data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "A cube has an edge length of 30. A rectangular solid has edge lengths 20, 30, and $L$. If the cube and the rectangular solid have equal surface areas, what is the value of $L$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 300,
"solution": "The cube has six identical square faces, each of which is 30 by 30. Therefore, the surface area of the cube is $6(30^{2})=5400$. The rectangular solid has two faces that are 20 by 30, two faces that are 20 by $L$, and two faces that are 30 by $L$. Thus, the surface a... |
deepscaler | [
{
"role": "user",
"content": "Find the integer $n,$ $-90 < n < 90,$ such that $\\tan n^\\circ = \\tan 312^\\circ.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-48"
} | {
"split": "train",
"index": 301,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A graph has $ 30$ vertices, $ 105$ edges and $ 4822$ unordered edge pairs whose endpoints are disjoint. Find the maximal possible difference of degrees of two vertices in this graph. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 302,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The shaded region formed by the two intersecting perpendicular rectangles, in square units, is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 303,
"solution": "1. **Identify the areas of the individual rectangles**: \n - The first rectangle has dimensions $2 \\times 10$. Therefore, its area is:\n \\[\n 2 \\cdot 10 = 20 \\text{ square units}\n \\]\n - The second rectangle has dimensions $3 \\times 8$. Therefo... |
deepscaler | [
{
"role": "user",
"content": "In the diagram below, lines $k$ and $\\ell$ are parallel. Find the measure of angle $x$ in degrees. [asy]\nsize(200);\nimport markers;\npair A = dir(-22)*(0,0);\npair B = dir(-22)*(4,0);\npair C = dir(-22)*(4,2);\npair D = dir(-22)*(0,2);\npair F = dir(-22)*(0,1.3);\npair G = ... | math | {
"style": "rule",
"ground_truth": "150^\\circ"
} | {
"split": "train",
"index": 304,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangle is divided into four smaller rectangles, labelled W, X, Y, and Z. The perimeters of rectangles W, X, and Y are 2, 3, and 5, respectively. What is the perimeter of rectangle Z? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 305,
"solution": "Label the lengths of the vertical and horizontal segments as $a, b, c, d$. Rectangle W is $b$ by $c$, so its perimeter is $2 b+2 c$, which equals 2. Rectangle X is $b$ by $d$, so its perimeter is $2 b+2 d$, which equals 3. Rectangle Y is $a$ by $c$, so its perimeter ... |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be angles such that $\\sin (a + b) = \\frac{3}{4}$ and $\\sin (a - b) = \\frac{1}{2}.$ Find $\\frac{\\tan a}{\\tan b}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 306,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\sin 2x + \\sqrt{3}\\cos 2x$, stretch the x-coordinates of all points on the graph to twice their original length, and then shift all points on the graph to the right by $\\frac{\\pi}{6}$ units, and find the equation of one of the axes of symmetry fo... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 307,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At a meeting with $25$ businessmen, twelve businessmen drank coffee, and ten businessmen drank tea. Five businessmen drank both coffee and tea. How many businessmen drank neither coffee nor tea? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 308,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $\\csc 225^\\circ.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\sqrt{2}"
} | {
"split": "train",
"index": 309,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "**Problem Statement**: Let $r$ and $k$ be integers such that $-5 < r < 8$ and $0 < k < 10$. What is the probability that the division $r \\div k$ results in an integer value? Express your answer as a common fraction. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "\\frac{33}{108}"
} | {
"split": "train",
"index": 310,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the sum \\[\\lfloor \\sqrt{1} \\rfloor + \\lfloor \\sqrt{2} \\rfloor + \\lfloor \\sqrt{3} \\rfloor + \\cdots + \\lfloor \\sqrt{25} \\rfloor.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 311,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a triangle in the plane with $AB=13, BC=14, AC=15$. Let $M_{n}$ denote the smallest possible value of $\\left(AP^{n}+BP^{n}+CP^{n}\\right)^{\\frac{1}{n}}$ over all points $P$ in the plane. Find $\\lim _{n \\rightarrow \\infty} M_{n}$. Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "\\frac{65}{8}"
} | {
"split": "train",
"index": 312,
