data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Let the function \n$$\nf(x) = A \\sin(\\omega x + \\varphi) \\quad (A>0, \\omega>0).\n$$\n\nIf \\( f(x) \\) is monotonic on the interval \\(\\left[\\frac{\\pi}{6}, \\frac{\\pi}{2}\\right]\\) and \n$$\nf\\left(\\frac{\\pi}{2}\\right) = f\\left(\\frac{2\\pi}{3}\\right) = -f\\left... | math | {
"style": "rule",
"ground_truth": "\\pi"
} | {
"split": "train",
"index": 400,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The expression $\\lfloor x\\rfloor$ denotes the greatest integer less than or equal to $x$. Find the value of $$\\left\\lfloor\\frac{2002!}{2001!+2000!+1999!+\\cdots+1!}\\right\\rfloor.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"split": "train",
"index": 401,
"solution": "2000 We break up 2002! = 2002(2001)! as $$2000(2001!)+2 \\cdot 2001(2000!)=2000(2001!)+2000(2000!)+2002 \\cdot 2000(1999!) >2000(2001!+2000!+1999!+\\cdots+1!)$$ On the other hand, $$2001(2001!+2000!+\\cdots+1!)>2001(2001!+2000!)=2001(2001!)+2001!=2002!$$ Thus we have... |
deepscaler | [
{
"role": "user",
"content": "Given an angle $α$ whose terminal side passes through the point $(-1, \\sqrt{2})$, find the values of $\\tan α$ and $\\cos 2α$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{3}"
} | {
"split": "train",
"index": 402,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 20 chairs arranged in a circle. There are \\(n\\) people sitting in \\(n\\) different chairs. These \\(n\\) people stand, move \\(k\\) chairs clockwise, and then sit again. After this happens, exactly the same set of chairs is occupied. For how many pairs \\((n, k)\\)... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 403,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)=x^{2}-2 x$. How many distinct real numbers $c$ satisfy $f(f(f(f(c))))=3$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 404,
"solution": "We see the size of the set $f^{-1}\\left(f^{-1}\\left(f^{-1}\\left(f^{-1}(3)\\right)\\right)\\right)$. Note that $f(x)=(x-1)^{2}-1=3$ has two solutions: $x=3$ and $x=-1$, and that the fixed points $f(x)=x$ are $x=3$ and $x=0$. Therefore, the number of real solutions ... |
deepscaler | [
{
"role": "user",
"content": "The average age of the three Wilson children is 7 years. If the two younger children are 4 years old and 7 years old, how many years old is the oldest child? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 405,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Equilateral triangle $T$ is inscribed in circle $A$, which has radius $10$. Circle $B$ with radius $3$ is internally tangent to circle $A$ at one vertex of $T$. Circles $C$ and $D$, both with radius $2$, are internally tangent to circle $A$ at the other two vertices of $T$. Cir... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 406,
"solution": "Let $X$ be the intersection of the circles with centers $B$ and $E$, and $Y$ be the intersection of the circles with centers $C$ and $E$. Since the radius of $B$ is $3$, $AX =4$. Assume $AE$ = $p$. Then $EX$ and $EY$ are radii of circle $E$ and have length $4+p$. $AC... |
deepscaler | [
{
"role": "user",
"content": "Given rectangle $R_1$ with one side $4$ inches and area $24$ square inches. Rectangle $R_2$ with diagonal $17$ inches is similar to $R_1$. Find the area of $R_2$ in square inches. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{433.5}{3.25}"
} | {
"split": "train",
"index": 407,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that\n$\\frac 1{2!17!}+\\frac 1{3!16!}+\\frac 1{4!15!}+\\frac 1{5!14!}+\\frac 1{6!13!}+\\frac 1{7!12!}+\\frac 1{8!11!}+\\frac 1{9!10!}=\\frac N{1!18!}$\nfind the greatest integer that is less than $\\frac N{100}$.\n Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "137"
} | {
"split": "train",
"index": 408,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle ABC$ be an isosceles triangle with $\\angle A = 90^\\circ.$ There exists a point $P$ inside $\\triangle ABC$ such that $\\angle PAB = \\angle PBC = \\angle PCA$ and $AP = 10.$ Find the area of $\\triangle ABC.$\nDiagram\n[asy] /* Made by MRENTHUSIASM */ size(200... | math | {
"style": "rule",
"ground_truth": "250"
} | {
"split": "train",
"index": 409,
"solution": "Denote the area of $X$ by $[X].$ As in previous solutions, we see that $\\angle APC = 90 ^\\circ, \\triangle BPC \\sim \\triangle APB$ with ratio $k = \\sqrt{2}\\implies$ \\[\\frac {PC}{PB} = \\frac {PB}{PA} = k \\implies PC = k^2 \\cdot AP = 20 \\implies [APC] = \\f... |
deepscaler | [
{
