[
{
"content": "Given a triangle ABC with points D, E, and F on sides BC, AC, and AB respectively, such that BD : DC = 1 : 1, CE : EA = 1 : 3, and AF : FB = 1 : 4. A line parallel to AB is drawn from D to G on side AC. Lines DG and EF intersect at X. If the area of triangle ABC is 120, find the area of triangle D... | {
"ground_truth": "13",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "22",
"problem": "Given a triangle ABC with points D, E, and F on sides BC, AC, and AB respectively, such that BD : DC = 1 : 1, CE : EA = 1 : 3, and AF : FB = 1 : 4. A line parallel to AB is drawn from D to G on side AC. Lines DG and EF intersect at X. If the area of triangle ABC is 120, find the area of... |
[
{
"content": "If matrix $A$ is $5 \\times 4$ and matrix $B$ is $4 \\times 5$, and matrix $B$ has at least one linearly dependent row, does the product matrix $AB$ have at least one linearly dependent row?",
"role": "user"
}
] | {
"ground_truth": "Yes",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "23",
"problem": "If matrix $A$ is $5 \\times 4$ and matrix $B$ is $4 \\times 5$, and matrix $B$ has at least one linearly dependent row, does the product matrix $AB$ have at least one linearly dependent row?",
"solution": "Okay, let me try to figure this out. So, we have matrix A which is 5x4 and matr... |
[
{
"content": "Find the smallest positive integer \\( n \\) such that \\( \\left(\\frac{1-i}{\\sqrt{2}}\\right)^n = 1 \\).",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "24",
"problem": "Find the smallest positive integer \\( n \\) such that \\( \\left(\\frac{1-i}{\\sqrt{2}}\\right)^n = 1 \\).",
"solution": "Alright, so I need to find the smallest positive integer \\( n \\) such that \\( \\left(\\frac{1-i}{\\sqrt{2}}\\right)^n = 1 \\). Hmm, okay. Let me start by recal... |
[
{
"content": "The duration $Y$ of long-distance telephone calls (in minutes) is a random variable with the following properties: $P(Y=3)=0.2$ and $P(Y=6)=0.1$. Otherwise, $Y$ has a continuous density function given by \\( f(y)= \\begin{cases} (1/4)ye^{-y/2}, & y>0 \\\\ 0, & \\text{elsewhere.} \\end{cases} \\) F... | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "25",
"problem": "The duration $Y$ of long-distance telephone calls (in minutes) is a random variable with the following properties: $P(Y=3)=0.2$ and $P(Y=6)=0.1$. Otherwise, $Y$ has a continuous density function given by \\( f(y)= \\begin{cases} (1/4)ye^{-y/2}, & y>0 \\\\ 0, & \\text{elsewhere.} \\end{c... |
[
{
"content": "True or false: If $f: \\mathbb{R}^n \\to \\mathbb{R}^m$ is a continuous function and $A$ is a convex set in $\\mathbb{R}^n$, then $f(A)$ is convex in $\\mathbb{R}^m$.",
"role": "user"
}
] | {
"ground_truth": "B",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "26",
"problem": "True or false: If $f: \\mathbb{R}^n \\to \\mathbb{R}^m$ is a continuous function and $A$ is a convex set in $\\mathbb{R}^n$, then $f(A)$ is convex in $\\mathbb{R}^m$.",
"solution": "Okay, so I need to figure out if the following statement is true or false: If f is a continuous functio... |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to \\infty }\\frac{(n!)^{1/n}}{n}. \\]",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{1}{e}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "27",
"problem": "Evaluate the limit: \\[ \\lim_{n \\to \\infty }\\frac{(n!)^{1/n}}{n}. \\]",
"solution": "Okay, so I need to find the limit as n approaches infinity of (n!)^(1/n) divided by n. Hmm, that looks a bit intimidating at first glance, but maybe I can break it down. Let me write it out again ... |
[
{
"content": "A function \\( f \\) is even. It is known that there are four values \\( x_1, x_2, x_3, x_4 \\) satisfying \\( f\\left(\\frac{x+1}{3-x}\\right) = f(x) \\). What is the value of \\( x_1 + x_2 + x_3 + x_4 \\)?\n\nA) 0\nB) 2\nC) 4\nD) 6",
"role": "user"
}
] | {
"ground_truth": "D",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "28",
"problem": "A function \\( f \\) is even. It is known that there are four values \\( x_1, x_2, x_3, x_4 \\) satisfying \\( f\\left(\\frac{x+1}{3-x}\\right) = f(x) \\). What is the value of \\( x_1 + x_2 + x_3 + x_4 \\)?\n\nA) 0\nB) 2\nC) 4\nD) 6",
"solution": "Alright, let's see. The problem says... |
[
{
"content": "Find the range of the function \\( f(x) = x\\sqrt{x} + \\frac{1}{x\\sqrt{x}} - 4\\left(x + \\frac{1}{x}\\right) \\), where \\( x > 0 \\).",
"role": "user"
}
] | {
"ground_truth": "[-10, \\infty)",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "29",
"problem": "Find the range of the function \\( f(x) = x\\sqrt{x} + \\frac{1}{x\\sqrt{x}} - 4\\left(x + \\frac{1}{x}\\right) \\), where \\( x > 0 \\).",
"solution": "Okay, so I need to find the range of the function \\( f(x) = x\\sqrt{x} + \\frac{1}{x\\sqrt{x}} - 4\\left(x + \\frac{1}{x}\\right) \... |
[
{
"content": "Determine the number of ways to distribute 15 fruits to 6 people such that each person receives at least 1 fruit and no more than 3 fruits.",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "30",
"problem": "Determine the number of ways to distribute 15 fruits to 6 people such that each person receives at least 1 fruit and no more than 3 fruits.",
"solution": "Okay, so I need to figure out how many ways there are to distribute 15 fruits to 6 people, with each person getting at least 1 fru... |
[
{
"content": "For the function \\( k(x) = 4\\sqrt{x} + \\frac{2}{\\sqrt{x}} \\) on the interval \\([\\frac{1}{4}, 1]\\), find the value \\( c \\) that satisfies the Mean Value Theorem.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{1}{2}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "31",
