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": "Find all functions \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) that satisfy the inequality\n\\[ \nf(y)-\\left(\\frac{z-y}{z-x} f(x)+\\frac{y-x}{z-x} f(z)\\right) \\leq f\\lef... | math | {
"ground_truth": "f(y) = ay^2 + by + c \\text{ where } a \\leq 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": "Given the arithmetic sequence $13, 16, 19, \\dotsc, 70, 73$, calculate the number of terms in this sequence.",
"role": "user"
}
] | math | {
"ground_truth": "21",
"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 a triangular pyramid \\(P-ABC\\) where \\(PA=PB=3\\) and \\(AB=BC=CA=2\\), find the maximum volume of the pyramid.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{2\\sqrt{6}}{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": "A series of lockers, numbered 1 through 100, are all initially closed. Student 1 goes through and opens every locker. Student 3 goes through and \"flips\" every 3rd locker (\"fip... | math | {
"ground_truth": "10",
"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 magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card. A member of... | 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": "In a right triangle $ABC$, $\\angle C=90^{\\circ}$, and $\\cos B = \\frac{1}{2}$. Determine the measure of $\\angle B$ in degrees.",
"role": "user"
}
] | math | {
"ground_truth": "60^{\\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": "Triangle $OAB$ has $O=(0,0)$, $B=(5,0)$, and $A$ in the first quadrant. In addition, $\\angle ABO=90^\\circ$ and $\\angle AOB=30^\\circ$. Suppose that $OA$ is rotated $90^\\circ$... | math | {
"ground_truth": "\\left(-\\frac{5\\sqrt{3}}{3}, 5\\right)",
"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 the notation $[a,b,c]$ defined as $\\frac {a+b}c$, where $c \\neq 0$, evaluate the expression $\\left[[80,40,120],[4,2,6],[50,25,75]\\right]$.",
"role": "user"
}
] | math | {
"ground_truth": "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": "Given the club has a total of 50 students, and a portion of the juniors and seniors are part of a special project, where 40% of the juniors and 20% of the seniors are part of the... | math | {
"ground_truth": "25",
"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 farmer had an enclosure with a fence 50 rods long, which could only hold 100 sheep. Suppose the farmer wanted to expand the enclosure so that it could hold twice as many sheep.... | math | {
"ground_truth": "21",
"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 equation of the circle that passes through point M (-1, 1) and has the same center as the given circle C: $x^2 + y^2 - 4x + 6y - 3 = 0$.",
"role": "user"
}
] | math | {
"ground_truth": "(x - 2)^2 + (y + 3)^2 = 25",
"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 \\(\\text{lcm}(x,y) = 180\\), \\(\\text{lcm}(x,z) = 504\\), and \\(\\text{lcm}(y,z) = 1260\\), find the number of ordered triples \\((x,y,z)\\) of positive integers that sa... | math | {
"ground_truth": "6",
"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 an isosceles triangle, if one angle is $80^{\\circ}$, find the measure of its vertex angle.",
"role": "user"
}
] | math | {
"ground_truth": "20^{\\circ} \\text{ or } 80^{\\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": "The solutions to the equation \\( (z-4)^8 = 256 \\) form a convex regular polygon in the complex plane, where three of its vertices are labeled \\( D, E, \\) and \\( F \\). What ... | math | {
"ground_truth": "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": "Given a sequence $\\{a_n\\}$ where each term is an integer and satisfies $a_n^2 \\leqslant 1$, $1 \\leqslant a_1^2 + a_2^2 + \\ldots + a_n^2 \\leqslant m$, where $m$, $n \\in \\m... | math | {
"ground_truth": "6",
"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": "There is more than one integer greater than $1$ which, when divided by any integer $k$ such that $2 \\le k \\le 7$, has a remainder of $2$. What is the difference between the two... | math | {
"ground_truth": "- **B)** 420",
"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 peasant woman sold eggs to two buyers: to the first one, she sold $\\frac{1}{3}$ of the eggs she had along with 15 more eggs, and to the second one, she sold $\\frac{7}{9}$ of ... | math | {
"ground_truth": "90 \\text{ eggs}",
