data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer $k$ such that the number $\\textstyle\\binom{2k}k$ ends in two zeros? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 3400,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the United States, coins have the following thicknesses: penny, 1.55 mm; nickel, 1.95 mm; dime, 1.35 mm; quarter, 1.75 mm. If a stack of these coins is exactly 14 mm high, how many coins are in the stack? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 3401,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the probability of rolling eight standard, six-sided dice and getting exactly three pairs of identical numbers, while the other two numbers are distinct from each other and from those in the pairs? Express your answer as a common fraction. Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "\\frac{525}{972}"
} | {
"split": "train",
"index": 3402,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A farmer has a rectangular field with dimensions $2m+7$ and $m-2$. If the field has an area of 51 square units, what is the value of $m$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 3403,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\{a_n\\}$ is an arithmetic sequence with the first term $a_1 > 0$, and $a_{1007} + a_{1008} > 0$, $a_{1007} \\cdot a_{1008} < 0$, determine the largest natural number $n$ for which the sum of the first $n$ terms $S_n > 0$. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "2014"
} | {
"split": "train",
"index": 3404,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal{S}$ be the set $\\lbrace1,2,3,\\ldots,10\\rbrace$ Let $n$ be the number of sets of two non-empty disjoint subsets of $\\mathcal{S}$. (Disjoint sets are defined as sets that have no common elements.) Find the remainder obtained when $n$ is divided by $1000$. Let's... | math | {
"style": "rule",
"ground_truth": "501"
} | {
"split": "train",
"index": 3405,
"solution": "Let the two disjoint subsets be $A$ and $B$, and let $C = S-(A+B)$. For each $i \\in S$, either $i \\in A$, $i \\in B$, or $i \\in C$. So there are $3^{10}$ ways to organize the elements of $S$ into disjoint $A$, $B$, and $C$.\n However, there are $2^{10}$ ways to o... |
deepscaler | [
{
"role": "user",
"content": "Rectangle $ABCD$ lies in a plane with $AB = CD = 3$ and $BC = DA = 8$. This rectangle is rotated $90^\\circ$ clockwise around $D$, followed by another $90^\\circ$ clockwise rotation around the new position of point $C$ after the first rotation. What is the length of the path tr... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi(8 + \\sqrt{73})}{2}"
} | {
"split": "train",
"index": 3406,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the distance between the points (0,15) and (8,0). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 3407,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many numbers in the set $\\{3,13,23,33, \\ldots\\}$ can be written as the difference of two primes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 3408,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangle having integer length and width has a perimeter of 100 units. What is the number of square units in the least possible area? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 3409,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many alphabetic sequences (that is, sequences containing only letters from $a\\cdots z$ ) of length $2013$ have letters in alphabetic order? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\binom{2038}{25}"
} | {
"split": "train",
"index": 3410,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest positive integer $n$ for which there is a unique integer $k$ such that $\\frac{8}{15} < \\frac{n}{n + k} < \\frac{7}{13}$?\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 3411,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be the point on line segment $\\overline{AB}$ such that $AP:PB = 2:7.$ Then\n\\[\\overrightarrow{P} = t \\overrightarrow{A} + u \\overrightarrow{B}\\]for some constants $t$ and $u.$ Enter the ordered pair $(t,u).$\n\n[asy]\nunitsize(1 cm);\n\npair A, B, P;\n\nA = (0,0... | math | {
"style": "rule",
"ground_truth": "\\left( \\frac{7}{9}, \\frac{2}{9} \\right)"
} | {
"split": "train",