"problem": "For the function \\( k(x) = 4\\sqrt{x} + \\frac{2}{\\sqrt{x}} \\) on the interval \\([\\frac{1}{4}, 1]\\), find the value \\( c \\) that satisfies the Mean Value Theorem.",
"solution": "Okay, so I need to find the value c in the interval [1/4, 1] that satisfies the Mean Value Theorem ... |
[
{
"content": "Given the double summation \\( \\sum_{i=1}^{n}\\sum_{j=i}^{i+1}(3i+j) \\), change it to the form \\( \\sum_{j}^{}\\sum_{i}^{}(3i+j) \\) and calculate the result.",
"role": "user"
}
] | {
"ground_truth": "4n^2 + 5n",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "32",
"problem": "Given the double summation \\( \\sum_{i=1}^{n}\\sum_{j=i}^{i+1}(3i+j) \\), change it to the form \\( \\sum_{j}^{}\\sum_{i}^{}(3i+j) \\) and calculate the result.",
"solution": "Okay, so I have this double summation problem here: \\( \\sum_{i=1}^{n}\\sum_{j=i}^{i+1}(3i+j) \\). The task... |
[
{
"content": "Calculate the limit: \\[ \\lim_{x \\to 0}((-2^x+1)^{\\sin(x)}) \\]",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "33",
"problem": "Calculate the limit: \\[ \\lim_{x \\to 0}((-2^x+1)^{\\sin(x)}) \\]",
"solution": "Okay, so I need to find the limit of (-2^x + 1)^(sin x) as x approaches 0. Hmm, let's start by understanding the components of this expression. The base is (-2^x + 1), and the exponent is sin x. Since th... |
[
{
"content": "Solve the system of equations: \\[ \\begin{cases} x + y^2 = 7 \\\\ x^2 + y = 11 \\end{cases} \\]",
"role": "user"
}
] | {
"ground_truth": "(3, 2)",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "34",
"problem": "Solve the system of equations: \\[ \\begin{cases} x + y^2 = 7 \\\\ x^2 + y = 11 \\end{cases} \\]",
"solution": "Alright, so I have this system of equations to solve:\n\n\\[\n\\begin{cases}\nx + y^2 = 7 \\\\\nx^2 + y = 11\n\\end{cases}\n\\]\n\nHmm, okay. Let's see. Both equations are n... |
[
{
"content": "Find the value of \\( \\lambda \\) such that \\( i \\) is a root of the quadratic equation \\( z^2 + \\lambda z - 4 = 0 \\).",
"role": "user"
}
] | {
"ground_truth": "-5i",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "35",
"problem": "Find the value of \\( \\lambda \\) such that \\( i \\) is a root of the quadratic equation \\( z^2 + \\lambda z - 4 = 0 \\).",
"solution": "Okay, so I need to find the value of λ such that the imaginary unit i is a root of the quadratic equation z² + λz - 4 = 0. Hmm, let's start by re... |
[
{
"content": "You have two coins: a fair coin with a probability of heads as 0.5, and a biased coin with a probability of heads as 0.8. You randomly select one of these coins and flip it repeatedly, obtaining heads each time. Determine the minimum number of consecutive heads required such that the probability o... | {
"ground_truth": "5",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "36",
"problem": "You have two coins: a fair coin with a probability of heads as 0.5, and a biased coin with a probability of heads as 0.8. You randomly select one of these coins and flip it repeatedly, obtaining heads each time. Determine the minimum number of consecutive heads required such that the pr... |
[
{
"content": "Let $x$ be a finite set such that $\\bigcup x = x$. Determine whether $x$ must be the empty set.",
"role": "user"
}
] | {
"ground_truth": "\\emptyset",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "37",
"problem": "Let $x$ be a finite set such that $\\bigcup x = x$. Determine whether $x$ must be the empty set.",
"solution": "Okay, so I need to figure out if a finite set x that satisfies the condition that the union of x is equal to x itself must necessarily be the empty set. Hmm, let's start by ... |
[
{
"content": "Given that \\( \\frac{\\cos x}{\\cos y} + \\frac{\\sin x}{\\sin y} = -1 \\), find the value of \\( 4\\left(\\frac{\\cos^3 y}{\\cos x} + \\frac{\\sin^3 y}{\\sin x}\\right) \\).",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "38",
"problem": "Given that \\( \\frac{\\cos x}{\\cos y} + \\frac{\\sin x}{\\sin y} = -1 \\), find the value of \\( 4\\left(\\frac{\\cos^3 y}{\\cos x} + \\frac{\\sin^3 y}{\\sin x}\\right) \\).",
"solution": "Okay, let me try to figure out how to solve this problem. So, we have the equation \\( \\frac{... |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\to \\frac{\\pi}{2}} \\frac{2^{-\\cos x}-1}{x-\\frac{\\pi}{2}} \\]",
"role": "user"
}
] | {
"ground_truth": "\\ln 2",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "39",
"problem": "Evaluate the limit: \\[ \\lim_{x\\to \\frac{\\pi}{2}} \\frac{2^{-\\cos x}-1}{x-\\frac{\\pi}{2}} \\]",
"solution": "Okay, so I need to find the limit as x approaches π/2 of [2^(-cos x) - 1] divided by (x - π/2). Hmm, let's see. First, when x approaches π/2, the denominator (x - π/2) ap... |
[
{
"content": "Given that $a = 1$, the random variable $m$ follows a uniform distribution $U(0.2, 1)$. If $a = 0$, then $m$ follows a uniform distribution $U(0, 0.5)$. If $m$ is observed to be 0.3, what is the probability that $a$ equals 1?",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{5}{13}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "40",
"problem": "Given that $a = 1$, the random variable $m$ follows a uniform distribution $U(0.2, 1)$. If $a = 0$, then $m$ follows a uniform distribution $U(0, 0.5)$. If $m$ is observed to be 0.3, what is the probability that $a$ equals 1?",
"solution": "Okay, so I have this probability problem her... |
[
{
"content": "Determine which is larger: $e^\\pi$ or $\\pi^e$. Provide your reasoning without using a calculator.",
"role": "user"
}
] | {
"ground_truth": "e^\\pi",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "41",
"problem": "Determine which is larger: $e^\\pi$ or $\\pi^e$. Provide your reasoning without using a calculator.",