"solution": "Let $R$ denote the circumradius of triangle $ABC$. As $ABC$ is an acute triangle, it isn't hard to check that for any point $P$, we have either $AP \\geq R, BP \\geq R$, or $CP \\geq R$. Also, note that if we choose $P=O$ (the circumcenter) then $\\left(AP^{n}+BP^{n}... |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system xOy, let there be a line $l$ with an inclination angle $\\alpha$ given by $$l: \\begin{cases} x=2+t\\cos\\alpha \\\\ y= \\sqrt {3}+t\\sin\\alpha \\end{cases}$$ (where $t$ is a parameter) that intersects with the curve $$C: \\begin{cases} x=2\\... | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {5}}{4}"
} | {
"split": "train",
"index": 313,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine $\\sqrt[6]{1061520150601}$ without a calculator. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "101"
} | {
"split": "train",
"index": 314,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a $3 \\times 3$ grid (like a Tic-Tac-Toe board), four randomly selected cells have been randomly placed with four tokens.\n\nFind the probability that among these four tokens, three are aligned in a row either vertically, horizontally, or diagonally. Let's think step by s... | math | {
"style": "rule",
"ground_truth": "8/21"
} | {
"split": "train",
"index": 315,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N$ denote the number of permutations of the $15$-character string $AAAABBBBBCCCCCC$ such that\nNone of the first four letters is an $A$.\nNone of the next five letters is a $B$.\nNone of the last six letters is a $C$.\nFind the remainder when $N$ is divided by $1000$.\n Le... | math | {
"style": "rule",
"ground_truth": "320"
} | {
"split": "train",
"index": 316,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the L-shaped region formed by three unit squares joined at their sides, as shown below. Two points $A$ and $B$ are chosen independently and uniformly at random from inside the region. The probability that the midpoint of $\\overline{AB}$ also lies inside this L-shaped ... | math | {
"style": "rule",
"ground_truth": "035"
} | {
"split": "train",
"index": 317,
"solution": "Consider this diagram:\n First, the one of points must be in the uppermost box and the other in the rightmost box. This happens with probability 2/3*1/3=2/9.\n We need the midpoints of the $x$ coordinates to be greater than $1$ but less than $2.$\n We need the midpoi... |
deepscaler | [
{
"role": "user",
"content": "Find the smallest number in which all digits are different and the sum of all digits equals 32. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26789"
} | {
"split": "train",
"index": 318,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bird discovered $543_{8}$ different ways to build a nest in each of its eight tree homes. How many ways are there in base 10? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "355"
} | {
"split": "train",
"index": 319,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two arithmetic sequences $\\{a_n\\}$ and $\\{b_n\\}$, the sum of the first $n$ terms of each sequence is denoted as $S_n$ and $T_n$ respectively. If $$\\frac {S_{n}}{T_{n}}= \\frac {7n+45}{n+3}$$, and $$\\frac {a_{n}}{b_{2n}}$$ is an integer, then the value of $n$ is \\_\... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 320,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The values of a function $f(x)$ are given below:\n\n\\begin{tabular}{|c||c|c|c|c|c|} \\hline $x$ & 3 & 4 & 5 & 6 & 7 \\\\ \\hline $f(x)$ & 10 & 17 & 26 & 37 & 50 \\\\ \\hline \\end{tabular}Evaluate $f^{-1}\\left(f^{-1}(50)\\times f^{-1}(10)+f^{-1}(26)\\right)$. Let's think step... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 321,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A dealer plans to sell a new type of air purifier. After market research, the following pattern was discovered: When the profit per purifier is $x$ (unit: Yuan, $x > 0$), the sales volume $q(x)$ (unit: hundred units) and $x$ satisfy the following relationship: If $x$ does not e... | math | {
"style": "rule",
"ground_truth": "240000"
} | {
"split": "train",
"index": 322,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a convex pentagon, all diagonals are drawn. For each pair of diagonals that intersect inside the pentagon, the smaller of the angles between them is found. What values can the sum of these five angles take? Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "180"
} | {
"split": "train",
"index": 323,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the probability that Robbie will win if he and Francine each roll a special six-sided die three times, and after two rolls each, Robbie has a score of 8 and Francine has a score of 10? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{55}{441}"
} | {
"split": "train",
"index": 324,