"role": "user",
"content": "Suppose that $x^{10} + x + 1 = 0$ and $x^100 = a_0 + a_1x +... + a_9x^9$ . Find $a_5$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-252"
} | {
"split": "train",
"index": 410,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that a website publishes pictures of five celebrities along with five photos of the celebrities when they were teenagers, but only three of the teen photos are correctly labeled, and two are unlabeled, determine the probability that a visitor guessing at random will match... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{20}"
} | {
"split": "train",
"index": 411,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given unit vectors $\\vec{a}$ and $\\vec{b}$ with an acute angle between them, for any $(x, y) \\in \\{(x, y) \\mid | x \\vec{a} + y \\vec{b} | = 1, xy \\geq 0 \\}$, it holds that $|x + 2y| \\leq \\frac{8}{\\sqrt{15}}$. Find the minimum possible value of $\\vec{a} \\cdot \\vec{... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 412,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to cut a 1 by 1 square into 8 congruent polygonal pieces such that all of the interior angles for each piece are either 45 or 90 degrees? Two ways are considered distinct if they require cutting the square in different locations. In particular, rotations... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 413,
"solution": "First note that only triangles and quadrilaterals are possible. There are 3 possibilities: - \\(1/2\\) by \\(1/2\\) right isosceles triangles - 1 by \\(1/8\\) rectangles - \\(1/2\\) by \\(1/4\\) rectangles The first case has 16 possibilities (there are 2 choices for ... |
deepscaler | [
{
"role": "user",
"content": "Given is a isosceles triangle ABC so that AB=BC. Point K is in ABC, so that CK=AB=BC and <KAC=30°.Find <AKB=? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 414,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\dfrac{\\pi}{2} < \\alpha < \\beta < \\dfrac{3\\pi}{4}, \\cos(\\alpha - \\beta) = \\dfrac{12}{13}, \\sin(\\alpha + \\beta) = -\\dfrac{3}{5}$, find the value of $\\sin 2\\alpha$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\dfrac{56}{65}"
} | {
"split": "train",
"index": 415,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a \\div b = 2$ and $b \\div c = \\frac{3}{4}$, what is the value of $c \\div a$? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 416,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( x \\neq y \\), and the two sequences \\( x, a_{1}, a_{2}, a_{3}, y \\) and \\( b_{1}, x, b_{2}, b_{3}, y, b_{4} \\) are both arithmetic sequences. Then \\(\\frac{b_{4}-b_{3}}{a_{2}-a_{1}}\\) equals $\\qquad$. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "2.6666666666666665"
} | {
"split": "train",
"index": 417,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cube is dissected into 6 pyramids by connecting a given point in the interior of the cube with each vertex of the cube, so that each face of the cube forms the base of a pyramid. The volumes of five of these pyramids are 200, 500, 1000, 1100, and 1400. What is the volume of t... | math | {
"style": "rule",
"ground_truth": "600"
} | {
"split": "train",
"index": 418,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "All positive integers whose digits add up to 12 are listed in increasing order. What is the eleventh number in that list? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "156"
} | {
"split": "train",
"index": 419,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of a square equals the square of a length of the side of the square. The perimeter of a square equals the sum of the lengths of all four sides. The sum of the areas of two squares is 65, while the difference in their areas is 33. Find the sum of their perimeters. Let'... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 420,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose a sequence $\\{a\\_n\\}$ satisfies $\\frac{1}{a\\_{n+1}} - \\frac{1}{a\\_n} = d (n \\in \\mathbb{N}^*, d$ is a constant), then the sequence $\\{a\\_n\\}$ is called a \"harmonic sequence\". It is known that the sequence $\\{\\frac{1}{x\\_n}\\}$ is a \"harmonic sequence\"... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 421,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Pizzas are sized by diameter. What percent increase in area results if Chantel's pizza increases from a 10-inch pizza to a 12-inch pizza? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "44\\%"