"index": 3412,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An engineer invested $\\$15,\\!000$ in a nine-month savings certificate that paid a simple annual interest rate of $9\\%$. After nine months, she invested the total value of her investment in another nine-month certificate. After another nine months, the investment was worth $\... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 3413,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The volume of a cylinder is $54\\pi$ $\\text{cm}^3$. How many cubic centimeters are in the volume of a cone with the same radius and height as the cylinder? Express your answer in terms of $\\pi$. [asy]\nimport solids; currentprojection=orthographic(0,100,25); defaultpen(linewi... | math | {
"style": "rule",
"ground_truth": "18\\pi"
} | {
"split": "train",
"index": 3414,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that sin(α + $\\frac {π}{4}$) = $\\frac { \\sqrt {3}}{3}$, and α ∈ (0, π), find the value of cosα. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {-2 \\sqrt {3}+ \\sqrt {6}}{6}"
} | {
"split": "train",
"index": 3415,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a triangle \\( \\triangle ABC \\), \\(a\\), \\(b\\), and \\(c\\) are the sides opposite to angles \\(A\\), \\(B\\), and \\(C\\) respectively, with \\(B= \\dfrac {2\\pi}{3}\\). If \\(a^{2}+c^{2}=4ac\\), then find the value of \\( \\dfrac {\\sin (A+C)}{\\sin A\\sin C} \\). Let... | math | {
"style": "rule",
"ground_truth": "\\dfrac{10\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 3416,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The average density of pathogenic microbes in one cubic meter of air is 100. A sample of 2 cubic decimeters of air is taken. Find the probability that at least one microbe will be found in the sample. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "0.181"
} | {
"split": "train",
"index": 3417,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the period of the repetend of the fraction $\\frac{39}{1428}$ by using *binary* numbers, i.e. its binary decimal representation.\n\n(Note: When a proper fraction is expressed as a decimal number (of any base), either the decimal number terminates after finite steps, or i... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 3418,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of the following expression: $3 - 8 + 13 - 18 + 23 - \\cdots - 98 + 103 - 108 + 113$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "58"
} | {
"split": "train",
"index": 3419,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In this problem only, assume that $s_{1}=4$ and that exactly one board square, say square number $n$, is marked with an arrow. Determine all choices of $n$ that maximize the average distance in squares the first player will travel in his first two turns. Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "n=4"
} | {
"split": "train",
"index": 3420,
"solution": "Because expectation is linear, the average distance the first player travels in his first two turns is the average sum of two rolls of his die (which does not depend on the board configuration) plus four times the probability that he lands on the arrow on one of his... |
deepscaler | [
{
"role": "user",
"content": "Let $A$ , $B$ , $C$ , and $P$ be points in the plane such that no three of them are collinear. Suppose that the areas of triangles $BPC$ , $CPA$ , and $APB$ are 13, 14, and 15, respectively. Compute the sum of all possible values for the area of triangle $ABC$ .\n\n*P... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 3421,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "1. Given $$\\cos\\left(\\alpha+ \\frac {\\pi}{6}\\right)-\\sin\\alpha= \\frac {3 \\sqrt {3}}{5}$$, find the value of $$\\sin\\left(\\alpha+ \\frac {5\\pi}{6}\\right)$$; \n2. Given $$\\sin\\alpha+\\sin\\beta= \\frac {1}{2}$$ and $$\\cos\\alpha+\\cos\\beta= \\frac {\\sqrt {2}}{2... | math | {
"style": "rule",
"ground_truth": "-\\frac {5}{8}"
} | {
"split": "train",
"index": 3422,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate\n\\[\\begin{vmatrix} 0 & \\sin \\alpha & -\\cos \\alpha \\\\ -\\sin \\alpha & 0 & \\sin \\beta \\\\ \\cos \\alpha & -\\sin \\beta & 0 \\end{vmatrix}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 3423,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many different ways can André form exactly \\( \\$10 \\) using \\( \\$1 \\) coins, \\( \\$2 \\) coins, and \\( \\$5 \\) bills? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 3424,
"solution": "Using combinations of \\( \\$5 \\) bills, \\( \\$2 \\) coins, and \\( \\$1 \\) coins, there are 10 ways to form \\( \\$10 \\)."
