data_source stringclasses 1
value | prompt listlengths 2 2 | ability stringclasses 1
value | reward_model dict | extra_info dict | sub_source stringclasses 8
values |
|---|---|---|---|---|---|
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Solve the equation $x^2 + (x+m)^2 = (x+n)^2 + 2mn$ for $x$, where $m \\neq n$ are constants.",
"role": "user"
}
] | math | {
"ground_truth": "x = -m + n \\pm \\sqrt{2(n^2 - mn + m^2)}",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The function \\( f(x)=x|x+a|+b \\) is an odd function if and only if \\( a^{2}+b^{2}=0 \\).",
"role": "user"
}
] | math | {
"ground_truth": "a^2 + b^2 = 0",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Calculate: $\\log_{\\sqrt{2}}4+e^{\\ln 3}+(0.125)^{-\\frac{2}{3}}=$ ______.",
"role": "user"
}
] | math | {
"ground_truth": "11",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Calculate the least common denominator needed for the sum $\\frac{1}{3} + \\frac{1}{4} + \\frac{1}{5} + \\frac{1}{8} + \\frac{1}{9} + \\frac{1}{11}$.",
"role": "user"
}
] | math | {
"ground_truth": "3960",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Define \\(Q(x) = (x-1^2)(x-2^2)\\cdots(x-50^2)\\). Determine the number of integers \\(n\\) for which \\(Q(n) \\leq 0\\).",
"role": "user"
}
] | math | {
"ground_truth": "1300",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given a line $l$ intersects with circle $C$ at points $P(1,0)$ and $Q(0,1)$.\n(1) Find the equation of the line where the center $C$ of the circle is located;\n(2) If the radius ... | math | {
"ground_truth": "x^2+y^2-2x-2y+1=0",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If the inequality \\( kx^2 + 2kx - (k+2) < 0 \\) always holds true, determine the range of real number \\( k \\).",
"role": "user"
}
] | math | {
"ground_truth": "-1 < k < 0",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $p$ and $q$ be the roots of $x^2 - 5x + 7 = 0$. Compute\n\\[ p^3 + p^4 q^2 + p^2 q^4 + q^3. \\]",
"role": "user"
}
] | math | {
"ground_truth": "559",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A railway turns in a circular arc with a radius of $2 \\, \\text{km}$. A train passes through at a speed of $30 \\, \\text{km/h}$. Calculate the number of radians the train turns... | math | {
"ground_truth": "\\frac{1}{24}",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Set \\(A\\) has 30 elements, and set \\(B\\) has 20 elements. What is the smallest possible number of elements in \\(A \\cup B\\) if it is known that at least 10 elements of \\(B... | math | {
"ground_truth": "40",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Henry and Alice are traveling to a park that is 2 miles away from their current location. Henry rides his scooter at a speed of 12 miles per hour, and Alice jogs at a speed of 6 ... | math | {
"ground_truth": "10",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A rectangular prism with dimensions $20 \\mathrm{~cm} \\times 30 \\mathrm{~cm} \\times 40 \\mathrm{~cm}$ is placed such that the edge of length $20 \\mathrm{~cm}$ is on the table... | math | {
"ground_truth": "\\text{18.8 liters}",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In a box containing 800 balls, 70% are red, and the rest are blue, calculate the number of red balls that must be removed so that 60% of the remaining balls are red.",
"role"... | math | {
"ground_truth": "200",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider the equations: \\(I: a + b + c = 55\\) where \\(a, b,\\) and \\(c\\) are positive integers, and \\(II: a + b + c + x + y = 71\\) where \\(a, b, c, x,\\) and \\(y\\) are ... | math | {
"ground_truth": "21465",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given circle $A$: $(x+3)^{2}+y^{2}=100$, and a fixed point $B(3,0)$ inside circle $A$. If circle $P$ passes through point $B$ and is tangent to circle $A$ internally, then the eq... | math | {
"ground_truth": "\\dfrac{x^{2}}{25}+ \\dfrac{y^{2}}{16}=1",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let \\( f(x) = \\frac{\\sin^m x}{a} + \\frac{b}{\\sin^n x} \\), where \\( a, b, m, n \\in \\mathbf{N}^{*} \\) and \\( x \\in (0, \\pi) \\). Find the minimum value of \\( f(x) \\)... | math | {
"ground_truth": "\\frac{1}{a} + b \\text{ or } (m+n) \\sqrt[m+n]{ \\left( \\frac{1}{n a} \\right)^n \\left( \\frac{b}{m} \\right)^m }.",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all functions $ f$ defined in the set of rational numbers and taking their values in the same set such that the equation $ f(x + y) + f(x - y) = 2f(x) + 2f(y)$ hold... | math | {