} | {
"split": "train",
"index": 422,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of children in the families $A$, $B$, $C$, $D$, and $E$ are as shown in the table below:\n\n| | $A$ | $B$ | $C$ | $D$ | $E$ |\n|---------|-----|-----|-----|-----|-----|\n| Boys | $0$ | $1$ | $0$ | $1$ | $1$ |\n| Girls | $0$ | $0$ | $1$ | $1$ | $2$ |\n\n$... | math | {
"style": "rule",
"ground_truth": "\\frac{6}{5}"
} | {
"split": "train",
"index": 423,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the smallest positive integer $n$ such that $n^2$ is divisible by 50 and $n^3$ is divisible by 294. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "210"
} | {
"split": "train",
"index": 424,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the definite integral:\n\n$$\n\\int_{0}^{\\pi}\\left(9 x^{2}+9 x+11\\right) \\cos 3 x \\, dx\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2\\pi - 2"
} | {
"split": "train",
"index": 425,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point $P$ is chosen at random in the interior of a unit square $S$. Let $d(P)$ denote the distance from $P$ to the closest side of $S$. The probability that $\\frac{1}{5}\\le d(P)\\le\\frac{1}{3}$ is equal to $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive int... | math | {
"style": "rule",
"ground_truth": "281"
} | {
"split": "train",
"index": 426,
"solution": "First, let's figure out $d(P) \\geq \\frac{1}{3}$ which is\\[\\left(\\frac{3}{5}\\right)^2=\\frac{9}{25}.\\]Then, $d(P) \\geq \\frac{1}{5}$ is a square inside $d(P) \\geq \\frac{1}{3}$, so\\[\\left(\\frac{1}{3}\\right)^2=\\frac{1}{9}.\\]Therefore, the probability tha... |
deepscaler | [
{
"role": "user",
"content": "Given that the function $f(x)$ is an odd function defined on $\\mathbb{R}$, and when $x > 0$, $f(x) = -x^{2} + 4x$.\n- (Ⅰ) Find the analytical expression of the function $f(x)$.\n- (Ⅱ) Find the minimum value of the function $f(x)$ on the interval $\\left[-2,a\\right]$ where $\\... | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 427,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the following equations using appropriate methods:<br/>$(1)2x^{2}-3x+1=0$;<br/>$(2)\\left(y-2\\right)^{2}=\\left(2y+3\\right)^{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{3}"
} | {
"split": "train",
"index": 428,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $3\\triangle_4=\\triangle2_{11}$, where $\\triangle$ represents a digit 0-9. What is the value of the missing digit? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 429,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function \\( f(x) = \\max \\left\\{\\sin x, \\cos x, \\frac{\\sin x + \\cos x}{\\sqrt{2}}\\right\\} \\) (for \\( x \\in \\mathbb{R} \\)) has a maximum value and a minimum value. Find the sum of these maximum and minimum values. Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "1 - \\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 430,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three $\\text{A's}$, three $\\text{B's}$, and three $\\text{C's}$ are placed in the nine spaces so that each row and column contains one of each letter. If $\\text{A}$ is placed in the upper left corner, how many arrangements are possible? Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 431,
"solution": "1. **Fixing A in the upper left corner**: We start by placing an A in the upper left corner of the grid. The grid now looks like this:\n \\[\n \\begin{array}{|c|c|c|}\n \\hline\n A & & \\\\\n \\hline\n & & \\\\\n \\hline\n & & \\\\\n ... |
deepscaler | [
{
"role": "user",
"content": "Let $d_1$, $d_2$, $d_3$, $d_4$, $e_1$, $e_2$, $e_3$, and $e_4$ be real numbers such that for every real number $x$, we have\n\\[\n x^8 - 2x^7 + 2x^6 - 2x^5 + 2x^4 - 2x^3 + 2x^2 - 2x + 1 = (x^2 + d_1 x + e_1)(x^2 + d_2 x + e_2)(x^2 + d_3 x + e_3)(x^2 + d_4 x + e_4). \n\\]\nComp... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 432,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all $c$ which satisfy $$\\frac{c}{3} \\le 2+c < -2(1+c).$$Express your answer in interval notation, simplifying any fractions which occur in your answer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left[-3,-\\frac{4}{3}\\right)"
} | {
"split": "train",
"index": 433,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When the two-digit integer $MM$, with equal digits, is multiplied by the one-digit integer $M$, the result is the three-digit integer $NPM$. What is the greatest possible value of $NPM$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "396"