"solution": "Robbie has a score of 8 and Francine has a score of 10 after two rolls each. Thus, in order for Robbie to win (that is, to have a higher total score), his third roll must be at least 3 larger than that of Francine. If Robbie rolls 1, 2 or 3, his roll cannot be 3 larg... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sin \\alpha = \\frac{4}{5}$ and $\\alpha$ is an angle in the second quadrant, find the value of $\\cot (\\frac{\\pi}{4} - \\frac{\\alpha}{2})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 325,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two right triangles share a side as follows: Triangle ABC and triangle ABD have AB as their common side. AB = 8 units, AC = 12 units, and BD = 8 units. There is a rectangle BCEF where point E is on line segment BD and point F is directly above E such that CF is parallel to AB. ... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 326,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest prime number $p$ such that $n^{2}+n+11$ is divisible by $p$ for some integer $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 327,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $1,2,2,2,2,1,2,2,2,2,2,1,2,2,2,2,2, \\cdots$ where the number of 2s between consecutive 1s increases by 1 each time, calculate the sum of the first 1234 terms. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2419"
} | {
"split": "train",
"index": 328,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute \\[ \\left\\lfloor \\dfrac {1007^3}{1005 \\cdot 1006} - \\dfrac {1005^3}{1006 \\cdot 1007} + 5 \\right\\rfloor,\\] where $\\lfloor x \\rfloor$ denotes the greatest integer less than or equal to $x.$ Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 329,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the sum of all 2-digit prime numbers $p$ such that there exists a prime number $q$ for which $100 q+p$ is a perfect square. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "179"
} | {
"split": "train",
"index": 330,
"solution": "All squares must end with $0,1,4,5,6$, or 9, meaning that $p$ must end with 1 and 9. Moreover, since all odd squares are $1 \\bmod 4$, we know that $p$ must be $1 \\bmod 4$. This rules all primes except for $41,61,29,89$. Since $17^{2}=289,19^{2}=361,23^{2}=529,89,61... |
deepscaler | [
{
"role": "user",
"content": "Let $U$ be a square with side length 1. Two points are randomly chosen on the sides of $U$. The probability that the distance between these two points is at least $\\frac{1}{2}$ is $\\frac{a - b \\pi}{c}\\left(a, b, c \\in \\mathbf{Z}_{+}, (a, b, c)=1\\right)$. Find the value o... | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 331,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Knights, who always tell the truth, and liars, who always lie, live on an island. One day, 30 inhabitants of this island sat around a round table. Each of them said one of two phrases: \"My neighbor on the left is a liar\" or \"My neighbor on the right is a liar.\" What is the ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 332,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A 3 by 2 rectangle is split into four congruent right-angled triangles. Those four triangles are rearranged to form a rhombus. What is the ratio of the perimeter of the rectangle to the perimeter of the rhombus? Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "1: 1"
} | {
"split": "train",
"index": 333,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a quadratic function $f(x) = ax^2 - 4bx + 1$.\n\n(1) Let set $P = \\{-1,1,2,3,4,5\\}$ and set $Q = \\{-2,-1,1,2,3,4\\}$. Randomly select a number from set $P$ as $a$ and from set $Q$ as $b$. Calculate the probability that the function $y = f(x)$ is increasing on the inter... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{3}"
} | {
"split": "train",
"index": 334,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five cards have the numbers 101, 102, 103, 104, and 105 on their fronts. On the reverse, each card has one of five different positive integers: \\(a, b, c, d\\), and \\(e\\) respectively. We know that:\n\n1. \\(c = b \\cdot e\\)\n2. \\(a + b = d\\)\n3. \\(e - d = a\\)\n\nFranki... | math | {
"style": "rule",
"ground_truth": "103"
} | {
"split": "train",
"index": 335,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is necessary to erect some public welfare billboards on one side of a road. The first billboard is erected at the beginning of the road, and then one billboard is erected every 5 meters, so that exactly one billboard can be erected at the end of the road. In this case, there... | math | {
"style": "rule",
"ground_truth": "1100"
} | {
"split": "train",
"index": 336,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three positive reals $x , y , z $ satisfy $x^2 + y^2 = 3^2 \ny^2 + yz + z^2 = 4^2 \nx^2 + \\sqrt{3}xz + z^2 = 5^2 .$ \nFind the value of $2xy + xz + \\sqrt{3}yz$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 337,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the roots of the polynomial $81x^3 - 162x^2 + 81x - 8 = 0$ are in arithmetic progression, find the difference between the largest and smallest roots. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{6}}{9}"
} | {