"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 right triangle $ABC$, we have $\\angle B = \\angle C$ and $AC = 10\\sqrt{2}$. What is the area of $\\triangle ABC$?",
"role": "user"
}
] | math | {
"ground_truth": "50",
"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 $n$ be an integer larger than $1$ and let $S$ be the set of $n$ -element subsets of the set $\\{1,2,\\ldots,2n\\}$ . Determine\n\\[\\max_{A\\in S}\\left (\\min_{x,y\\... | math | {
"ground_truth": "24",
"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 all positive integers $n$ such that we can divide the set $\\{1,2,3,\\ldots,n\\}$ into three sets with the same sum of members.",
"role": "user"
}
] | math | {
"ground_truth": "n \\equiv 0 \\pmod{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": "Given that $\\cos(75^\\circ + \\alpha) = \\frac{1}{2}$, where $\\alpha$ is an angle in the third quadrant, find the value of $\\cos(105^\\circ - \\alpha) + \\sin(\\alpha - 105^\\... | math | {
"ground_truth": "\\frac{1}{2} + \\frac{\\sqrt{3}}{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": "Vova covered the distance of 300 meters from school to home by taking 400 steps, with the length of each step being no more than \\(a\\) cm, and the sum of the lengths of any two... | math | {
"ground_truth": "a \\in \\left[75, \\frac{10000}{133}\\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": "Square $PQRS$ lies in the first quadrant. Points $(3,0), (5,0), (7,0),$ and $(13,0)$ lie on lines $SP, RQ, PQ$, and $SR$, respectively. What is the sum of the coordinates of the ... | math | {
"ground_truth": "\\frac{32}{5}",
"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": "On a peculiar bicycle, the front wheel has a radius of 3 feet, and the back wheel has a radius of 5 inches. If there is no slippage, how many revolutions will the back wheel make... | math | {
"ground_truth": "1080",
"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 number $x$ has the property that $x\\%$ of $x$ is $4$. Solve for $x$.",
"role": "user"
}
] | math | {
"ground_truth": "20",
"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": "Points $A$ and $B$ are fixed points in the plane such that $AB = 1$ . Find the area of the region consisting of all points $P$ such that $\\angle APB > 120^o$",
"role... | math | {
"ground_truth": "\\frac{4\\pi - 3\\sqrt{3}}{18}",
"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": "Define the function $f(n)$ as follows:\n\\[ f(n) = \\left\\{\n\\begin{array}{cl}\nn^2 + 2n & \\text{ if } n < -1, \\\\\ne^n - 10 & \\text{ if } n \\geq -1.\n\\end{array}\n\\right... | math | {
"ground_truth": "5.772",
"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 particle moves so that its speed for the second and subsequent miles varies inversely as the integral number of miles already traveled. If the second mile is traversed in $2$ h... | math | {
"ground_truth": "2(n-1)",
"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 the equation of a circle $x^2 + y^2 - 6x - 8y = 0$, if the longest chord AC and the shortest chord BD both pass through the point (-1, 4) on the circle, find the area of th... | math | {
"ground_truth": "30",
"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 all positive integers that can be written in the following way $\\frac{b}{a}+\\frac{c}{a}+\\frac{c}{b}+\\frac{a}{b}+\\frac{a}{c}+\\frac{b}{c}$ .\nAlso, $a,b, c$ are posi... | math | {
"ground_truth": "6, 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": "When studying the operation of a new type of cyclic thermal engine, it was found that during part of the period it receives heat, and the absolute power of heat supply is express... | math | {
"ground_truth": "\\frac{1}{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": "Hey, \r\n\r\nThis problem is from the VTRMC 2006. \r\n\r\n3. Recall that the Fibonacci numbers $ F(n)$ are defined by $ F(0) \\equal{} 0$ , $ F(1) \\equal{} 1$ and $ F(n) \... | math | {
"ground_truth": "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": "Find the number of ordered triples of positive integers $(a, b, c)$ , where $a, b,c$ is a strictly increasing arithmetic progression, $a + b + c = 2019$ , and there is a tria... | math | {
"ground_truth": "672",
"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": "Evaluate $\\sqrt{20 + \\sqrt{20 + \\sqrt{20 + \\sqrt{20 + \\cdots}}}}$.",
"role": "user"
}
] | math | {