} | {
"split": "train",
"index": 434,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_{0}, a_{1}, \\ldots$ be a sequence such that $a_{0}=3, a_{1}=2$, and $a_{n+2}=a_{n+1}+a_{n}$ for all $n \\geq 0$. Find $\\sum_{n=0}^{8} \\frac{a_{n}}{a_{n+1} a_{n+2}}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{105}{212}"
} | {
"split": "train",
"index": 435,
"solution": "We can re-write $\\frac{a_{n}}{a_{n+1} a_{n+2}}$ as $\\frac{a_{n+2}-a_{n+1}}{a_{n+1} a_{n+2}}=\\frac{1}{a_{n+1}}-\\frac{1}{a_{n+2}}$. We can thus re-write the sum as $$\\left(\\frac{1}{a_{1}}-\\frac{1}{a_{2}}\\right)+\\left(\\frac{1}{a_{2}}-\\frac{1}{a_{3}}\\right)+\... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\frac{\\ln x}{x}$.\n(1) Find the monotonic intervals of the function $f(x)$;\n(2) Let $m > 0$, find the maximum value of $f(x)$ on $[m, 2m]$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{e}"
} | {
"split": "train",
"index": 436,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given real numbers $a$ and $b$ satisfying the equation $\\sqrt{(a-1)^2} + \\sqrt{(a-6)^2} = 10 - |b+3| - |b-2|$, find the maximum value of $a^2 + b^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 437,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, if $a + c = 2b$, then find the value of $\\tan \\frac{A}{2} \\cdot \\tan \\frac{C}{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"split": "train",
"index": 438,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)=x^{3}-3x$. Compute the number of positive divisors of $$\\left\\lfloor f\\left(f\\left(f\\left(f\\left(f\\left(f\\left(f\\left(f\\left(\\frac{5}{2}\\right)\\right)\\right)\\right)\\right)\\right)\\right)\\right)\\right)\\rfloor$$ where $f$ is applied 8 times. Let's th... | math | {
"style": "rule",
"ground_truth": "6562"
} | {
"split": "train",
"index": 439,
"solution": "Note that $f\\left(y+\\frac{1}{y}\\right)=\\left(y+\\frac{1}{y}\\right)^{3}-3\\left(y+\\frac{1}{y}\\right)=y^{3}+\\frac{1}{y^{3}}$. Thus, $f\\left(2+\\frac{1}{2}\\right)=2^{3}+\\frac{1}{2^{3}}$, and in general $f^{k}\\left(2+\\frac{1}{2}\\right)=2^{3^{k}}+\\frac{1}{2... |
deepscaler | [
{
"role": "user",
"content": "The inclination angle of the line $\\sqrt {3}x-y+1=0$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {\\pi}{3}"
} | {
"split": "train",
"index": 440,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the terminal side of angle $\\alpha$ passes through point $P(m, 2\\sqrt{2})$, $\\sin{\\alpha} = \\frac{2\\sqrt{2}}{3}$, and $\\alpha$ is in the second quadrant.\n\n(1) Find the value of $m$;\n\n(2) If $\\tan{\\beta} = \\sqrt{2}$, find the value of $\\frac{\\sin{\\alp... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{11}"
} | {
"split": "train",
"index": 441,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\log_3 (x+5)^2 + \\log_{1/3} (x - 1) = 4,$ compute $x.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{71 + \\sqrt{4617}}{2}"
} | {
"split": "train",
"index": 442,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation \\[\\frac{x}{x+1} + \\frac{x}{x+2} = kx\\]has exactly two complex roots. Find all possible complex values for $k.$\n\nEnter all the possible values, separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0,\\tfrac32, 2i, -2i"
} | {
"split": "train",
"index": 443,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cyclist traveled from point A to point B, stayed there for 30 minutes, and then returned to A. On the way to B, he overtook a pedestrian, and met him again 2 hours later on his way back. The pedestrian arrived at point B at the same time the cyclist returned to point A. How m... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 444,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( a_{1}, a_{2}, \\cdots, a_{2014} \\) be a permutation of the positive integers \\( 1, 2, \\cdots, 2014 \\). Define\n\\[ S_{k} = a_{1} + a_{2} + \\cdots + a_{k} \\quad (k=1, 2, \\cdots, 2014). \\]\n\nWhat is the maximum number of odd numbers among \\( S_{1}, S_{2}, \\cdot... | math | {
"style": "rule",
"ground_truth": "1511"
} | {
"split": "train",