"solution": "Okay, so I need to figure out whether e raised to the power of pi is larger than pi raised to the power of e, or vice versa. And I can't use a calculator. Hmm, interestin... |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\rightarrow0} \\frac{\\tan(x)-x}{x^3} \\]",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{1}{3}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "42",
"problem": "Evaluate the limit: \\[ \\lim_{x\\rightarrow0} \\frac{\\tan(x)-x}{x^3} \\]",
"solution": "Okay, so I need to find the limit as x approaches 0 of [tan(x) - x] divided by x cubed. Hmm, let's see. I remember that when dealing with limits like this, especially when it's 0/0 indeterminate ... |
[
{
"content": "Determine the order of the pole of the function \\( f(z) = \\frac{\\sin(3z)}{z^2} \\) at \\( z = 0 \\).",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "43",
"problem": "Determine the order of the pole of the function \\( f(z) = \\frac{\\sin(3z)}{z^2} \\) at \\( z = 0 \\).",
"solution": "Okay, so I need to figure out the order of the pole of the function \\( f(z) = \\frac{\\sin(3z)}{z^2} \\) at \\( z = 0 \\). Hmm, let me start by recalling what a pole... |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\to\\infty}\\left(\\sqrt[4]{x^4+x^3}-x\\right) \\]",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{1}{4}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "44",
"problem": "Evaluate the limit: \\[ \\lim_{x\\to\\infty}\\left(\\sqrt[4]{x^4+x^3}-x\\right) \\]",
"solution": "Okay, so I need to evaluate the limit as x approaches infinity of the fourth root of (x^4 + x^3) minus x. Hmm. Let me write that down again to make sure I got it right:\n\n\\[\n\\lim_{x\... |
[
{
"content": "Compute the line integral \\( \\int_\\gamma x \\, dx \\) where \\( \\gamma \\) is parameterized by \\( g(t) = (\\cos t, \\sin t) \\) for \\( 0 \\leq t \\leq 2\\pi \\).",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "45",
"problem": "Compute the line integral \\( \\int_\\gamma x \\, dx \\) where \\( \\gamma \\) is parameterized by \\( g(t) = (\\cos t, \\sin t) \\) for \\( 0 \\leq t \\leq 2\\pi \\).",
"solution": "Okay, so I need to compute the line integral ∫γ x dx, where γ is parameterized by g(t) = (cos t, sin t... |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to 0} \\frac{e^{\\tan^2(x)} - \\cos(x)}{\\tan^2(x)} \\]",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{3}{2}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "46",
"problem": "Evaluate the limit: \\[ \\lim_{x \\to 0} \\frac{e^{\\tan^2(x)} - \\cos(x)}{\\tan^2(x)} \\]",
"solution": "Alright, so I need to evaluate the limit as x approaches 0 of [e^(tan²x) - cosx] divided by tan²x. Hmm, okay. Let me start by recalling some basic limit properties and maybe Taylo... |
[
{
"content": "Find the limit \\( \\lim_{n \\to \\infty} a_n \\) for the sequence defined by the recurrence relation \\( a_{n+1} = \\sin a_n \\).",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "47",
"problem": "Find the limit \\( \\lim_{n \\to \\infty} a_n \\) for the sequence defined by the recurrence relation \\( a_{n+1} = \\sin a_n \\).",
"solution": "Okay, so I need to find the limit of the sequence defined by \\( a_{n+1} = \\sin a_n \\) as \\( n \\) approaches infinity. Hmm, let's start... |
[
{
"content": "Evaluate the integral \\( \\int_{-1}^{1} \\text{sgn}(x) \\, dx \\).",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "48",
"problem": "Evaluate the integral \\( \\int_{-1}^{1} \\text{sgn}(x) \\, dx \\).",
"solution": "Okay, so I have this integral to solve: the integral from -1 to 1 of the sign function of x, dx. Hmm, let's start by recalling what the sign function is. If I remember correctly, the sign function, ofte... |
[
{
"content": "Find the closed form of the recursive function defined by:\n\\[ f(n) = 2f(n-1) + n-1 \\text{ for } n > 1, \\]\nwith the initial condition \\( f(n) = 0 \\) for \\( n < 1 \\).",
"role": "user"
}
] | {
"ground_truth": "2^n - n - 1",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "49",
"problem": "Find the closed form of the recursive function defined by:\n\\[ f(n) = 2f(n-1) + n-1 \\text{ for } n > 1, \\]\nwith the initial condition \\( f(n) = 0 \\) for \\( n < 1 \\).",
"solution": "Okay, so I need to find the closed-form expression for this recursive function. Let me start by ... |
[
{
"content": "In a class trial with 15 participants, each participant votes either \"guilty\", \"not guilty\", or \"abstain\". The voting stops as soon as at least half of the votes are \"guilty\". Determine the total number of possible voting sequences that can occur.",
"role": "user"
}
] | {
"ground_truth": "13663745",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "50",
"problem": "In a class trial with 15 participants, each participant votes either \"guilty\", \"not guilty\", or \"abstain\". The voting stops as soon as at least half of the votes are \"guilty\". Determine the total number of possible voting sequences that can occur.",
"solution": "Okay, let me t... |
[
{
"content": "Find the number of integer solutions to the equation \\(a + b + c = 0\\) where \\(a, b, c\\) are integers in the range \\([-k, k]\\) and \\(a \\neq 0\\). Here, \\(k\\) is a positive integer.",
"role": "user"
}
] | {
"ground_truth": "3k^2 + k",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "51",
"problem": "Find the number of integer solutions to the equation \\(a + b + c = 0\\) where \\(a, b, c\\) are integers in the range \\([-k, k]\\) and \\(a \\neq 0\\). Here, \\(k\\) is a positive integer.",
"solution": "Okay, so I need to find the number of integer solutions to the equation \\(a + ... |
[
{