} |
deepscaler | [
{
"role": "user",
"content": "The maximum and minimum values of the function $y=2x^3-3x^2-12x+5$ in the interval $[0,3]$ are respectively what are the values? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-15"
} | {
"split": "train",
"index": 3425,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest whole number that must be a divisor of the product of any three consecutive positive integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 3426,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the value of $\\frac{3b^{-1} - \\frac{b^{-1}}{3}}{b^2}$ when $b = \\tfrac{1}{3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 3427,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A path of length $n$ is a sequence of points $\\left(x_{1}, y_{1}\\right),\\left(x_{2}, y_{2}\\right), \\ldots,\\left(x_{n}, y_{n}\\right)$ with integer coordinates such that for all $i$ between 1 and $n-1$ inclusive, either (1) $x_{i+1}=x_{i}+1$ and $y_{i+1}=y_{i}$ (in which c... | math | {
"style": "rule",
"ground_truth": "1024"
} | {
"split": "train",
"index": 3428,
"solution": "This is just the number of paths of length 10. The $i$th step can be either upward or rightward, so there are $2^{10}=1024$ such paths."
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the squares of the solutions to\n\\[\\left| x^2 - x + \\frac{1}{2016} \\right| = \\frac{1}{2016}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1007}{504}"
} | {
"split": "train",
"index": 3429,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the greatest common divisor of 957 and 1537. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 3430,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sofia has forgotten the passcode of her phone. She only remembers that it has four digits and that the product of its digits is $18$ . How many passcodes satisfy these conditions? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 3431,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "With $400$ members voting the House of Representatives defeated a bill. A re-vote, with the same members voting, resulted in the passage of the bill by twice the margin by which it was originally defeated. The number voting for the bill on the revote was $\\frac{12}{11}$ of the... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 3432,
"solution": "Let $x$ be the number of members who voted for the bill the first time, and $y$ be the number of members who voted against it the first time. Since the total number of members voting is $400$, we have:\n\\[ x + y = 400 \\]\n\nThe bill was defeated the first time, so... |
deepscaler | [
{
"role": "user",
"content": "Given Ms. Grace Swift leaves her house at 7:30 AM to reach her office, she arrives 5 minutes late at an average speed of 50 miles per hour, and 5 minutes early at an average speed of 70 miles per hour, determine the average speed she should maintain to arrive exactly on time. L... | math | {
"style": "rule",
"ground_truth": "58"
} | {
"split": "train",
"index": 3433,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let a binary operation $\\star$ on ordered pairs of integers be defined by $(a,b)\\star (c,d)=(a-c,b+d)$. Then, if $(3,3)\\star (0,0)$ and $(x,y)\\star (3,2)$ represent identical pairs, $x$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 3434,
"solution": "1. **Define the operation $\\star$:** Given $(a,b) \\star (c,d) = (a-c, b+d)$.\n\n2. **Calculate $(3,3) \\star (0,0)$:**\n \\[\n (3,3) \\star (0,0) = (3-0, 3+0) = (3,3)\n \\]\n\n3. **Calculate $(x,y) \\star (3,2)$:**\n \\[\n (x,y) \\star (3,2) = (x-3, y+2)... |
deepscaler | [
{
"role": "user",
"content": "If the line $l_{1}$: $x+my-2=0$ intersects the line $l_{2}$: $mx-y+2=0$ at point $P$, and a tangent line passing through point $P$ is drawn to the circle $C: (x+2)^{2} + (y+2)^{2} = 1$, with the point of tangency being $M$, then the maximum value of $|PM|$ is ____. Let's think ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{31}"
} | {
"split": "train",
"index": 3435,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alli rolls a standard 8-sided die twice. What is the probability of rolling integers that differ by 3 on her first two rolls? 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": "\\dfrac{1}{8}"
} | {
"split": "train",
"index": 3436,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$ with a common difference $d \\neq 0$, and $a_1=1$, $a_3$, $a_{13}$ form a geometric sequence. Find the minimum value of $\\frac{2S_n+8}{a_n+3}$ for all positive integers $n$. Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"split": "train",
"index": 3437,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $\\begin{cases} x=t \\\\ y= \\sqrt {2}+2t \\end{cases}$ (where $t$ is the parameter), with point $O$ as the pole and the positive $x$-axis as the polar axis, the polar coordinate equation of curve ... | math | {
"style": "rule",
"ground_truth": "\\frac {3 \\sqrt {10}}{5}"
} | {
"split": "train",
"index": 3438,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A=(0,9)$ and $B=(0,12)$. Points $A'$ and $B'$ are on the line $y=x$, and $\\overline{AA'}$ and $\\overline{BB'}$ intersect at $C=(2,8)$. What is the length of $\\overline{A'B'}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 3439,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of\n\\[x^2 + xy + y^2\\]over all real numbers $x$ and $y.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 3440,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Anton writes down all positive integers that are divisible by 2. Berta writes down all positive integers that are divisible by 3. Clara writes down all positive integers that are divisible by 4. The orderly Dora notes the numbers written by the others. She arranges these number... | math | {