"index": 445,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The price of an item is an integer number of yuan. With 100 yuan, you can buy up to 3 items. Person A and Person B each have a certain number of 100-yuan bills. The amount of money Person A has can buy at most 7 items, and the amount of money Person B has can buy at most 14 ite... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 446,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p$ and $q$ be real numbers, and suppose that the roots of the equation \\[x^3 - 10x^2 + px - q = 0\\] are three distinct positive integers. Compute $p + q.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 447,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A permutation of the set of positive integers $[n] = \\{1, 2, \\ldots, n\\}$ is a sequence $(a_1, a_2, \\ldots, a_n)$ such that each element of $[n]$ appears precisely one time as a term of the sequence. For example, $(3, 5, 1, 2, 4)$ is a permutation of $[5]$ . Let $P(n)$ be t... | math | {
"style": "rule",
"ground_truth": "4489"
} | {
"split": "train",
"index": 448,
"solution": "We claim that the smallest $n$ is $67^2 = \\boxed{4489}$ .\nSolution 1\nLet $S = \\{1, 4, 9, \\ldots\\}$ be the set of positive perfect squares. We claim that the relation $R = \\{(j, k)\\in [n]\\times[n]\\mid jk\\in S\\}$ is an equivalence relation on $[n]$ .\nIt is... |
deepscaler | [
{
"role": "user",
"content": "A natural number $n$ is said to be $good$ if $n$ is the sum or $r$ consecutive positive integers, for some $r \\geq 2 $ . Find the number of good numbers in the set $\\{1,2 \\dots , 100\\}$ . Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "93"
} | {
"split": "train",
"index": 449,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal{F}$ be the set of all the functions $f : \\mathcal{P}(S) \\longrightarrow \\mathbb{R}$ such that for all $X, Y \\subseteq S$, we have $f(X \\cap Y) = \\min (f(X), f(Y))$, where $S$ is a finite set (and $\\mathcal{P}(S)$ is the set of its subsets). Find\n\\[\\max_... | math | {
"style": "rule",
"ground_truth": "n+1"
} | {
"split": "train",
"index": 450,
"solution": "\nLet \\( S \\) be a finite set with \\( |S| = n \\). We are asked to find the maximum size of the image of a function \\( f \\) in the set \\(\\mathcal{F}\\), where \\(\\mathcal{F}\\) is the set of all functions \\( f : \\mathcal{P}(S) \\to \\mathbb{R} \\) satisfyin... |
deepscaler | [
{
"role": "user",
"content": "Given points \\( A(3,1) \\) and \\( B\\left(\\frac{5}{3}, 2\\right) \\), and the four vertices of quadrilateral \\( \\square ABCD \\) are on the graph of the function \\( f(x)=\\log _{2} \\frac{a x+b}{x-1} \\), find the area of \\( \\square ABCD \\). Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "\\frac{26}{3}"
} | {
"split": "train",
"index": 451,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the number \\( 2016****02* \\), each of the 5 asterisks needs to be replaced with any of the digits \\( 0, 2, 4, 5, 7, 9 \\) (digits can be repeated) so that the resulting 11-digit number is divisible by 15. In how many ways can this be done? Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "864"
} | {
"split": "train",
"index": 452,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Chelsea is leading by 60 points halfway through a 120-shot archery competition, each shot can score 10, 7, 3, or 0 points, and Chelsea always scores at least 3 points. If Chelsea's next \\(n\\) shots are all for 10 points, she will secure her victory regardless of he... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 453,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Inside a cube with edge length 1, an inscribed sphere \\( O_1 \\) is drawn. Another smaller sphere \\( O_2 \\) is drawn inside the cube such that it is externally tangent to the larger sphere and simultaneously tangent to three faces of the cube. What is the surface area of the... | math | {
"style": "rule",
"ground_truth": "(7-4\\sqrt{3})\\pi"
} | {
"split": "train",
"index": 454,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sector with a radius of 16, and the arc length of the sector is $16\\pi$, calculate the central angle and the area of the sector. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "128\\pi"
} | {
"split": "train",
"index": 455,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $cos(\\frac{π}{4}-α)=\\frac{3}{5}$ and $sin(\\frac{5π}{4}+β)=-\\frac{12}{13}$, where $α∈(\\frac{π}{4},\\frac{3π}{4})$ and $β∈(0,\\frac{π}{4})$, find $\\frac{tanα}{tanβ}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-17"
} | {
"split": "train",
"index": 456,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the distance that the origin $O(0,0)$ moves under the dilation transformation that sends the circle of radius $4$ centered at $B(3,1)$ to the circle of radius $6$ centered at $B'(7,9)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.5\\sqrt{10}"