"ground_truth": "x = 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": "A classroom has $30$ seats arranged into $5$ rows of $6$ seats. Thirty students of distinct heights come to class every day, each sitting in a random seat. The teacher stan... | math | {
"ground_truth": "\\frac{163}{10}",
"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": "Solve the equation:\n\na) $\\log _{3}(x-2)+\\log _{3} x=\\log _{3} 8$;\n\nb) $\\lg (x-9)+2 \\lg \\sqrt{2 x-1}=2$;\n\nc) $\\frac{1}{2} \\log _{5}(x+5)+\\log _{5} \\sqrt{x-3}=\\fra... | math | {
"ground_truth": "4",
"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 all pairs of integers $(p,q)$ for which all roots of the trinomials $x^2+px+q$ and $x^2+qx+p$ are integers.",
"role": "user"
}
] | math | {
"ground_truth": "(4, 4), (9, 8), (8, 9)",
"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": "There are two urns containing balls of three colors. The first urn contains 2 blue, 3 red, and 5 green balls, while the second urn contains 4 blue, 2 red, and 4 green balls. One ... | math | {
"ground_truth": "0.34",
"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 $f(x)=(x^2+3x+2)^{\\cos(\\pi x)}$. Find the sum of all positive integers $n$ for which \n\\[\\left |\\sum_{k=1}^n\\log_{10}f(k)\\right|=1.\\]",
"role": "user"
}
] | math | {
"ground_truth": "21",
"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": "On the table stand three cones on their bases, touching each other. The heights of the cones are the same, and the radii of their bases are 1, 2, and 3. A sphere is placed on the... | math | {
"ground_truth": "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": "How many squares whose sides are parallel to the axes and whose vertices have integer coordinates lie entirely within the region bounded by the line $y=\\pi x$, the line $y=-0.1$... | math | {
"ground_truth": "50",
"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": "In a revival competition, two contestants, A and B, are vying for the final spot in the finals based on the number of votes they receive. The number of voters is fixed, and each ... | math | {
"ground_truth": "\\text{Candidate A: 147 votes, Candidate B: 140 votes or Candidate A: 126 votes, Candidate B: 120 votes}",
"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": "The equation of line $l$ passes through the point $(3,4)$ and is equidistant from points $A(-2,2)$ and $B(4,-2)$. Find the equation of line $l$.",
"role": "user"
}
] | math | {
"ground_truth": "2x - y - 2 = 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": "Fill in each blank with a digit not equal to 1, such that the equation holds true. How many different ways can this be done?\n$$\n[\\mathrm{A} \\times(\\overline{1 \\mathrm{~B}}+... | math | {
"ground_truth": "8",
"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 sequence of real numbers $ x_0, x_1, x_2, \\ldots$ is defined as follows: $ x_0 \\equal{} 1989$ and for each $ n \\geq 1$ \r\n\r\n\\[ x_n \\equal{} \\minus{} \\frac{1989}{... | math | {
"ground_truth": "-1989",
"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": "Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $105, Dorothy paid $125, and Sammy paid $175. In order to share costs ... | math | {
"ground_truth": "20",
"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": "Points $E$ and $F$ lie on line segment $\\overline{GH}$. The length of $\\overline{GE}$ is $3$ times the length of $\\overline{EH}$, and the length of $\\overline{GF}$ is $7$ tim... | math | {
"ground_truth": "\\textbf{(D)}\\ \\frac{1}{8}",
"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 vectors $\\overrightarrow{m}=(\\sin x,1), \\overrightarrow{n}=(\\sqrt{3}\\cos x,\\frac{1}{2}\\cos 2x)$, and the function $f(x)=\\overrightarrow{m}\\cdot\\overrightarrow{n}$... | 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": "The $\\emph{Stooge sort}$ is a particularly inefficient recursive sorting algorithm defined as follows: given an array $A$ of size $n$ , we swap the first and last elements ... | math | {
"ground_truth": "243",
"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": "In a scalene triangle one angle is exactly two times as big as another one and some angle in this triangle is $36^o$ . Find all possibilities, how big the angles of this triangl... | math | {
"ground_truth": "(36^\\circ, 48^\\circ, 96^\\circ)",
"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 two coprime numbers \\(a\\) and \\(b\\) that satisfy the equation:\n\n$$\n\\frac{5a + 7b}{6a + 5b} = \\frac{29}{28}\n$$",