"ground_truth": "f(x) = ax^2 \\quad \\forall x \\in \\mathbb{Q}",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Each of the positive integers $1,2,\\dots,n$ is colored in one of the colors red, blue or yellow regarding the following rules:\n(1) A Number $x$ and the smallest number larg... | math | {
"ground_truth": "3 \\cdot 2^n - 3",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $g(x)$ be a polynomial of degree 2010 with real coefficients, and let its roots be $s_1, s_2, \\dots, s_{2010}$. There are exactly 1005 distinct values among\n\\[|s_1|, |s_2|... | math | {
"ground_truth": "5",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In the sequence $\\{a_{n}\\}$, $a_{2}=2$, $a_{3}+a_{7}=10$, and $a_{n+1}=2a_{n}-a_{n-1}$ for $n\\geqslant 2$. \n$(1)$ Find the general formula for the sequence $\\{a_{n}\\}$. \... | math | {
"ground_truth": "T_{k} = 2^{k+1} - \\frac{k(k+1)}{2} + k - 2",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider the following function \\(h(x)\\) defined as\n\\[(x^{2^{2009}-1}-1)h(x) = (x+1)(x^2+1)(x^4+1)\\cdots (x^{2^{2008}}+1) - 1.\\]\nFind \\(h(3)\\).",
"role": "user"
}
... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the minimum value of the function $y=x+2\\cos x$ on the interval $[0,\\pi]$.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{5\\pi}{6}-\\sqrt{3}",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Evaluate the product \n\\[\n(n-2) \\cdot (n-1) \\cdot n \\cdot (n+1) \\cdot (n+2) \\cdot (n+3) \\cdot (n+4),\n\\]\nwhere \\( n = 3 \\).",
"role": "user"
}
] | math | {
"ground_truth": "5040",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The probability of a patient being cured after taking a certain new medicine is 0.9. The probability that at least 3 out of 4 patients taking this new medicine are cured is (answ... | math | {
"ground_truth": "0.9477",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The function $f\\left(x\\right)=\\left[x\\right]$ represents the greatest integer not exceeding $x$. For example, $\\left[-3.5\\right]=-4$, $\\left[2.1\\right]=2$. When $x\\in \\... | math | {
"ground_truth": "[0,1)",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given a right triangular prism $ABC-A_{1}B_{1}C_{1}$ whose side edge length is equal to the base edge length, find the sine value of the angle formed by $AB_{1}$ and the side fac... | math | {
"ground_truth": "\\frac{\\sqrt{6}}{4}",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "On a trip from the United States to France, Julia took $d'$ U.S. dollars. At the border, she exchanged all her money, receiving $8$ Euros for every $5$ U.S. dollars. After spendi... | math | {
"ground_truth": "\\text{C) 7}",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A digital watch now displays time in a 24-hour format, showing hours and minutes. Find the largest possible sum of the digits when it displays time in this format, where the hour... | math | {
"ground_truth": "24",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "An integer is to be written in each circle of the network shown. The integers must be written so that the sum of the numbers at the end of each line segment is the same. Two of t... | math | {
"ground_truth": "132",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Four of the following test scores are John's and the other three are Sarah's: 78, 80, 85, 87, 90, 95, 100. John's mean score is 86. What is Sarah's mean score?",
"role": "use... | math | {
"ground_truth": "90.\\overline{3}",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A coin is tossed $10$ times. Compute the probability that two heads will turn up in succession somewhere in the sequence of throws.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{55}{64}",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given $\\frac{1}{a}+\\frac{2}{b}=1$, and $a\\neq -b$, then the value of $\\frac{ab-a}{a+b}$ is ______.",
"role": "user"
}
] | math | {
"ground_truth": "1",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "$100$ distinct natural numbers $a_1, a_2, a_3, \\ldots, a_{100}$ are written on the board. Then, under each number $a_i$ , someone wrote a number $b_i$ , such that $b_i$ i... | math | {
"ground_truth": "100",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that \\(a, b, c\\) are real numbers and satisfy the equations \\(a + b + c = 15\\) and \\(a^2 + b^2 + c^2 = 100\\), find the product of the maximum and minimum values of \\... | math | {