} | {
"split": "train",
"index": 457,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Convex quadrilateral $ABCD$ has $AB = 9$ and $CD = 12$. Diagonals $AC$ and $BD$ intersect at $E$, $AC = 14$, and $\\triangle AED$ and $\\triangle BEC$ have equal areas. What is $AE$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 458,
"solution": "1. **Given Information and Assumptions**:\n - Convex quadrilateral $ABCD$ with $AB = 9$, $CD = 12$, and diagonals $AC = 14$.\n - Diagonals $AC$ and $BD$ intersect at $E$.\n - $\\triangle AED$ and $\\triangle BEC$ have equal areas.\n\n2. **Using Equal Areas to I... |
deepscaler | [
{
"role": "user",
"content": "What is the largest $4$ digit integer congruent to $15 \\pmod{22}?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9981"
} | {
"split": "train",
"index": 459,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest 4-digit base 7 positive integer that is divisible by 7? (Express your answer in base 7.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6660_7"
} | {
"split": "train",
"index": 460,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rectangle $EFGH$ has sides $\\overline {EF}$ of length 5 and $\\overline {FG}$ of length 4. Divide $\\overline {EF}$ into 196 congruent segments with points $E=R_0, R_1, \\ldots, R_{196}=F$, and divide $\\overline {FG}$ into 196 congruent segments with points $F=S_0, S_1, \\ldo... | math | {
"style": "rule",
"ground_truth": "195 \\sqrt{41}"
} | {
"split": "train",
"index": 461,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)=\\frac{3x^2+5x+8}{x^2-x+4}$ and $g(x)=x-1$. Find $f(g(x))+g(f(x))$, evaluated when $x=1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 462,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $\\frac{1357_{9}}{100_{4}}-2460_{8}+5678_{9}$? Express your answer in base 10. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2938"
} | {
"split": "train",
"index": 463,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Elmo bakes cookies at a rate of one per 5 minutes. Big Bird bakes cookies at a rate of one per 6 minutes. Cookie Monster *consumes* cookies at a rate of one per 4 minutes. Together Elmo, Big Bird, Cookie Monster, and Oscar the Grouch produce cookies at a net rate of one per 8 m... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 464,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Nyusha has 2022 coins, and Barash has 2023. Nyusha and Barash toss all their coins simultaneously and count how many heads each gets. The one who gets more heads wins, and in case of a tie, Nyusha wins. What is the probability that Nyusha wins? Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"split": "train",
"index": 465,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_{1}, a_{2}, \\ldots$ be a sequence defined by $a_{1}=a_{2}=1$ and $a_{n+2}=a_{n+1}+a_{n}$ for $n \\geq 1$. Find $$\\sum_{n=1}^{\\infty} \\frac{a_{n}}{4^{n+1}}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{11}"
} | {
"split": "train",
"index": 466,
"solution": "Let $X$ denote the desired sum. Note that $$\\begin{array}{rlr} X & = & \\frac{1}{4^{2}}+\\frac{1}{4^{3}}+\\frac{2}{4^{4}}+\\frac{3}{4^{5}}+\\frac{5}{4^{6}}+\\ldots \\\\ 4 X & =\\quad \\frac{1}{4^{1}}+\\frac{1}{4^{2}}+\\frac{2}{4^{3}}+\\frac{3}{4^{4}}+\\frac{5}{4^{5}... |
deepscaler | [
{
"role": "user",
"content": "The probability of missing the target at least once in 4 shots is $\\frac{1}{81}$, calculate the shooter's hit rate. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 467,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the line $l_{1}: 2x + 5y = 1$ and the circle $C: x^{2} + y^{2} - 2x + 4y = 4$ with center $O_{1}$, let a moving line $l_{2}$ which is parallel to $l_{1}$ intersect the circle $C$ at points $A$ and $B$. Find the maximum value of the area $S_{\\triangle ABB_{1}}$. Let's thi... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"split": "train",
"index": 468,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer $n$ such that $3n \\equiv 1356 \\pmod{22}?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 469,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When three consecutive integers are added, the total is 27. What is the result when the same three integers are multiplied? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "720"
} | {
"split": "train",
"index": 470,
"solution": "If the sum of three consecutive integers is 27, then the numbers must be 8, 9, and 10. Their product is $8 \times 9 \times 10=720$."