"content": "Given that \\( z_1=2\\sqrt{3}\\operatorname{cis}\\left(\\frac{3\\pi}{2}\\right) \\) and \\( z_2=2\\operatorname{cis}\\left(\\frac{2\\pi}{3}\\right) \\), find the smallest positive integer \\( n \\) such that \\( \\left(\\frac{z_1}{z_2}\\right)^n \\in \\mathbb{R}^+ \\).",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "52",
"problem": "Given that \\( z_1=2\\sqrt{3}\\operatorname{cis}\\left(\\frac{3\\pi}{2}\\right) \\) and \\( z_2=2\\operatorname{cis}\\left(\\frac{2\\pi}{3}\\right) \\), find the smallest positive integer \\( n \\) such that \\( \\left(\\frac{z_1}{z_2}\\right)^n \\in \\mathbb{R}^+ \\).",
"solution": "... |
[
{
"content": "Find the acute dihedral angle between two faces of a regular tetrahedron, rounded to the nearest tenth of a degree.",
"role": "user"
}
] | {
"ground_truth": "70.5",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "53",
"problem": "Find the acute dihedral angle between two faces of a regular tetrahedron, rounded to the nearest tenth of a degree.",
"solution": "Okay, so I need to find the acute dihedral angle between two faces of a regular tetrahedron and round it to the nearest tenth of a degree. Hmm, let me rec... |
[
{
"content": "At a party, there are 20 girls and 20 boys. How many dance pairs can be formed if boys may dance with boys and girls with girls?",
"role": "user"
}
] | {
"ground_truth": "780",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "54",
"problem": "At a party, there are 20 girls and 20 boys. How many dance pairs can be formed if boys may dance with boys and girls with girls?",
"solution": "Okay, so there's this problem: at a party with 20 girls and 20 boys, how many dance pairs can be formed if boys may dance with boys and girls... |
[
{
"content": "Evaluate the limit: \\[ \\lim_{n \\to +\\infty} \\left(0.999 + \\frac{1}{n}\\right)^{n+3} \\]",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "55",
"problem": "Evaluate the limit: \\[ \\lim_{n \\to +\\infty} \\left(0.999 + \\frac{1}{n}\\right)^{n+3} \\]",
"solution": "Okay, so I need to evaluate the limit as n approaches infinity of (0.999 + 1/n) raised to the power of (n + 3). Hmm. Let me write that down again to make sure I have it right: ... |
[
{
"content": "Factorize the expression \\( \\Sigma a^{2}(b^{4}-c^{4}) \\). Choose the correct factorization from the options below:\n\n(a) \\((a-b)^{2}(b-c)^{2}(c-a)^{2}\\)\n\n(b) \\((a-b)(a+b)(b-c)(b+c)(c-a)(c+a)\\)\n\n(c) \\((a+b)^{2}(b+c)^{2}(c+a)^{2}\\)\n\n(d) None of these",
"role": "user"
}
] | {
"ground_truth": "B",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "56",
"problem": "Factorize the expression \\( \\Sigma a^{2}(b^{4}-c^{4}) \\). Choose the correct factorization from the options below:\n\n(a) \\((a-b)^{2}(b-c)^{2}(c-a)^{2}\\)\n\n(b) \\((a-b)(a+b)(b-c)(b+c)(c-a)(c+a)\\)\n\n(c) \\((a+b)^{2}(b+c)^{2}(c+a)^{2}\\)\n\n(d) None of these",
"solution": "Okay,... |
[
{
"content": "Calculate the limit: \\( \\lim_{x \\to 0} \\sqrt[x]{1+\\sin x} \\).",
"role": "user"
}
] | {
"ground_truth": "e",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "57",
"problem": "Calculate the limit: \\( \\lim_{x \\to 0} \\sqrt[x]{1+\\sin x} \\).",
"solution": "Okay, so I need to find the limit as x approaches 0 of the (1 + sin x) raised to the power of 1/x. Hmm. Let me write that down properly: \n\n\\( \\lim_{x \\to 0} \\sqrt[x]{1 + \\sin x} \\)\n\nAlright, s... |
[
{
"content": "Three traders have payoffs at the end of the day that are uniformly distributed between $0 and $100. What is the probability that the payoff of trader A is greater than the payoff of trader B, and the payoff of trader B is greater than the payoff of trader C?",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{1}{6}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "58",
"problem": "Three traders have payoffs at the end of the day that are uniformly distributed between $0 and $100. What is the probability that the payoff of trader A is greater than the payoff of trader B, and the payoff of trader B is greater than the payoff of trader C?",
"solution": "Okay, so I... |
[
{
"content": "In \\(\\Delta ABC\\), \\(AC = BC\\) and \\(\\angle C = 120^\\circ\\). Points \\(M\\) and \\(N\\) are on sides \\(AC\\) and \\(BC\\) respectively, such that \\(\\angle BAN = 50^\\circ\\) and \\(\\angle ABM = 60^\\circ\\). Find \\(\\angle NMB\\).",
"role": "user"
}
] | {
"ground_truth": "30^\\circ",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "59",
"problem": "In \\(\\Delta ABC\\), \\(AC = BC\\) and \\(\\angle C = 120^\\circ\\). Points \\(M\\) and \\(N\\) are on sides \\(AC\\) and \\(BC\\) respectively, such that \\(\\angle BAN = 50^\\circ\\) and \\(\\angle ABM = 60^\\circ\\). Find \\(\\angle NMB\\).",
"solution": "Okay, let me try to solve... |
[
{
"content": "Evaluate the limit: \\[ \\lim _{x\\to 0^+}\\frac{\\left(e^x+e^{2x}\\right)^2-4}{\\sqrt{9+\\sin x}-3} \\]",
"role": "user"
}
] | {
"ground_truth": "72",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "60",
"problem": "Evaluate the limit: \\[ \\lim _{x\\to 0^+}\\frac{\\left(e^x+e^{2x}\\right)^2-4}{\\sqrt{9+\\sin x}-3} \\]",
"solution": "Okay, so I have this limit to evaluate: the limit as x approaches 0 from the positive side of [(e^x + e^{2x})^2 - 4] divided by [sqrt(9 + sin x) - 3]. Hmm, let me tr... |
[
{
"content": "Evaluate \\( \\lim_{x\\to\\infty} ((x^5+x^4)^{1/6}-(x^5-x^4)^{1/6}) \\).",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "61",
"problem": "Evaluate \\( \\lim_{x\\to\\infty} ((x^5+x^4)^{1/6}-(x^5-x^4)^{1/6}) \\).",
"solution": "Okay, so I need to evaluate the limit as x approaches infinity of ( (x^5 + x^4)^(1/6) - (x^5 - x^4)^(1/6) ). Hmm, limits at infinity often involve looking at the highest degree terms, right? Let me... |
[
{