"ground_truth": "\\frac{25}{3}",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "There are finite many coins in David’s purse. The values of these coins are pair wisely distinct positive integers. Is that possible to make such a purse, such that David has exa... | math | {
"ground_truth": "2020",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Five students, including A, B, C, D, and E, are participating in a speech contest. Neither student A nor B is the first to speak, and B is not the last to speak. Determine the pr... | math | {
"ground_truth": "\\frac {1}{3}",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "$4^4 \\cdot 9^4 \\cdot 4^9 \\cdot 9^9=$",
"role": "user"
}
] | math | {
"ground_truth": "36^{13}",
"style": "rule"
} | {
"id": "numina_amc_aime-0",
"rating": null,
"split": "test",
"sub_source": "amc_aime"
} | amc_aime |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Ms. Daily Commuter needs to reach her office by 9:00 AM each day. If she drives at an average speed of 30 miles per hour, she will be late by 5 minutes. If she drives at 50 miles... | math | {
"ground_truth": "\\text{B) 37.5 mph}",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The amount of algae in a small lake doubles in size every day. By the end of the 20th day of a particular month, the lake is completely covered with algae. Determine on which day... | math | {
"ground_truth": "17",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Add \\(5326_{8}\\) to \\(1473_{8}\\). Express your answer in base 8.",
"role": "user"
}
] | math | {
"ground_truth": "7043_{8}",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $I_{m}=\\textstyle\\int_{0}^{2 \\pi} \\cos (x) \\cos (2 x) \\cdots \\cos (m x) d x .$ For which integers $m, 1 \\leq m \\leq 10$ is $I_{m} \\neq 0 ?$",
"role": "user... | math | {
"ground_truth": "m = 3, 4, 7, 8",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Three swimmers must swim in a pool on a lane 50 meters long, immediately turn back, and return to the starting point. The first swimmer starts first, the second swimmer starts af... | math | {
"ground_truth": "x = \\frac{100 s - 50 r - rs}{(3 s - r) a}, \\quad z = \\frac{100 s - 50 r - rs}{(r - s) a}, \\quad \\text{for } s < r < \\frac{100 s}{50 + s}",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that $\\cos (\\frac{\\pi}{2}+\\phi)=\\frac{2}{3}$ and $|\\phi| < \\frac{\\pi}{2}$, find the value of $\\tan \\phi$.\n\nA) $-\\frac{\\sqrt{5}}{2}$\nB) $\\frac{\\sqrt{5}}{2}$... | math | {
"ground_truth": "-\\frac{2\\sqrt{5}}{5}",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In a math competition, 30 students participated. From these students:\n\n- 20 solved the 1st problem,\n- 16 solved the 2nd problem,\n- 10 solved the 3rd problem,\n- 11 solved bot... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $x=-1000$. Evaluate the expression $\\bigg|$ $||x|-x|-|x|$ $\\bigg|$ $+x$.",
"role": "user"
}
] | math | {
"ground_truth": "0",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In any finite grid of squares, some shaded and some not, for each unshaded square, record the number of shaded squares horizontally or vertically adjacent to it; this grid's *sco... | math | {
"ground_truth": "51",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Compute the indefinite integral:\n\n$$\n\\int \\frac{2 x^{3}+6 x^{2}+7 x}{(x-2)(x+1)^{3}} \\, dx\n$$",
"role": "user"
}
] | math | {
"ground_truth": "2 \\ln |x-2| - \\frac{1}{2(x+1)^2} + C.",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given triangle \\( \\triangle ABC \\) with circumcenter \\( O \\) and orthocenter \\( H \\), and \\( O \\neq H \\). Let \\( D \\) and \\( E \\) be the midpoints of sides \\( BC \... | math | {
"ground_truth": "\\frac{3}{2}",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Janez wants to make an $m\\times n$ grid (consisting of unit squares) using equal elements of the form $\\llcorner$ , where each leg of an element has the unit length. No two ... | math | {
"ground_truth": "6 \\mid mn",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A square with side length 2R is rolled into the lateral surface of a cylinder, and a semicircle with radius R is rolled into the lateral surface of a cone. The ratio of the volum... | math | {
"ground_truth": "\\frac{16\\sqrt{3}}{\\pi^2}",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the 2020th term of the following sequence:\n$$\n1, 1, 3, 1, 3, 5, 1, 3, 5, 7, 1, 3, 5, 7, 9, 1, 3, 5, 7, 9, 11, \\ldots\n$$",