"role": "user"
}
] | math | {
"ground_truth": "a = 3, \\quad b = 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": "Given the function \\( f(x) = a x^{2} - c \\) satisfies:\n\\[ -4 \\leqslant f(1) \\leqslant -1 \\]\n\\[ -1 \\leqslant f(2) \\leqslant 5 \\]\nDetermine the range of values for \\(... | math | {
"ground_truth": "-1 \\leq f(3) \\leq 20",
"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 constants $a_1$ and $a_2$ such that\n\\[\na_1 \\begin{pmatrix} 3 \\\\ 4 \\end{pmatrix} + a_2 \\begin{pmatrix} -3 \\\\ 7 \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ 5 \\end{... | math | {
"ground_truth": "\\left(\\frac{5}{11}, \\frac{5}{11}\\right)",
"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": "Express the number $10,030,000,000$ in scientific notation.",
"role": "user"
}
] | math | {
"ground_truth": "1.003 \\times 10^{10}",
"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 value of $(\\sqrt{81} - \\sqrt{77}) + 1$.",
"role": "user"
}
] | math | {
"ground_truth": "1.225",
"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 lamp post is supported by a steel cable which extends from the top of the lamp post to a point on the ground 4 meters away from its base. When Mike walks 3 meters away from the... | math | {
"ground_truth": "6.4",
"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": "An artist has $19$ cubes, each with an edge length of $1$ meter. He arranges them to form a pyramid-like sculpture with a broader base: 1 cube on the top layer, 6 cubes on the mi... | math | {
"ground_truth": "C) 35 square meters",
"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 \\(41 \\times 41\\) battlefield is divided into 1681 unit squares. A tank is hidden in one of these unit squares. A bomber conducts airstrikes on the battlefield, bombing one u... | math | {
"ground_truth": "2521",
"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 functions $f(x)=mx-\\frac{m}{x}$ and $g(x)=3\\ln x$, \n(1) When $m=4$, find the equation of the tangent line to the curve $y=f(x)$ at the point $(2,f(2))$; \n(2) If f... | math | {
"ground_truth": "m \\in (-\\infty,\\frac{9\\sqrt{e}}{2(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": "Consider a sequence of numbers where each integer \\( n \\) from 1 to \\( 150 \\) appears \\( n \\) times in the list: \\[1, 2, 2, 3, 3, 3, \\ldots, 150, 150, \\ldots, 150\\]. Wh... | math | {
"ground_truth": "C) 106",
"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 line is parameterized by a parameter $t$, so that the vector on the line at $t = 1$ is $\\begin{pmatrix} 2 \\\\ 5 \\end{pmatrix},$ and the vector on the line at $t = 4$ is $\\b... | math | {
"ground_truth": "\\begin{pmatrix} 1 \\\\ 9 \\end{pmatrix}",
"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": "How many numbers are in the list $5, 8, 11, \\ldots, 164, 167$?",
"role": "user"
}
] | math | {
"ground_truth": "55",
"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": "**p1.** Given $x + y = 7$ , find the value of x that minimizes $4x^2 + 12xy + 9y^2$ .**p2.** There are real numbers $b$ and $c$ such that the only $x$ -intercept of $8y =... | math | {
"ground_truth": "6",
"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": "The mass of a container completely filled with kerosene is 20 kg. If this container is completely filled with water, its mass will be 24 kg. Determine the mass of the empty conta... | math | {
"ground_truth": "4 \\ \\text{kg}",
"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 \\(k\\) such that the polynomial \\(4x^3 - 8x^2 + kx - 16\\) is divisible by \\(x - 2\\). Then, identify which of the given options is also a factor of the polynomial.\... | math | {
"ground_truth": "The correct answer is **D) \\(x^2 + 1\\)**.",
"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": "Find the values of $p,\\ q,\\ r\\ (-1<p<q<r<1)$ such that for any polynomials with degree $\\leq 2$ , the following equation holds:\n\n\\[\\int_{-1}^p f(x)\\ dx-\\int_p^q f(x)\... | math | {
"ground_truth": "p = \\frac{1}{\\sqrt{2}}, q = 0, r = -\\frac{1}{\\sqrt{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": "There are constants $\\alpha$ and $\\beta$ such that $\\frac{x-\\alpha}{x+\\beta} = \\frac{x^2-96x+2209}{x^2+66x-3969}$. What is $\\alpha+\\beta$?",
"role": "user"
}
] | math | {
"ground_truth": "128",