"split": "train",
"index": 338,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that vectors $\\overrightarrow{α}$ and $\\overrightarrow{β}$ are two mutually perpendicular unit vectors in a plane, and $(5\\overrightarrow{α} - 2\\overrightarrow{γ}) \\cdot (12\\overrightarrow{β} - 2\\overrightarrow{γ}) = 0$, find the maximum value of $|\\overrightarrow... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{2}"
} | {
"split": "train",
"index": 339,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If\n\\[\n\\begin{pmatrix}\n1 & 3 & b \\\\\n0 & 1 & 5 \\\\\n0 & 0 & 1\n\\end{pmatrix}^m = \\begin{pmatrix}\n1 & 33 & 6006 \\\\\n0 & 1 & 55 \\\\\n0 & 0 & 1\n\\end{pmatrix},\n\\]\nthen find $b + m.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "432"
} | {
"split": "train",
"index": 340,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify \\[\\frac{1}{\\dfrac{3}{\\sqrt{5}+2} + \\dfrac{4}{\\sqrt{7}-2}}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{9\\sqrt{5} + 4\\sqrt{7} - 10}"
} | {
"split": "train",
"index": 341,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The polynomial $(x+y)^9$ is expanded in decreasing powers of $x$. The second and third terms have equal values \nwhen evaluated at $x=p$ and $y=q$, where $p$ and $q$ are positive numbers whose sum is one. What is the value of $p$? Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "4/5"
} | {
"split": "train",
"index": 342,
"solution": "1. **Identify the Terms in the Expansion**: \n The polynomial $(x+y)^9$ can be expanded using the binomial theorem, which states that:\n \\[\n (x+y)^n = \\sum_{k=0}^n \\binom{n}{k} x^{n-k} y^k\n \\]\n For $n=9$, the second term (where $k=1$) is:\n \\[\n ... |
deepscaler | [
{
"role": "user",
"content": "The projection of $\\begin{pmatrix} -8 \\\\ b \\end{pmatrix}$ onto $\\begin{pmatrix} 2 \\\\ 1 \\end{pmatrix}$ is\n\\[-\\frac{13}{5} \\begin{pmatrix} 2 \\\\ 1 \\end{pmatrix}.\\]Find $b.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 343,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For $\\{1, 2, 3, \\ldots, 10\\}$ and each of its non-empty subsets, a unique alternating sum is defined similarly as before. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. Find the sum... | math | {
"style": "rule",
"ground_truth": "5120"
} | {
"split": "train",
"index": 344,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $n\\neq 2004$ for which there exists a polynomial $f\\in\\mathbb{Z}[x]$ such that the equation $f(x)=2004$ has at least one, and the equation $f(x)=n$ has at least $2004$ different integer solutions. Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "(1002!)^2 + 2004"
} | {
"split": "train",
"index": 345,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the values of:\n(1) $8^{\\frac{2}{3}} - (0.5)^{-3} + \\left(\\frac{1}{\\sqrt{3}}\\right)^{-2} \\times \\left(\\frac{81}{16}\\right)^{-\\frac{1}{4}}$;\n(2) $\\log 5 \\cdot \\log 8000 + (\\log 2^{\\sqrt{3}})^2 + e^{\\ln 1} + \\ln(e \\sqrt{e})$. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{2}"
} | {
"split": "train",
"index": 346,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The volume of a hemisphere is $\\frac{500}{3}\\pi$. What is the total surface area of the hemisphere including its base? Express your answer in terms of $\\pi$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\pi \\times 250^{2/3}"
} | {
"split": "train",
"index": 347,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a certain value of $k,$ the system\n\\begin{align*}\nx + ky + 3z &= 0, \\\\\n3x + ky - 2z &= 0, \\\\\n2x + 4y - 3z &= 0\n\\end{align*}has a solution where $x,$ $y,$ and $z$ are all nonzero. Find $\\frac{xz}{y^2}.$ Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 348,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\n\\left( 1 + \\sin \\frac {\\pi}{12} \\right) \\left( 1 + \\sin \\frac {5\\pi}{12} \\right) \\left( 1 + \\sin \\frac {7\\pi}{12} \\right) \\left( 1 + \\sin \\frac {11\\pi}{12} \\right).\n\\] Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"split": "train",
"index": 349,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rectangle $EFGH$ has area $4032$. An ellipse with area $4032\\pi$ passes through $E$ and $G$ and has foci at $F$ and $H$. What is the perimeter of the rectangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8\\sqrt{2016}"
} | {
"split": "train",
"index": 350,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct diagonals of a convex nonagon (9-sided polygon) can be drawn such that none of the diagonals is parallel to any side of the polygon? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 351,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a = -2$, the largest number in the set $\\{ -3a, 4a, \\frac{24}{a}, a^2, 1\\}$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3a"
} | {
"split": "train",
"index": 352,