} |
deepscaler | [
{
"role": "user",
"content": "All positive integers whose digits add up to 12 are listed in increasing order: $39, 48, 57, ...$. What is the tenth number in that list? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "147"
} | {
"split": "train",
"index": 471,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square has a 6x6 grid, where every third square in each row following a checkerboard pattern is shaded. What percent of the six-by-six square is shaded? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "33.33\\%"
} | {
"split": "train",
"index": 472,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two lines $l_1: ax+3y-1=0$ and $l_2: 2x+(a^2-a)y+3=0$, and $l_1$ is perpendicular to $l_2$, find the value of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "a = \\frac{1}{3}"
} | {
"split": "train",
"index": 473,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are seven students taking a graduation photo in a row. Student A must stand in the middle, and students B and C must stand together. How many different arrangements are possible? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "192"
} | {
"split": "train",
"index": 474,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $3^p + 3^4 = 90$, $2^r + 44 = 76$, and $5^3 + 6^s = 1421$, what is the product of $p$, $r$, and $s$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 475,
"solution": "We are given three equations involving exponents and asked to find the product of $p$, $r$, and $s$.\n\n1. **Solving for $p$:**\n \\[\n 3^p + 3^4 = 90\n \\]\n We know that $3^4 = 81$. Substituting this into the equation, we get:\n \\[\n 3^p + 81 = 90\n ... |
deepscaler | [
{
"role": "user",
"content": "Given $a^2 = 16$, $|b| = 3$, $ab < 0$, find the value of $(a - b)^2 + ab^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 476,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a polynomial $p(x),$ define its munificence as the maximum value of $|p(x)|$ on the interval $-1 \\le x \\le 1.$ For example, the munificence of the polynomial $p(x) = -x^2 + 3x - 17$ is 21, since the maximum value of $|-x^2 + 3x - 17|$ for $-1 \\le x \\le 1$ is 21, occurr... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 477,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system xOy, an ellipse C is given by the equation $$\\frac {x^{2}}{a^{2}}+ \\frac {y^{2}}{b^{2}}=1$$ ($$a>b>0$$), with left and right foci $$F_1$$ and $$F_2$$, respectively. The left vertex's coordinates are ($$-\\sqrt {2}$$, 0), and point M lies o... | math | {
"style": "rule",
"ground_truth": "\\frac {2\\sqrt {3}}{5}"
} | {
"split": "train",
"index": 478,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two $4 \\times 4$ squares are randomly placed on an $8 \\times 8$ chessboard so that their sides lie along the grid lines of the board. What is the probability that the two squares overlap? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "529/625"
} | {
"split": "train",
"index": 479,
"solution": "$529 / 625$. Each square has 5 horizontal $\\cdot 5$ vertical $=25$ possible positions, so there are 625 possible placements of the squares. If they do not overlap, then either one square lies in the top four rows and the other square lies in the bottom four rows, or... |
deepscaler | [
{
"role": "user",
"content": "The product $8 \\cdot 78$ and the increasing sequence of integers ${3, 15, 24, 48, \\cdots}$, which consists of numbers that are divisible by 3 and are one less than a perfect square, are given. What is the remainder when the 1994th term in this sequence is divided by 1000? Let... | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 480,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$-14-(-2)^{3}\\times \\dfrac{1}{4}-16\\times \\left(\\dfrac{1}{2}-\\dfrac{1}{4}+\\dfrac{3}{8}\\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-22"
} | {
"split": "train",
"index": 481,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the first term of the sequence $\\{a_n\\}$ is $a_{1}= \\frac {1}{8}$, and the sequence $\\{b_n\\}$ is a geometric sequence with $b_{5}=2$. If $$b_{n}= \\frac {a_{n+1}}{a_{n}}$$, find the value of $a_{10}$. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 482,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Andy, Beth, Charlie, and Daniel take a test with thirty questions. Andy and Beth together get the same number of questions wrong as Charlie and Daniel together. Andy and Daniel together get four more questions wrong than Beth and Charlie do together. If Charlie gets five questi... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 483,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle of radius 1 is randomly placed in a 15-by-36 rectangle $ABCD$ so that the circle lies completely within the rectangle. Given that the probability that the circle will not touch diagonal $AC$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m + ... | math | {
"style": "rule",
"ground_truth": "817"
} | {
"split": "train",
"index": 484,
"solution": "Let the bisector of $\\angle CAD$ be $AE$, with $E$ on $CD$. By the angle bisector theorem, $DE = 36/5$. Since $\\triangle AOR \\sim \\triangle AED$ ($O$ is the center of the circle), we find that $AR = 5$ since $OR = 1$. Also $AT = 35$ so $RT = OQ = 30$.\n We can ap... |
deepscaler | [