"content": "In triangle ABC, the areas of the shaded regions are denoted as \\( S_1 \\) and \\( S_2 \\). Given that \\( S_1 \\cdot S_2 = 16 \\text{ cm}^4 \\), calculate the length of segment \\( MN \\).",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "62",
"problem": "In triangle ABC, the areas of the shaded regions are denoted as \\( S_1 \\) and \\( S_2 \\). Given that \\( S_1 \\cdot S_2 = 16 \\text{ cm}^4 \\), calculate the length of segment \\( MN \\).",
"solution": "Okay, so I need to find the length of segment MN in triangle ABC, given that th... |
[
{
"content": "How many lines in a three-dimensional rectangular coordinate system pass through four distinct points of the form \\((i, j, k)\\) where \\(i\\), \\(j\\), and \\(k\\) are positive integers not exceeding four? \n\\(\\text{(A)} \\ 60 \\qquad \\text{(B)} \\ 64 \\qquad \\text{(C)} \\ 72 \\qquad \\text{... | {
"ground_truth": "D",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "63",
"problem": "How many lines in a three-dimensional rectangular coordinate system pass through four distinct points of the form \\((i, j, k)\\) where \\(i\\), \\(j\\), and \\(k\\) are positive integers not exceeding four? \n\\(\\text{(A)} \\ 60 \\qquad \\text{(B)} \\ 64 \\qquad \\text{(C)} \\ 72 \\qq... |
[
{
"content": "Given two points $A$ and $B$ on the same side of a line $l$, with orthogonal projections $A'$ and $B'$ on $l$, where $d(A,A') = 5$, $d(B,B') = 4$, and $d(A',B') = 10$, find a point $C$ on line $l$ such that the sum $d(A,C) + d(B,C)$ is minimized.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{50}{9}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "64",
"problem": "Given two points $A$ and $B$ on the same side of a line $l$, with orthogonal projections $A'$ and $B'$ on $l$, where $d(A,A') = 5$, $d(B,B') = 4$, and $d(A',B') = 10$, find a point $C$ on line $l$ such that the sum $d(A,C) + d(B,C)$ is minimized.",
"solution": "Okay, so I have this ge... |
[
{
"content": "Find the minimum value of the function \\( f(x,y,z) = x + 3y + 5z \\) subject to the constraint \\( x^2 + y^2 + z^2 = 1 \\) using the method of Lagrange multipliers.",
"role": "user"
}
] | {
"ground_truth": "-\\sqrt{35}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "65",
"problem": "Find the minimum value of the function \\( f(x,y,z) = x + 3y + 5z \\) subject to the constraint \\( x^2 + y^2 + z^2 = 1 \\) using the method of Lagrange multipliers.",
"solution": "Okay, so I need to find the minimum value of the function f(x, y, z) = x + 3y + 5z subject to the constr... |
[
{
"content": "Evaluate the limit \\(\\lim_{x\\to\\infty} (\\sqrt{x+1} + \\sqrt{x-1} - \\sqrt{2x})\\) using the little o notation.",
"role": "user"
}
] | {
"ground_truth": "\\infty",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "66",
"problem": "Evaluate the limit \\(\\lim_{x\\to\\infty} (\\sqrt{x+1} + \\sqrt{x-1} - \\sqrt{2x})\\) using the little o notation.",
"solution": "Alright, so I need to evaluate the limit as x approaches infinity of sqrt(x+1) + sqrt(x-1) - sqrt(2x). Hmm, okay. The problem suggests using little o nota... |
[
{
"content": "How many 10-digit numbers can be formed using the digits \\{5, 6, 7\\} such that no two consecutive digits are the same and the first and last digits are the same?",
"role": "user"
}
] | {
"ground_truth": "510",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "67",
"problem": "How many 10-digit numbers can be formed using the digits \\{5, 6, 7\\} such that no two consecutive digits are the same and the first and last digits are the same?",
"solution": "Okay, so I need to figure out how many 10-digit numbers can be formed using the digits 5, 6, and 7, with t... |
[
{
"content": "Let $f(x)$ be a function that is infinitely differentiable on $\\mathbb{R}$. Determine whether the following statement is true: \\[ \\int_{a}^{b} f'(x) \\, dx = f(b) - f(a) \\]",
"role": "user"
}
] | {
"ground_truth": "True",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "68",
"problem": "Let $f(x)$ be a function that is infinitely differentiable on $\\mathbb{R}$. Determine whether the following statement is true: \\[ \\int_{a}^{b} f'(x) \\, dx = f(b) - f(a) \\]",
"solution": "Okay, so I need to figure out if the integral of f'(x) from a to b equals f(b) minus f(a). Hm... |
[
{
"content": "Determine if the expression \\( \\sqrt{-i \\sqrt{5}} \\) is equal to \\( -i\\sqrt[4]{-5} \\).",
"role": "user"
}
] | {
"ground_truth": "Equal",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "69",
"problem": "Determine if the expression \\( \\sqrt{-i \\sqrt{5}} \\) is equal to \\( -i\\sqrt[4]{-5} \\).",
"solution": "Okay, so I need to figure out if the square root of negative i times the square root of 5 is equal to negative i times the fourth root of negative 5. Hmm, let's break this down... |
[
{
"content": "Calculate the limit: \\[ \\lim_{n \\to \\infty} \\left( \\frac{1}{n+1} + \\frac{1}{n+2} + \\cdots + \\frac{1}{n+n} \\right) \\]",
"role": "user"
}
] | {
"ground_truth": "\\ln 2",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "70",
"problem": "Calculate the limit: \\[ \\lim_{n \\to \\infty} \\left( \\frac{1}{n+1} + \\frac{1}{n+2} + \\cdots + \\frac{1}{n+n} \\right) \\]",
"solution": "Alright, so I need to find the limit as n approaches infinity of the sum from k = 1 to n of 1/(n + k). Hmm, okay. Let me write that down prope... |
[
{
"content": "Calculate the area under the curve of the function \\( f(x) = \\sqrt{x} \\) over the interval \\([1, 9]\\) using the Riemann sum approach. Divide the interval into \\( n \\) equal parts and express the area as a limit of a sum. Evaluate this limit to find the area.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{52}{3}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "71",