"style": "rule",
"ground_truth": "3026"
} | {
"split": "train",
"index": 3441,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $x$ if $x=\\frac{2009^2-2009}{2009}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2008"
} | {
"split": "train",
"index": 3442,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A game of drawing balls involves a non-transparent paper box containing $6$ identical-sized, differently colored glass balls. Participants pay $1$ unit of fee to play the game once, drawing balls with replacement three times. Participants must specify a color from the box befor... | math | {
"style": "rule",
"ground_truth": "110"
} | {
"split": "train",
"index": 3443,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Problem\nFind all pairs of primes $(p,q)$ for which $p-q$ and $pq-q$ are both perfect squares. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(p, q) = (3, 2)"
} | {
"split": "train",
"index": 3444,
"solution": "We first consider the case where one of $p,q$ is even. If $p=2$ , $p-q=0$ and $pq-q=2$ which doesn't satisfy the problem restraints. If $q=2$ , we can set $p-2=x^2$ and $2p-2=y^2$ giving us $p=y^2-x^2=(y+x)(y-x)$ . This forces $y-x=1$ so $p=2x+1\\rightarrow 2x+1=x^2... |
deepscaler | [
{
"role": "user",
"content": "Given triangle $ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively. It is known that $a=5$ and $\\sin A= \\frac{\\sqrt{5}}{5}$.\n(1) If the area of triangle $ABC$ is $\\sqrt{5}$, find the minimum value of the perimeter $l$.\n(2) If $\\cos B= \\... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 3445,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There were four space stations in the three-dimensional space, each pair spaced 1 light year away from each other. Determine the volume, in cubic light years, of the set of all possible locations for a base such that the sum of squares of the distances from the base to each of ... | math | {
"style": "rule",
"ground_truth": "\\frac{27 \\sqrt{6} \\pi}{8}"
} | {
"split": "train",
"index": 3446,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The matrices\n\\[\\begin{pmatrix} a & 2 \\\\ 1 & 4 \\end{pmatrix} \\quad \\text{and} \\quad \\begin{pmatrix} -\\frac{2}{7} & \\frac{1}{7} \\\\ b & \\frac{3}{14} \\end{pmatrix}\\]are inverses. Enter the ordered pair $(a,b).$ Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "\\left( -3, \\frac{1}{14} \\right)"
} | {
"split": "train",
"index": 3447,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Convert the point $( -2, -2 \\sqrt{3}, -1)$ in rectangular coordinates to cylindrical coordinates. Enter your answer in the form $(r,\\theta,z),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left( 4, \\frac{4 \\pi}{3}, -1 \\right)"
} | {
"split": "train",
"index": 3448,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The whole numbers from 1 to 1000 are written. How many of these numbers have at least two 7's appearing side-by-side? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 3449,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest value of $n$ such that $3x^2 +nx + 72$ can be factored as the product of two linear factors with integer coefficients. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "217"
} | {
"split": "train",
"index": 3450,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ and $c$ be distinct real numbers such that\n\\[\\frac{a^3 + 6}{a} = \\frac{b^3 + 6}{b} = \\frac{c^3 + 6}{c}.\\]Find $a^3 + b^3 + c^3.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-18"
} | {
"split": "train",
"index": 3451,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rabbit and a hedgehog participated in a running race on a 550 m long circular track, both starting and finishing at the same point. The rabbit ran clockwise at a speed of 10 m/s and the hedgehog ran anticlockwise at a speed of 1 m/s. When they met, the rabbit continued as bef... | math | {
"style": "rule",
"ground_truth": "545"
} | {
"split": "train",
"index": 3452,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\angle C=90^{\\circ}$, \\(A D=D B\\), \\(D E \\perp A B\\), \\(A B=20\\), and \\(A C=12\\), find the area of quadrilateral \\(AD EC\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "58 \\frac{1}{2}"
} | {
"split": "train",
"index": 3453,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(The full score for this question is 8 points) There are 4 red cards labeled with the numbers 1, 2, 3, 4, and 2 blue cards labeled with the numbers 1, 2. Four different cards are drawn from these 6 cards. \n(1) If it is required that at least one blue card is drawn, how many d... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 3454,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the paths from \\((0,0)\\) to \\((6,3)\\) that only take steps of unit length up and right. Compute the sum of the areas bounded by the path, the \\(x\\)-axis, and the line \\(x=6\\) over all such paths. Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "756"
} | {
"split": "train",
"index": 3455,
"solution": "We see that the sum of the areas under the path is equal the sum of the areas above the path. Thus, the sum of the areas under the path is half the area of the rectangle times the number of paths, which is \\(\\frac{18\\binom{9}{3}}{2}=756\\)."