{
"role": "user",
"content": "If $y = -x^2 + 5$ and $x$ is a real number, then what is the maximum value possible for $y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 485,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $PQRS$ be an isosceles trapezoid with bases $PQ=120$ and $RS=25$. Suppose $PR=QS=y$ and a circle with center on $\\overline{PQ}$ is tangent to segments $\\overline{PR}$ and $\\overline{QS}$. If $n$ is the smallest possible value of $y$, then $n^2$ equals what? Let's think s... | math | {
"style": "rule",
"ground_truth": "2850"
} | {
"split": "train",
"index": 486,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a right triangle, right at $B$ , and let $M$ be the midpoint of the side $BC$ . Let $P$ be the point in\nbisector of the angle $ \\angle BAC$ such that $PM$ is perpendicular to $BC (P$ is outside the triangle $ABC$ ). Determine the triangle area $AB... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 487,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P(x)$ be a polynomial with degree 2008 and leading coefficient 1 such that $P(0)=2007, P(1)=2006, P(2)=2005, \\ldots, P(2007)=0$. Determine the value of $P(2008)$. You may use factorials in your answer. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "2008!-1"
} | {
"split": "train",
"index": 488,
"solution": "Consider the polynomial $Q(x)=P(x)+x-2007$. The given conditions tell us that $Q(x)=0$ for $x=0,1,2, \\ldots, 2007$, so these are the roots of $Q(x)$. On the other hand, we know that $Q(x)$ is also a polynomial with degree 2008 and leading coefficient 1 . It follows ... |
deepscaler | [
{
"role": "user",
"content": "A robotic grasshopper jumps 1 cm to the east, then 2 cm to the north, then 3 cm to the west, then 4 cm to the south. After every fourth jump, the grasshopper restarts the sequence of jumps: 1 cm to the east, then 2 cm to the north, then 3 cm to the west, then 4 cm to the south.... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 489,
"solution": "Each group of four jumps takes the grasshopper 1 cm to the east and 3 cm to the west, which is a net movement of 2 cm to the west, and 2 cm to the north and 4 cm to the south, which is a net movement of 2 cm to the south. In other words, we can consider each group of... |
deepscaler | [
{
"role": "user",
"content": "In \\( \\triangle ABC \\), \\( AB = AC = 26 \\) and \\( BC = 24 \\). Points \\( D, E, \\) and \\( F \\) are on sides \\( \\overline{AB}, \\overline{BC}, \\) and \\( \\overline{AC}, \\) respectively, such that \\( \\overline{DE} \\) and \\( \\overline{EF} \\) are parallel to \\(... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 490,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In front of you is a clock face. Divide it into three parts using two straight lines so that the sum of the numbers in each part is equal. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 491,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a 50-cent piece. What is the probability that at least 40 cents worth of coins come up heads? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{16}"
} | {
"split": "train",
"index": 492,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\\times 5$ square array of dots? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 493,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum number $n$ such that for any coloring of the integers from $1$ to $n$ into two colors, one can find monochromatic $a$ , $b$ , $c$ , and $d$ (not necessarily distinct) such that $a+b+c=d$ . Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 494,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $\\sqrt{36 \\times \\sqrt{16}}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 495,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $M$ be a set consisting of $n$ points in the plane, satisfying:\ni) there exist $7$ points in $M$ which constitute the vertices of a convex heptagon; \nii) if for any $5$ points in $M$ which constitute the vertices of a convex pentagon, then there is a point in ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 496,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the figure below, a 3-inch by 3-inch square adjoins a 10-inch by 10-inch square. What is the area of the shaded region? Express your answer in square inches as a common fraction. [asy]\nunitsize(2mm);\ndefaultpen(linewidth(0.7pt)+fontsize(12pt));\n\npair H=(0,0), A=(0,10), B... | math | {
"style": "rule",
"ground_truth": "\\frac{72}{13}"
} | {
"split": "train",
"index": 497,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two mathematics teachers administer a geometry test, assessing the ability to solve problems and knowledge of theory for each 10th-grade student. The first teacher spends 5 and 7 minutes per student, and the second teacher spends 3 and 4 minutes per student. What is the minimum... | math | {
"style": "rule",
"ground_truth": "110"
} | {
"split": "train",
"index": 498,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Select 3 numbers from the set $\\{0,1,2,3,4,5,6,7,8,9\\}$ such that their sum is an even number not less than 10. How many different ways are there to achieve this? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 499,
"solution": ""
} |
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