"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": "Come up with five different natural numbers whose product is equal to 1000.",
"role": "user"
}
] | math | {
"ground_truth": "1, 2, 4, 5, 25",
"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 five test scores have a mean (average score) of $90$, a median (middle score) of $91$ and a mode (most frequent score) of $94$, find the sum of the two lowest test scores."... | math | {
"ground_truth": "171",
"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 $\\frac{2^{2001}\\cdot3^{2003}}{6^{2002}}$, simplify the expression.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{3}{2}",
"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": "We have a rectangular prism, one of whose edges is five times longer than another of its edges. If we were to increase the height of the prism by $2 \\mathrm{~cm}$, its volume wo... | math | {
"ground_truth": "0.9 \\ \\text{cm} \\times 50 \\ \\text{cm} \\times 10 \\ \\text{cm},\\ \\ 2 \\ \\text{cm} \\times 22.5 \\ \\text{cm} \\times 10 \\ \\text{cm},\\ \\ 3 \\ \\text{cm} \\times 15 \\ \\text{cm} \\times 10 \\ \\text{cm}",
"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": "Grogg and Winnie are playing a game using a deck of $50$ cards numbered $1$ through $50$ . They take turns with Grogg going first. On each turn a player chooses a card from ... | math | {
"ground_truth": "75",
"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 $a,$ $b,$ and $c$ be nonnegative real numbers such that $a^2 + b^2 + 2c^2 = 1.$ Find the maximum value of:\n\\[ ab\\sqrt{3} + 3bc. \\]",
"role": "user"
}
] | math | {
"ground_truth": "\\sqrt{7}",
"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 the following instructions are carried out by a computer, which value of $X$ will be printed due to instruction 5?\n\n1. START $X$ AT $5$ AND $S$ AT $0$. \n2. INCREASE THE VA... | math | {
"ground_truth": "C) 299",
"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 three positive numbers \\( p \\), \\( q \\), and \\( M \\) such that \\( q < p \\), the number derived by first increasing \\( M \\) by \\( p\\% \\) and then increasing the... | math | {
"ground_truth": "\\(\\textbf{(C)}\\ p > 0 \\text{ and } q > 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": "Let \\( n \\) be a natural number and \\( p \\) a prime number.\n\nShow using Legendre's formula (Exercise 8)\n\n\\[ \nv_{p}\\left(\\binom{2n}{n}\\right) = \\sum_{k=1}^{\\left\\l... | math | {
"ground_truth": "v_p\\left(\\binom{2n}{n}\\right) \\leq \\left\\lfloor \\frac{\\ln (2n)}{\\ln p} \\right\\rfloor",
"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 four primes $p, q, s$, and $r$ satisfy $p + q + s = r$ and $2 < p < q < s$, determine the smallest value of $p$.",
"role": "user"
}
] | math | {
"ground_truth": "3",
"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 diameter of a circle is divided into n equal parts. On each part, a quarter circle is constructed. As n becomes very large, find the sum of the lengths of the arcs of the qua... | math | {
"ground_truth": "\\frac{\\pi D}{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": "Find the greatest value of $x$ such that $-x^2 + 9x - 20 \\geq 0$ and determine if this quadratic has real roots.",
"role": "user"
}
] | 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": "Let sequence \\( b_1, b_2, \\ldots \\) be defined by the following properties:\n\n(i) \\( b_1 = 2 \\),\n\n(ii) \\( b_{2n} = n \\cdot b_n \\) for any positive integer \\( n \\),\n... | math | {
"ground_truth": "2^{101}",
"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 $f(x)=25x+4$. Find the sum of all $x$ that satisfy the equation $f^{-1}(x)=f(x^{-2})$.",
"role": "user"
}
] | math | {
"ground_truth": "104",
"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": "Three positive integers $x$, $y$, and $z$ satisfy $x\\cdot y\\cdot z=9!$ and $x<y<z$. What is the smallest possible value of $z-x$?",
"role": "user"
}
] | math | {
"ground_truth": "396",
"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 area of the triangle formed by the lines $y = 4$, $y = 2 + x$, and $y = 2 - x$, can be determined by calculating the appropriate expression for area.",
"role": "user"
}... | math | {
"ground_truth": "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": "Find the number of non-negative integer solutions \\(\\left(x_{1}, x_{2}, \\cdots, x_{25}\\right)\\) that satisfy the equation\n$$\nx_{1}^{2} + x_{2}^{2} + \\cdots + x_{25}^{2} =... | math | {