"problem": "Calculate the area under the curve of the function \\( f(x) = \\sqrt{x} \\) over the interval \\([1, 9]\\) using the Riemann sum approach. Divide the interval into \\( n \\) equal parts and express the area as a limit of a sum. Evaluate this limit to find the area.",
"solution": "Okay... |
[
{
"content": "The quadratic equation \\(x^2 - 3ax + b = 0\\) does not have distinct real roots. Find the least possible value of \\(\\frac{b}{a-2}\\), given that \\(a > 2\\).",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "72",
"problem": "The quadratic equation \\(x^2 - 3ax + b = 0\\) does not have distinct real roots. Find the least possible value of \\(\\frac{b}{a-2}\\), given that \\(a > 2\\).",
"solution": "Alright, let's tackle this quadratic equation problem. So, the equation is \\(x^2 - 3a x + b = 0\\), and we'r... |
[
{
"content": "Let $X$ and $Y$ be sets such that the cardinality of their power sets are equal, i.e., $|\\mathscr{P}(X)| = |\\mathscr{P}(Y)|$. Does it follow that the cardinality of $X$ is equal to the cardinality of $Y$, i.e., $|X| = |Y|$? Justify your answer.",
"role": "user"
}
] | {
"ground_truth": "No",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "73",
"problem": "Let $X$ and $Y$ be sets such that the cardinality of their power sets are equal, i.e., $|\\mathscr{P}(X)| = |\\mathscr{P}(Y)|$. Does it follow that the cardinality of $X$ is equal to the cardinality of $Y$, i.e., $|X| = |Y|$? Justify your answer.",
"solution": "Okay, so I have this pr... |
[
{
"content": "Find the volume of the region \\( X = \\{(x, y, z) \\in \\mathbb{R}^{3} : x+y+z<1 ; 0<x<1 ; 0<y<1 ; z>-1\\} \\) using the triple integral form: \n\\[ \\int \\left( \\int \\left( \\int dx \\right) dy \\right) dz. \\]",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "74",
"problem": "Find the volume of the region \\( X = \\{(x, y, z) \\in \\mathbb{R}^{3} : x+y+z<1 ; 0<x<1 ; 0<y<1 ; z>-1\\} \\) using the triple integral form: \n\\[ \\int \\left( \\int \\left( \\int dx \\right) dy \\right) dz. \\]",
"solution": "Okay, so I need to find the volume of this region X de... |
[
{
"content": "Calculate the volume of the solid formed by revolving the triangular region bounded by the lines $y = |x|$ and $y = 1$ around the line $x = -2$.",
"role": "user"
}
] | {
"ground_truth": "4\\pi",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "75",
"problem": "Calculate the volume of the solid formed by revolving the triangular region bounded by the lines $y = |x|$ and $y = 1$ around the line $x = -2$.",
"solution": "Okay, so I need to find the volume of the solid formed by revolving the triangular region bounded by y = |x| and y = 1 around... |
[
{
"content": "Given a probability space \\(\\left([0,5], \\mathcal{B}([0,5]), \\frac{\\lambda_{1}}{5}\\right)\\) and the random variable \\(X(\\omega)=I_{[0,1]}(\\omega)-2 I_{[2,4]}(\\omega)\\), determine the expected value \\(E(X)\\).",
"role": "user"
}
] | {
"ground_truth": "-\\dfrac{3}{5}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "76",
"problem": "Given a probability space \\(\\left([0,5], \\mathcal{B}([0,5]), \\frac{\\lambda_{1}}{5}\\right)\\) and the random variable \\(X(\\omega)=I_{[0,1]}(\\omega)-2 I_{[2,4]}(\\omega)\\), determine the expected value \\(E(X)\\).",
"solution": "Okay, so I need to find the expected value E(X) ... |
[
{
"content": "Provide a counterexample to the statement: If \\(G\\) is a group with operation \\(*\\) and \\(a\\) and \\(b\\) are elements of \\(G\\), then \\((a*b)^2 = a^2*b^2\\).",
"role": "user"
}
] | {
"ground_truth": "(a * b)^2 \\neq a^2 * b^2",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "77",
"problem": "Provide a counterexample to the statement: If \\(G\\) is a group with operation \\(*\\) and \\(a\\) and \\(b\\) are elements of \\(G\\), then \\((a*b)^2 = a^2*b^2\\).",
"solution": "Okay, so I need to find a counterexample to the statement that in any group G, for elements a and b, (a... |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x\\rightarrow +\\infty} \\left(\\sqrt[3]{x^3+3x^2} - \\sqrt{x^2-2x}\\right) \\]",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "78",
"problem": "Evaluate the limit: \\[ \\lim_{x\\rightarrow +\\infty} \\left(\\sqrt[3]{x^3+3x^2} - \\sqrt{x^2-2x}\\right) \\]",
"solution": "Okay, so I need to evaluate the limit as x approaches positive infinity of the expression cube root of (x³ + 3x²) minus the square root of (x² - 2x). Hmm, both... |
[
{
"content": "Determine the number of solutions to the equation $t \\sin(2t) = 2$ for $0 < t < 3\\pi$. Provide your answer as a single integer.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "79",
"problem": "Determine the number of solutions to the equation $t \\sin(2t) = 2$ for $0 < t < 3\\pi$. Provide your answer as a single integer.",
"solution": "Okay, let's try to figure out how many solutions the equation \\( t \\sin(2t) = 2 \\) has between \\( 0 \\) and \\( 3\\pi \\). Hmm, this see... |
[
{
"content": "Given a list of numbers where every number except one appears twice, find the number that appears only once. For example, in the list $2, 3, 4, 3, 2, 1, 1$, identify the number that does not have a pair.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "80",
"problem": "Given a list of numbers where every number except one appears twice, find the number that appears only once. For example, in the list $2, 3, 4, 3, 2, 1, 1$, identify the number that does not have a pair.",
"solution": "Okay, so I need to find the number that appears only once in a lis... |
[
{
"content": "Use a Riemann sum to compute the integral \\( \\int_0^\\frac{\\pi}{2} \\sin(x)\\, dx \\).",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "81",