"role": "user"
}
] | math | {
"ground_truth": "7",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider a configuration involving two unit squares placed adjacent to each other vertically. If triangle ABC is formed such that A = (0, 0), B = (1, 0), and C = (0, 1), find the... | math | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let \\( A = (0, 3), \\) \\( B = (4, 3), \\) and \\( C = (p, q) \\) be three points on the parabola \\( y = x^2 - 4x + 3, \\) where \\( 0 \\leq p \\leq 4 \\). Find the largest pos... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If $f(x, y) = 3x^2 + 3xy + 1$ and $f(a, b) + 1 = f(b, a) = 42$ , then determine $|a + b|$ .",
"role": "user"
}
] | math | {
"ground_truth": "3 \\sqrt{3}",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "$f(x)$ is an odd function with a period of 4, and $f(-1) = 2$. Find $f(13)$.",
"role": "user"
}
] | math | {
"ground_truth": "-2",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The problem is to find the slope angle of the line passing through points A(2,0) and B(5,3).",
"role": "user"
}
] | math | {
"ground_truth": "45^\\circ",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that the product of two positive integers $a$ and $b$ is $161$ after reversing the digits of the two-digit number $a$, find the correct value of the product of $a$ and $b$.... | math | {
"ground_truth": "224",
"style": "rule"
} | {
"id": "numina_amc_aime-0",
"rating": null,
"split": "test",
"sub_source": "amc_aime"
} | amc_aime |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given circle $C$: $(x-3)^2+(y-4)^2=4$, and line $l$ passes through the fixed point $A(1,0)$.\n\n(I) If $l$ is tangent to circle $C$, find the equation of $l$;\n\n(II) If $l$ inte... | math | {
"ground_truth": "y=x-1 \\text{ or } y=7x-7",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathbb{N}_0=\\{0,1,2 \\cdots \\}$ . Does there exist a function $f: \\mathbb{N}__0 \\to \\mathbb{N}_0$ such that:\r\n\\[ f^{2003}(n)=5n, \\forall n \\in \\mathbb{N}_0 ... | math | {
"ground_truth": "f^{2003}(n) = 5n",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider the fraction $\\frac{4a^2 + 2c^2 - 4b^2 - 8bc}{3a^2 + 6ac - 3c^2 - 6ab}$. Is it reducible to a simpler form, and if so, to which of the following?\nA) $\\frac{1}{3} \\fr... | math | {
"ground_truth": "\\frac{4}{3} \\frac{(a-2b+c)(a-c)}{(a-b+c)(a-b-c)}",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Solve the following system of equations:\n\\begin{align*}\n4x + 3y &= 6.4, \\\\\n5x - 6y &= -1.5.\n\\end{align*}\nExpress your answer as an ordered pair $(x,y)$.",
"role": "u... | math | {
"ground_truth": "(0.8692, 0.9744)",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that $\\frac{p}{q} = \\frac{5}{4}$ and $\\frac{s}{u} = \\frac{8}{15}$, find the value of $\\frac{2ps - 3qu}{5qu - 4ps}$.\nA) $-\\frac{5}{7}$ \nB) $\\frac{5}{7}$ \nC) $-\\... | math | {
"ground_truth": "-\\frac{5}{7}",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A cube is divided into $27$ unit cubes. A sphere is inscribed in each of the corner unit cubes, and another sphere is placed tangent to these $8$ spheres. What is the smalles... | math | {
"ground_truth": "\\frac{3\\sqrt{3}}{2} - 1",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given $3\\sin 2\\theta = 4\\tan \\theta$, and $\\theta \\neq k\\pi$ ($k \\in \\mathbb{Z}$), find the value of $\\cos 2\\theta$.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac {1}{3}",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Two (not necessarily different) integers between $1$ and $60$ , inclusive, are chosen independently and at random. What is the probability that their product is a multiple of ... | math | {
"ground_truth": "\\frac{61}{300}",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given a geometric sequence $\\{a_n\\}$, the sum of its first $n$ terms $S_n = 2^{n+1} - 2$,\n\n1. Find the first term and the common ratio of the sequence;\n2. If $b_n = \\log_{2... | math | {
"ground_truth": "\\frac{n(n+1)}{2}",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If an integer of two digits is \\( k \\) times the difference between its digits, find how many times the difference between its digits is the number formed by interchanging the ... | math | {