"ground_truth": "29900",
"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": "Arrange 10 balls labeled with the numbers 1 to 10 in a row such that the sum of the numbers on any three consecutive balls is a multiple of 3. How many ways can this be done?",
... | math | {
"ground_truth": "1728",
"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": "Three fair coins are tossed once. For each head that results, one fair die is rolled. What is the probability that the sum of the die rolls is even?\nA) $\\frac{27}{64}$\nB) $\\f... | math | {
"ground_truth": "\\frac{37}{64}",
"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 ${\\cal C}_1$ and ${\\cal C}_2$ be concentric circles, with ${\\cal C}_2$ in the interior of ${\\cal C}_1$. From a point $A$ on ${\\cal C}_1$ one draws the tangent $AB$ to ... | math | {
"ground_truth": "\\frac{5}{3}",
"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": "Find $r$ such that $\\log_{64} (3r-2) = -\\frac{1}{3}$.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{3}{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": "In triangle $ABC$, the sides $AB$ and $AC$ are 5 inches and 8 inches, respectively. The median $AM$ from vertex $A$ to side $BC$ measures 5 inches. Additionally, the height from ... | math | {
"ground_truth": "\\text{B}) \\sqrt{78} \\text{ inches}",
"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": "Observe the pattern in the following sequence of numbers and fill in the blank: 0, 3, 8, 15, 24, …, then the 2002nd number is ___.",
"role": "user"
}
] | math | {
"ground_truth": "4008003",
"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 derivative.\n\n$$\ny=\\frac{5^{x}(\\sin 3 x \\cdot \\ln 5-3 \\cos 3 x)}{9+\\ln ^{2} 5}\n$$",
"role": "user"
}
] | math | {
"ground_truth": "5^x \\sin 3x",
"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": "On graph paper, a right-angled triangle with legs of 7 units each is drawn. Then all the grid lines located inside the triangle were traced. What is the maximum number of triangl... | math | {
"ground_truth": "28",
"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 the complex sum \\[\\sin{30^\\circ} + i\\sin{150^\\circ} + \\cdots + i^n\\sin{(30 + 120n)^\\circ} + \\cdots + i^{20}\\sin{2430^\\circ}\\]\nDetermine the value of this s... | math | {
"ground_truth": "\\frac{5}{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": "A 0-1 sequence of length $2^k$ is given. Alice can pick a member from the sequence, and reveal it (its place and its value) to Bob. Find the largest number $s$ for which Bob ... | math | {
"ground_truth": "s = k+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": "Tulips cost $4 each, and lilies cost $6 each. The club collects $120 and wants to purchase exactly 25 flowers. Determine the number of ways they can arrange the number of tulips ... | math | {
"ground_truth": "(15, 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 splitting of a planar polygon is a fi\fnite set of triangles whose interiors are pairwise disjoint, and whose union is the polygon in question. Given an integer $n \\ge 3$ , d... | math | {
"ground_truth": "\\lceil \\frac{n}{3} \\rceil",
"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 cone is formed from a 270-degree sector of a circle with a radius of 20 units by aligning the two straight edges. Calculate the result of dividing the volume of the cone by $\\... | math | {
"ground_truth": "1125\\sqrt{7}",
"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 standard equation of the circle whose center lies on the parabola $x^2=2y$ and which is tangent to the directrix of the parabola and the y-axis.",
"role": "user"
}... | math | {
"ground_truth": "(x+1)^2+\\left(y-\\frac{1}{2}\\right)^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": "Calculate the minimum number of digits to the right of the decimal point needed to express the fraction $\\frac{987654321}{2^{30}\\cdot 5^5}$ as a decimal.\nA) 10\nB) 25\nC) 30\n... | math | {
"ground_truth": "C) 30",
"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 radius of the circle is $\\sqrt{50}$ cm, the length of $AB$ is 6 cm, and that of $BC$ is 2 cm. Find the square of the distance from $B$ to the center of the circle.",
"ro... | math | {
"ground_truth": "26",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "test",
"sub_source": "aops_forum"
} | aops_forum |
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