"problem": "Use a Riemann sum to compute the integral \\( \\int_0^\\frac{\\pi}{2} \\sin(x)\\, dx \\).",
"solution": "Okay, so I need to compute the integral of sin(x) from 0 to π/2 using a Riemann sum. Hmm, Riemann sums are those approximations where you divide the area under the curve into recta... |
[
{
"content": "Which simple interest rate over six years is closest to being equivalent to the following: an effective rate of discount of 3% for the first year, an effective rate of discount of 6% for the second year, an effective rate of discount of 9% for the third year, and an effective rate of interest of 5... | {
"ground_truth": "D",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "82",
"problem": "Which simple interest rate over six years is closest to being equivalent to the following: an effective rate of discount of 3% for the first year, an effective rate of discount of 6% for the second year, an effective rate of discount of 9% for the third year, and an effective rate of in... |
[
{
"content": "Find the total number of non-similar triangles that can be formed such that all the angles of the triangles are integers.",
"role": "user"
}
] | {
"ground_truth": "2700",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "83",
"problem": "Find the total number of non-similar triangles that can be formed such that all the angles of the triangles are integers.",
"solution": "Okay, so I need to figure out how many non-similar triangles exist where all the angles are integers. Hmm, let me start by recalling some basics abo... |
[
{
"content": "Use polar coordinates to find the limit:\n\\[\\lim_{(x, y)\\rightarrow(0, 0)}\\dfrac{3e^{-x^2 - y^2}-3}{x^2+y^2}\\]",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "84",
"problem": "Use polar coordinates to find the limit:\n\\[\\lim_{(x, y)\\rightarrow(0, 0)}\\dfrac{3e^{-x^2 - y^2}-3}{x^2+y^2}\\]",
"solution": "Okay, so I need to find the limit as (x, y) approaches (0, 0) of [3e^(-x² - y²) - 3] divided by (x² + y²). Hmm, the problem says to use polar coordinates.... |
[
{
"content": "A triangle has two sides of lengths 4 cm and 9 cm. The radius of the inscribed circle that touches all three sides of the triangle is 1 cm. Determine the length of the third side of the triangle.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "85",
"problem": "A triangle has two sides of lengths 4 cm and 9 cm. The radius of the inscribed circle that touches all three sides of the triangle is 1 cm. Determine the length of the third side of the triangle.",
"solution": "Okay, let's try to solve this problem. So, we have a triangle with two sid... |
[
{
"content": "You have 100 boxes, and C of them contain a gift. You can open up to 16 boxes. What is the smallest number of C that will give you a probability greater than 0.5 of finding at least one gift? Assume that once you find a gift, you stop opening boxes.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "86",
"problem": "You have 100 boxes, and C of them contain a gift. You can open up to 16 boxes. What is the smallest number of C that will give you a probability greater than 0.5 of finding at least one gift? Assume that once you find a gift, you stop opening boxes.",
"solution": "Okay, so here's the ... |
[
{
"content": "In a complete graph with 5 vertices, labeled \\( \\{1, 2, 3, 4, 5\\} \\), how many Hamiltonian cycles include both the edges \\( \\{1, 2\\} \\) and \\( \\{3, 4\\} \\)? Assume that a Hamiltonian cycle is defined solely by the collection of edges it contains, without regard to orientation or startin... | {
"ground_truth": "4",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "87",
"problem": "In a complete graph with 5 vertices, labeled \\( \\{1, 2, 3, 4, 5\\} \\), how many Hamiltonian cycles include both the edges \\( \\{1, 2\\} \\) and \\( \\{3, 4\\} \\)? Assume that a Hamiltonian cycle is defined solely by the collection of edges it contains, without regard to orientation... |
[
{
"content": "A tea merchant has five cubical tin tea boxes, each with a picture on each of its six sides, totaling thirty pictures. However, one picture on box No. 1 is repeated on box No. 4, and two pictures on box No. 4 are repeated on box No. 3, resulting in only twenty-seven different pictures. The merchan... | {
"ground_truth": "186624",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "88",
"problem": "A tea merchant has five cubical tin tea boxes, each with a picture on each of its six sides, totaling thirty pictures. However, one picture on box No. 1 is repeated on box No. 4, and two pictures on box No. 4 are repeated on box No. 3, resulting in only twenty-seven different pictures. ... |
[
{
"content": "Given the equations:\n\\[ y = \\left\\lfloor\\frac{ x \\mod 790}{10}\\right\\rfloor + 48 \\]\n\\[ z = (x \\mod 790) \\mod 10 + 10\\left\\lfloor\\frac{x}{790}\\right\\rfloor + 48 \\]\nIf \\(y = 69\\) and \\(z = 60\\), find the value of \\(x\\).",
"role": "user"
}
] | {
"ground_truth": "1002",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "89",
"problem": "Given the equations:\n\\[ y = \\left\\lfloor\\frac{ x \\mod 790}{10}\\right\\rfloor + 48 \\]\n\\[ z = (x \\mod 790) \\mod 10 + 10\\left\\lfloor\\frac{x}{790}\\right\\rfloor + 48 \\]\nIf \\(y = 69\\) and \\(z = 60\\), find the value of \\(x\\).",
"solution": "Okay, let's see. I need to... |
[
{
"content": "Evaluate the limit: \\[ \\lim_{x \\to \\frac{\\pi}{6}}{(1-2\\sin(x))}^{\\tan(\\frac{\\pi}{6}-x)} \\]",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "90",
"problem": "Evaluate the limit: \\[ \\lim_{x \\to \\frac{\\pi}{6}}{(1-2\\sin(x))}^{\\tan(\\frac{\\pi}{6}-x)} \\]",