"solution": "1. **Substitute $a = -2$ into each expression in the set**:\n - $-3a = -3(-2) = 6$\n - $4a = 4(-2) = -8$\n - $\\frac{24}{a} = \\frac{24}{-2} = -12$\n - $a^2 = (-2)^2 = 4$\n - The last element in the set is $1$.\n\n2. **List the evaluated set**:\n \\[\n ... |
deepscaler | [
{
"role": "user",
"content": "A total of $960$ people are randomly numbered from $1$ to $960$. Using systematic sampling, $32$ people are selected for a survey. Find the number of people to be selected from those with numbers falling within $[450,750]$. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 353,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Entrepreneurs Vasiliy Petrovich and Petr Gennadievich opened a clothing factory \"ViP.\" Vasiliy Petrovich invested 200 thousand rubles, while Petr Gennadievich invested 350 thousand rubles. The factory was successful, and after a year, Anastasia Alekseevna approached them with... | math | {
"style": "rule",
"ground_truth": "1000000"
} | {
"split": "train",
"index": 354,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ana, Bob, and Cao bike at constant rates of $8.6$ meters per second, $6.2$ meters per second, and $5$ meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking alon... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"split": "train",
"index": 355,
"solution": "Let P, Q, and R be the east-west distance of the field, the north-south distance, and the distance from the southeast corner to point D, respectively.\n Ana's distance to point D = $P + Q + (P - R) = 2P + Q - R$\n Bob's distance to point D = $Q + R$\n Cao's distance ... |
deepscaler | [
{
"role": "user",
"content": "The line $y = 2$ intersects the graph of $y = 3x^2 + 2x - 5$ at the points $C$ and $D$. Determine the distance between $C$ and $D$ and express this distance in the form $\\frac{\\sqrt{p}}{q}$, where $p$ and $q$ are coprime positive integers. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{22}}{3}"
} | {
"split": "train",
"index": 356,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Observing the equations:<br/>$\\frac{1}{1×2}=1-\\frac{1}{2}$; $\\frac{1}{2×3}=\\frac{1}{2}-\\frac{1}{3}$; $\\frac{1}{3×4}=\\frac{1}{3}-\\frac{1}{4}$.<br/>By adding both sides of the above three equations, we get:<br/>$\\frac{1}{1×2}+\\frac{1}{2×3}+\\frac{1}{3×4}=1-\\frac{1}{2}+... | math | {
"style": "rule",
"ground_truth": "\\frac{674}{2023}"
} | {
"split": "train",
"index": 357,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of two teams, Team A and Team B, sends 7 players in a predetermined order to participate in a Go contest. The players from both teams compete sequentially starting with Player 1 from each team. The loser of each match is eliminated, and the winner continues to compete with... | math | {
"style": "rule",
"ground_truth": "3432"
} | {
"split": "train",
"index": 358,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An old clock's minute and hour hands overlap every 66 minutes of standard time. Calculate how much the old clock's 24 hours differ from the standard 24 hours. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 359,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the numbers 0, 1, 2, 3, 4, select three different digits to form a three-digit number. What is the sum of the units digit of all these three-digit numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 360,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When $3z^3-4z^2-14z+3$ is divided by $3z+5$, the quotient is $z^2-3z+\\frac{1}{3}$. What is the remainder? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 361,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of numbers of the form $n^5 - 5n^3 + 4n$ , where $n$ is an integer that is not a multiple of $3$ . What is the largest integer that is a divisor of every number in $S$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "360"
} | {
"split": "train",
"index": 362,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among a group of 120 people, some pairs are friends. A [i]weak quartet[/i] is a set of four people containing exactly one pair of friends. What is the maximum possible number of weak quartets ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4769280"
} | {
"split": "train",
"index": 363,
"solution": "\nGiven a group of 120 people, where some pairs are friends, we need to determine the maximum possible number of weak quartets. A weak quartet is defined as a set of four people containing exactly one pair of friends. \n\nTo solve this, we need to analyze the structu... |
deepscaler | [
{
"role": "user",
"content": "Calculate the number of minutes in a week. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10000"
} | {
"split": "train",
"index": 364,
"solution": "There are 60 minutes in an hour and 24 hours in a day. Thus, there are $60 \\cdot 24=1440$ minutes in a day. Since there are 7 days in a week, the number of minutes in a week is $7 \\cdot 1440=10080$. Of the given choices, this is closest to 10000."