"ground_truth": "k - 9",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Two spheres with radius $r$ are externally tangent. Three other spheres, each with radius $R$, are mutually externally tangent in pairs. Each of these three spheres is also exter... | math | {
"ground_truth": "R = 6r",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the integral \\(\\int \\frac{dx}{x^{2}-6x-16}\\).",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{1}{10} \\ln \\left| \\frac{x - 8}{x + 2} \\right| + C",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In their base $10$ representations, the integer $a$ consists of a sequence of $1000$ sixes and the integer $b$ consists of a sequence of $1000$ sevens. What is the sum of the dig... | math | {
"ground_truth": "19986",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Evaluate: $(47^2 - 28^2) + 100$",
"role": "user"
}
] | math | {
"ground_truth": "1525",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $A$ be a finite set, and $A_1,A_2,\\cdots, A_n$ are subsets of $A$ with the following conditions:\n(1) $|A_1|=|A_2|=\\cdots=|A_n|=k$ , and $k>\\frac{|A|}{2}$ ;\n(2) f... | math | {
"ground_truth": "n = 11",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the area enclosed between the parabola $y = x^2 - 2$, the line $y = 1$, and the circle $x^2 + y^2 = 9$.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{9\\pi}{4} - 4",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $T_1$ be a triangle with side lengths $1011$, $1012$, and $1013$. For $n \\geq 1$, if $T_n = \\Delta ABC$ and $D, E$, and $F$ are the points of tangency of the incircle of $\... | math | {
"ground_truth": "\\frac{759}{128}",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "After the tourists walked 1 km and half of the remaining distance, they still had to cover a third of the entire distance and 1 km. What is the total distance?",
"role": "use... | math | {
"ground_truth": "9 \\text{ km}",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Two parabolas have equations $y = x^2 + ax + b$ and $y = x^2 + px^2 + cx + d$, where $a, b, c, p,$ and $d$ are integers, $p \\neq 1$. $a, b, c,$ and $d$ are each chosen independe... | math | {
"ground_truth": "\\frac{209}{216}",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that the two foci of a hyperbola are $F_1(- \\sqrt {10},0)$ and $F_2(\\sqrt {10},0)$, $M$ is a point on this hyperbola and satisfies $\\overrightarrow{MF_1} \\cdot \\overri... | math | {
"ground_truth": "\\frac {x^{2}}{9}-y^{2}=1",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "3. Let $ABC$ be a triangle with $AB=30$ , $BC=14$ , and $CA=26$ . Let $N$ be the center of the equilateral triangle constructed externally on side $AB$ . Let $M$ be the... | math | {
"ground_truth": "49 + 145\\sqrt{3}",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If the function \\( f(x) = -\\frac{1}{2} x^{2} + \\frac{13}{2} \\) attains its minimum value of \\( 2a \\) and maximum value of \\( 2b \\) on the interval \\([a, b]\\), determine... | math | {
"ground_truth": "[1, 3] \\text{ or } \\left[-2 - \\sqrt{17}, \\frac{13}{4}\\right]",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the solution to the Cauchy problem: \\( y^{\\prime \\prime}-3 y^{\\prime}+2 y=0 \\), with the initial conditions \\( y(0)=1 \\) and \\( y^{\\prime}(0)=0 \\).",
"role": "... | math | {
"ground_truth": "y(x) = 2 e^{x} - e^{2 x}",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine \\( g(x) \\) if \\[4x^5 + 3x^3 - 2x + 1 + g(x) = 7x^3 - 5x^2 + 4x - 3.\\]",
"role": "user"
}
] | math | {
"ground_truth": "-4x^5 + 4x^3 - 5x^2 + 6x - 4",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Square \\( ABCD \\) has center \\( O \\). Points \\( P \\) and \\( Q \\) are on \\( AB \\), \\( R \\) and \\( S \\) are on \\( BC \\), \\( T \\) and \\( U \\) are on \\( CD \\), ... | math | {
"ground_truth": "1:1",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let the function \\( f(x) = \\sqrt{x^{2} + 1} - a x \\), where \\( a > 0 \\).\n1. Solve the inequality \\( f(x) \\leq 1 \\).\n2. Determine the range of values for \\( a \\) such ... | math | {
"ground_truth": "a \\geq 1",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In the plane rectangular coordinate system $xOy$, given two points $P_{1}(0,2)$ and $P_{2}(4,4)$, find the standard equation of the circle with diameter $P_{1}P_{2}$.",
"role... | math | {
"ground_truth": "(x-2)^{2}+(y-3)^{2}=5",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In $\\triangle ABC$, we have $\\angle A = 90^\\circ$ and $\\tan C = 3$. If $AB = 90$, then what is $AC$?",