"solution": "Okay, so I have this limit to evaluate: the limit as x approaches π/6 of (1 - 2 sin x) raised to the power of tan(π/6 - x). Hmm, that seems a bit tricky, but maybe I can... |
[
{
"content": "Find the number of integer solutions to the equation \\(x_1 + x_2 - x_3 - x_4 = 0\\) where each variable \\(x_1, x_2, x_3, x_4\\) is an integer between -4 and 4, inclusive.",
"role": "user"
}
] | {
"ground_truth": "489",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "91",
"problem": "Find the number of integer solutions to the equation \\(x_1 + x_2 - x_3 - x_4 = 0\\) where each variable \\(x_1, x_2, x_3, x_4\\) is an integer between -4 and 4, inclusive.",
"solution": "Okay, so I need to find the number of integer solutions to the equation \\(x_1 + x_2 - x_3 - x_4 ... |
[
{
"content": "Given that \\(x+y+z=3\\), \\(x^2 + y^2 + z^2 = 5\\), and \\(x^3+y^3+z^3=7\\), find the value of \\(x^4+y^4+z^4\\).",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "92",
"problem": "Given that \\(x+y+z=3\\), \\(x^2 + y^2 + z^2 = 5\\), and \\(x^3+y^3+z^3=7\\), find the value of \\(x^4+y^4+z^4\\).",
"solution": "<think>\nOkay, so I need to find the value of \\(x^4 + y^4 + z^4\\) given the equations \\(x + y + z = 3\\), \\(x^2 + y^2 + z^2 = 5\\), and \\(x^3 + y^3 + ... |
[
{
"content": "A music school has 40 students, with 10 each studying violin, viola, cello, and string bass. The director wants to form 10 string quartets, each consisting of one student from each instrument group. In how many ways can this be done?",
"role": "user"
}
] | {
"ground_truth": "(10!)^3",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "93",
"problem": "A music school has 40 students, with 10 each studying violin, viola, cello, and string bass. The director wants to form 10 string quartets, each consisting of one student from each instrument group. In how many ways can this be done?",
"solution": "Alright, so we have this music schoo... |
[
{
"content": "Determine the subspace $X \\subseteq \\mathbb{R}$ with the usual topology, including $\\{0,1\\} \\subseteq X$, for which there exists a continuous function $f: X \\to \\{0,1\\}$ such that $f(0)=0$ and $f(1)=1$. Consider the following options:\n\n1. $X = [0,1]$\n2. $X = [-1,1]$\n3. $X = \\mathbb{R}... | {
"ground_truth": "D",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "94",
"problem": "Determine the subspace $X \\subseteq \\mathbb{R}$ with the usual topology, including $\\{0,1\\} \\subseteq X$, for which there exists a continuous function $f: X \\to \\{0,1\\}$ such that $f(0)=0$ and $f(1)=1$. Consider the following options:\n\n1. $X = [0,1]$\n2. $X = [-1,1]$\n3. $X = ... |
[
{
"content": "Given that $90^a = 2$ and $90^b = 5$, evaluate $45^{\\frac{1-a-b}{2-2a}}$ without using logarithms or calculators.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "95",
"problem": "Given that $90^a = 2$ and $90^b = 5$, evaluate $45^{\\frac{1-a-b}{2-2a}}$ without using logarithms or calculators.",
"solution": "Okay, so I have this problem here: given that \\(90^a = 2\\) and \\(90^b = 5\\), I need to evaluate \\(45^{\\frac{1 - a - b}{2 - 2a}}\\) without using loga... |
[
{
"content": "For an arithmetic sequence $a_1, a_2, a_3, \\ldots$, where $(a_1)^2 + (a_{11})^2 \\leq 100$, determine the maximum value of $S = a_{11} + a_{12} + \\ldots + a_{21}$. What is this maximum value?",
"role": "user"
}
] | {
"ground_truth": "55\\sqrt{10}",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "96",
"problem": "For an arithmetic sequence $a_1, a_2, a_3, \\ldots$, where $(a_1)^2 + (a_{11})^2 \\leq 100$, determine the maximum value of $S = a_{11} + a_{12} + \\ldots + a_{21}$. What is this maximum value?",
"solution": "Okay, let's see. I need to find the maximum value of S, which is the sum of ... |
[
{
"content": "Find all solutions to the equation \\(x^x = \\frac{1}{256}\\).",
"role": "user"
}
] | {
"ground_truth": "-4",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "97",
"problem": "Find all solutions to the equation \\(x^x = \\frac{1}{256}\\).",
"solution": "Okay, so I need to solve the equation \\(x^x = \\frac{1}{256}\\). Hmm, let's see. I remember that equations where the variable is in both the base and the exponent can sometimes be tricky. Maybe I should sta... |
[
{
"content": "Calculate the value of \\( M = F(7, 2) + F(7, 3) + F(7, 5) - F(7, 6) \\) where \\( F(x, y) = \\cos^y\\left(\\frac{\\pi}{x}\\right) + \\cos^y\\left(\\frac{3\\pi}{x}\\right) + \\cos^y\\left(\\frac{5\\pi}{x}\\right) \\). Choose the correct answer from the following options:\n\n(a) \\( \\frac{37}{32} ... | {
"ground_truth": "D",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "98",
"problem": "Calculate the value of \\( M = F(7, 2) + F(7, 3) + F(7, 5) - F(7, 6) \\) where \\( F(x, y) = \\cos^y\\left(\\frac{\\pi}{x}\\right) + \\cos^y\\left(\\frac{3\\pi}{x}\\right) + \\cos^y\\left(\\frac{5\\pi}{x}\\right) \\). Choose the correct answer from the following options:\n\n(a) \\( \\fr... |
[
{
"content": "In triangle \\(\\triangle{ABC}\\), the side lengths are \\(\\overline{AB}=62\\), \\(\\overline{AC}=50\\), and \\(\\overline{BC}=105\\). Let \\(M\\) be the midpoint of segment \\(AB\\), and \\(N\\) be the midpoint of the median from \\(C\\) to \\(AB\\). Define \\(P_1\\) as the midpoint of \\(CN\\) ... | {
"ground_truth": "48",
"style": "rule-lighteval/MATH_v2"
} | deepmath | MATH | {
"index": "99",
"problem": "In triangle \\(\\triangle{ABC}\\), the side lengths are \\(\\overline{AB}=62\\), \\(\\overline{AC}=50\\), and \\(\\overline{BC}=105\\). Let \\(M\\) be the midpoint of segment \\(AB\\), and \\(N\\) be the midpoint of the median from \\(C\\) to \\(AB\\). Define \\(P_1\\) as the midpoint o... |
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