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest real number $c$ such that $$\\sum_{i=1}^{101} x_{i}^{2} \\geq c M^{2}$$ whenever $x_{1}, \\ldots, x_{101}$ are real numbers such that $x_{1}+\\cdots+x_{101}=0$ and $M$ is the median of $x_{1}, \\ldots, x_{101}$. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "\\frac{5151}{50}"
} | {
"split": "train",
"index": 365,
"solution": "Suppose without loss of generality that $x_{1} \\leq \\cdots \\leq x_{101}$ and $M=x_{51} \\geq 0$. Note that $f(t)=t^{2}$ is a convex function over the reals, so we may \"smooth\" to the case $x_{1}=\\cdots=x_{50}=-51 r$ and $x_{51}=\\cdots=x_{101}=50 r$ for some $r... |
deepscaler | [
{
"role": "user",
"content": "Max and Minnie each add up sets of three-digit positive integers. Each of them adds three different three-digit integers whose nine digits are all different. Max creates the largest possible sum. Minnie creates the smallest possible sum. What is the difference between Max's sum... | math | {
"style": "rule",
"ground_truth": "1845"
} | {
"split": "train",
"index": 366,
"solution": "Consider three three-digit numbers with digits \\( RST, UVW \\) and \\( XYZ \\). The integer with digits \\( RST \\) equals \\( 100R+10S+T \\), the integer with digits \\( UVW \\) equals \\( 100U+10V+W \\), and the integer with digits \\( XYZ \\) equals \\( 100X+10Y+... |
deepscaler | [
{
"role": "user",
"content": "Given any point $P$ on the line $l: x-y+4=0$, two tangent lines $AB$ are drawn to the circle $O: x^{2}+y^{2}=4$ with tangent points $A$ and $B$. The line $AB$ passes through a fixed point ______; let the midpoint of segment $AB$ be $Q$. The minimum distance from point $Q$ to th... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 367,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers have cube roots that are less than 20? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7999"
} | {
"split": "train",
"index": 368,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\\(\\cos \\frac{\\pi}{15} - \\cos \\frac{2\\pi}{15} - \\cos \\frac{4\\pi}{15} + \\cos \\frac{7\\pi}{15} =\\) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 369,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $r$, $s$, and $t$ be the three roots of the equation \\[8x^3 + 1001x + 2008 = 0.\\] Find $(r + s)^3 + (s + t)^3 + (t + r)^3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "753"
} | {
"split": "train",
"index": 370,
"solution": "Write $(r+s)^3+(s+t)^3+(t+r)^3=-(r^3+s^3+t^3)$ and let $f(x)=8x^3+1001x+2008$. Then \\[f(r)+f(s)+f(t)=8(r^3+s^3+t^3)+1001(r+s+t)+6024=8(r^3+s^3+t^3)+6024=0.\\] Solving for $r^3+s^3+t^3$ and negating the result yields the answer $\\boxed{753}.$"
} |
deepscaler | [
{
"role": "user",
"content": "Five people can paint a house in 10 hours. How many hours would it take four people to paint the same house and mow the lawn if mowing the lawn takes an additional 3 hours per person, assuming that each person works at the same rate for painting and different rate for mowing? L... | math | {
"style": "rule",
"ground_truth": "15.5"
} | {
"split": "train",
"index": 371,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the coefficient of $x^3$ in the expansion of $(1-x)^5(3+x)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-20"
} | {
"split": "train",
"index": 372,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the ones digit of the largest power of $2$ that divides into $(2^4)!$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 373,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a $9 \\times 9$ chess board, we consider all the rectangles whose edges lie along grid lines (the board consists of 81 unit squares, and the grid lines lie on the borders of the unit squares). For each such rectangle, we put a mark in every one of the unit squares inside ... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 374,
"solution": "56. Consider the rectangles which contain the square in the $i$th row and $j$th column. There are $i$ possible positions for the upper edge of such a rectangle, $10-i$ for the lower edge, $j$ for the left edge, and $10-j$ for the right edge; thus we have $i(10-i) j(1... |
deepscaler | [
{
"role": "user",