"role": "user"
}
] | math | {
"ground_truth": "27\\sqrt{10}",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "\\(ABCD\\) is a square with a side length of \\(20 \\sqrt{5} x\\). \\(P\\) and \\(Q\\) are the midpoints of \\(DC\\) and \\(BC\\) respectively.\n\n1. If \\(AP = ax\\), find \\(a\... | math | {
"ground_truth": "60",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If the ratio of $3x-2y$ to $2x+y$ is $\\frac{5}{4}$, what is the ratio of $y$ to $x$?\nA) $\\frac{1}{7}$ \nB) $\\frac{1}{4}$\nC) $\\frac{2}{13}$\nD) $\\frac{4}{25}$\nE) $\\frac{5... | math | {
"ground_truth": "\\frac{2}{13}",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that exactly two of the nine squares are shaded, determine the number of distinct patterns assuming rotations and reflections are considered equivalent.",
"role": "user... | math | {
"ground_truth": "8",
"style": "rule"
} | {
"id": "numina_amc_aime-0",
"rating": null,
"split": "test",
"sub_source": "amc_aime"
} | amc_aime |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that a town's vehicle license plate numbers consist of four characters, with the first character being a vowel (A, E, I, O, U), the second and third characters being differ... | math | {
"ground_truth": "\\frac{1}{21000}",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that $ABC$ is a triangle with $m(\\widehat{ABC}) = 90^{\\circ}$, the circle with diameter $AB$ intersects the side $[AC]$ at $D$, and the tangent to the circle at $D$ meets... | math | {
"ground_truth": "12",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A circle $ \\Gamma$ is inscribed in a quadrilateral $ ABCD$ . If $ \\angle A\\equal{}\\angle B\\equal{}120^{\\circ}, \\angle D\\equal{}90^{\\circ}$ and $ BC\\equal{}1$ , fi... | math | {
"ground_truth": "\\frac{\\sqrt{3} - 1}{2}",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that $\\sum_{k=1}^{18}\\sin 10k = \\tan \\frac{p}{q},$ where angles are measured in degrees, and $p$ and $q$ are relatively prime positive integers that satisfy $\\frac{p}{... | math | {
"ground_truth": "86",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A circle with its center on the side \\( AC \\) of an isosceles triangle \\( ABC \\) (\\( AB = BC \\)) is tangent to the sides \\( AB \\) and \\( BC \\). Find the radius of the c... | math | {
"ground_truth": "2\\sqrt{3}",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the function $f(x) = e^x + 2x - a$, where $a \\in \\mathbb{R}$, if there exists a point $(x_0, y_0)$ on the curve $y = \\sin x$ such that $f(f(y_0)) = y_0$, then the range ... | math | {
"ground_truth": "[-1 + e^{-1}, e + 1]",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Calculate\n\n$$\n\\frac{1}{1 \\times 2}+\\frac{1}{2 \\times 3}+\\frac{1}{3 \\times 4}+\\ldots+\\frac{1}{2014 \\times 2015}\n$$",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{2014}{2015}",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all positive integers $a$ for which there exist exactly $2014$ positive integers $b$ such that $\\displaystyle2\\leq\\frac{a}{b}\\leq5$ .",
"role": "user"
... | math | {
"ground_truth": "6710, 6712, 6713",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that $\\overrightarrow{m}$ and $\\overrightarrow{n}$ are unit vectors with an angle of $60^{\\circ}$ between them, find the value of $|2\\overrightarrow{m} - \\overrightarr... | math | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "test",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let \\( a_{1} = 3 \\), and for \\( n > 1 \\), let \\( a_{n} \\) be the largest real number such that:\n\\[ 4\\left(a_{n-1}^{2} + a_{n}^{2}\\right) = 10 a_{n-1} a_{n} - 9 \\]\n\nW... | math | {
"ground_truth": "335",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of ordered pairs \\((m, n)\\) of positive integers \\(m\\) and \\(n\\) such that \\(m+n=190\\) and \\(m\\) and \\(n\\) are relatively prime.",
"role": "user"
... | math | {
"ground_truth": "26",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "test",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A closed rectangular box with a base of length $l$ and width $w$ and height $h$ needs to be wrapped with a single piece of square wrapping paper. The wrapping paper must extend 2... | math | {
"ground_truth": "l^2 + 8l + 16 + 4lh + 16h + 4h^2",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "test",
"sub_source": "synthetic_amc"
} | synthetic_amc |
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