"content": "Let \\( p, q, r, s, t, u, v, w \\) be real numbers such that \\( pqrs = 16 \\) and \\( tuvw = 25 \\). Find the minimum value of\n\\[ (pt)^2 + (qu)^2 + (rv)^2 + (sw)^2. \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 375,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Travel along the alley clockwise.\n\nIn 1 hour of walking, the pedestrian walked 6 kilometers and did not reach point $B$ (a whole $2 \\pi - 6$ km!), so the third option is clearly longer than the first and can be excluded.\n\nIn the first case, when moving along the alley, the... | math | {
"style": "rule",
"ground_truth": "0.21"
} | {
"split": "train",
"index": 376,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of all rational numbers that can be expressed as a repeating decimal in the form $0.\\overline{abcd},$ where at least one of the digits $a,$ $b,$ $c,$ or $d$ is nonzero. Let $N$ be the number of distinct numerators obtained when numbers in $S$ are written as ... | math | {
"style": "rule",
"ground_truth": "392"
} | {
"split": "train",
"index": 377,
"solution": "$0.abcd=\\frac{\\overline{abcd}}{9999}$, $9999=9\\times 11\\times 101$.\n Then we need to find the number of positive integers less than $10000$ that can meet the requirement. Suppose the number is $x$.\n Case $1$: $(9999, x)=1$. Clearly $x$ satisfies. \\[\\varphi \\... |
deepscaler | [
{
"role": "user",
"content": "For a natural number \\( x \\), five statements are made:\n\n$$\n3x > 91\n$$\n\n$$\n\\begin{aligned}\n& x < 120 \\\\\n& 4x > 37 \\\\\n& 2x \\geq 21 \\\\\n& x > 7\n\\end{aligned}\n$$\n\nIt is known that only three of these statements are true, and two are false. Determine \\( x ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 378,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Homewood Middle School has 1200 students, and 730 of these students attend a summer picnic. If two-thirds of the girls in the school and one-half of the boys in the school attend the picnic, how many girls attend the picnic? (Assume that each student in the school is either a ... | math | {
"style": "rule",
"ground_truth": "520"
} | {
"split": "train",
"index": 379,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A number is formed using the digits 1, 2, ..., 9. Any digit can be used more than once, but adjacent digits cannot be the same. Once a pair of adjacent digits has occurred, that pair, in that order, cannot be used again. How many digits are in the largest such number? Let's thi... | math | {
"style": "rule",
"ground_truth": "73"
} | {
"split": "train",
"index": 380,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all odd natural numbers greater than 500 but less than 1000, each of which has the sum of the last digits of all its divisors (including 1 and the number itself) equal to 33. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "729"
} | {
"split": "train",
"index": 381,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The apex of a regular pyramid with a square base $ABCD$ of unit side length is $E$. Point $P$ lies on the base edge $AB$ and point $Q$ lies on the lateral edge $EC$ such that $PQ$ is perpendicular to both $AB$ and $EC$. Additionally, we know that $AP : PB = 6 : 1$. What are the... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 382,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $a^2 + b^2 + c^2,$ given that $a,$ $b,$ and $c$ are the roots of \\[2x^3 - x^2 + 4x + 10 = 0.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{15}4"
} | {
"split": "train",
"index": 383,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An investor placed $\\$15,000$ in a three-month term deposit that offered a simple annual interest rate of $8\\%$. After three months, the total value of the deposit was reinvested in another three-month term deposit. After these three months, the investment amounted to $\\$15,... | math | {
"style": "rule",
"ground_truth": "11.36"
} | {
"split": "train",
"index": 384,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Vaccination is one of the important means to protect one's own and others' health and lives. In order to test the immune effect of a certain vaccine on the $C$ virus, researchers used white rabbits as experimental subjects and conducted the following experiments:<br/>Experiment... | math | {
"style": "rule",
"ground_truth": "80\\%"
} | {
"split": "train",
"index": 385,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are two targets, A and B. A shooter shoots at target A once, with a probability of $\\frac{3}{4}$ of hitting it and scoring $1$ point, or missing it and scoring $-1$ point. The shooter shoots at target B twice, with a probability of $\\frac{2}{3}$ of hitting it and scorin... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"split": "train",
"index": 386,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least positive integer that can be added to 412 to yield a multiple of 3? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 387,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The polynomial $P(x) = x^3 + ax^2 + bx +c$ has the property that the mean of its zeros, the product of its zeros, and the sum of its coefficients are all equal. If the $y$-intercept of the graph of $y=\nP(x)$ is 2, what is $b$? Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "-11"
} | {
"split": "train",
"index": 388,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $2.43-1.2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1.23"
} | {
"split": "train",
"index": 389,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\mathbf{a}$ and $\\mathbf{b}$ are nonzero vectors such that $\\|\\mathbf{a} + \\mathbf{b}\\| = \\|\\mathbf{a} - \\mathbf{b}\\|,$ find the angle between $\\mathbf{a}$ and $\\mathbf{b},$ in degrees. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "90^\\circ"
} | {
"split": "train",
"index": 390,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $M$ be the second smallest positive integer that is divisible by every positive integer less than 9, find the sum of the digits of $M$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 391,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Form a four-digit number using the digits 0, 1, 2, 3, 4, 5 without repetition.\n(I) How many different four-digit numbers can be formed?\n(II) How many of these four-digit numbers have a tens digit that is larger than both the units digit and the hundreds digit?\n(III) Arrange ... | math | {
"style": "rule",
"ground_truth": "2301"
} | {
"split": "train",
"index": 392,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\\\(\\alpha \\in (0^{\\circ}, 90^{\\circ})\\\\) and \\\\(\\sin (75^{\\circ} + 2\\alpha) = -\\frac{3}{5}\\\\), calculate \\\\(\\sin (15^{\\circ} + \\alpha) \\cdot \\sin (75^{\\circ} - \\alpha)\\\\). Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{20}"
} | {
"split": "train",
"index": 393,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive four-digit integers of the form $\\_\\_35$ are divisible by 35? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 394,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We call a four-digit number with the following property a \"centered four-digit number\": Arrange the four digits of this four-digit number in any order, and when all the resulting four-digit numbers (at least 2) are sorted from smallest to largest, the original four-digit numb... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 395,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n \\ge 3$ be an integer. What is the largest possible number of interior angles greater than $180^\\circ$ in an $n$-gon in the plane, given that the $n$-gon does not intersect itself and all its sides have the same length? Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 396,
"solution": "\nLet \\( n \\ge 3 \\) be an integer, and consider an \\( n \\)-gon in the plane with equal side lengths. We are asked to find the largest possible number of interior angles greater than \\( 180^\\circ \\), given that the \\( n \\)-gon does not intersect itself.\n\nT... |
deepscaler | [
{
"role": "user",
"content": "What is the remainder when $11065+11067+11069+11071+11073+11075+11077$ is divided by $14$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 397,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The operation $\\star$ is defined as $a \\star b = a^2 \\div b$. For how many negative integer values of $x$ will the value of $12 \\star x$ be a positive integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 398,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the expression $-20 + 15 \\times (4^{\\div -1} \\times 2)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-12.5"
} | {
"split": "train",
"index": 399,
"solution": ""
} |
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