} |
deepscaler | [
{
"role": "user",
"content": "Four points are on a line segment.\n\nIf \\( A B : B C = 1 : 2 \\) and \\( B C : C D = 8 : 5 \\), then \\( A B : B D \\) equals Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4 : 13"
} | {
"split": "train",
"index": 3456,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of $0$, and the score increases by $1$ for each like vote and decreases by $1$ for each dislike vote. At one point Sangho saw that his video had a score... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 3457,
"solution": "1. **Define Variables:**\n Let $x$ be the total number of votes cast. Since $65\\%$ of the votes are likes and $35\\%$ are dislikes, we can denote the number of like votes as $0.65x$ and the number of dislike votes as $0.35x$.\n\n2. **Set Up the Equation:**\n Ea... |
deepscaler | [
{
"role": "user",
"content": "Pentagon $ANDD'Y$ has $AN \\parallel DY$ and $AY \\parallel D'N$ with $AN = D'Y$ and $AY = DN$ . If the area of $ANDY$ is 20, the area of $AND'Y$ is 24, and the area of $ADD'$ is 26, the area of $ANDD'Y$ can be expressed in the form $\\frac{m}{n}$ for relativ... | math | {
"style": "rule",
"ground_truth": "71"
} | {
"split": "train",
"index": 3458,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The line $y=kx$ intersects the graph of the function $y=\\tan x$ ($-\\frac{π}{2}＜x＜\\frac{π}{2}$) at points $M$ and $N$ (not coinciding with the origin $O$). The coordinates of point $A$ are $(-\\frac{π}{2},0)$. Find $(\\overrightarrow{AM}+\\overrightarrow{AN})\\cdot\\overright... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi^2}{2}"
} | {
"split": "train",
"index": 3459,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)={e}^{-x}+ \\frac{nx}{mx+n}$.\n\n(1) If $m=0$, $n=1$, find the minimum value of the function $f(x)$;\n\n(2) If $m > 0$, $n > 0$, and the minimum value of $f(x)$ on $[0,+\\infty)$ is $1$, find the maximum value of $\\frac{m}{n}$. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 3460,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many roots does $\\arctan x=x^{2}-1.6$ have, where the arctan function is defined in the range $-\\frac{p i}{2}<\\arctan x<\\frac{p i}{2}$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 3461,
"solution": "2."
} |
deepscaler | [
{
"role": "user",
"content": "A flag in the shape of a square has a symmetrical cross of uniform width, featuring a center decorated with a green square, and set against a yellow background. The entire cross (comprising both the arms and the green center) occupies 49% of the total area of the flag. What per... | math | {
"style": "rule",
"ground_truth": "25\\%"
} | {
"split": "train",
"index": 3462,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It takes $5$ seconds for a clock to strike $6$ o'clock beginning at $6:00$ o'clock precisely. If the strikings are uniformly spaced, how long, in seconds, does it take to strike $12$ o'clock? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 3463,
"solution": "1. **Understanding the Problem:** The problem states that it takes 5 seconds for a clock to strike 6 times at 6 o'clock. We need to find out how long it takes for the clock to strike 12 times.\n\n2. **Analyzing the 6 o'clock strikes:** When the clock strikes 6 times... |
deepscaler | [
{
"role": "user",
"content": "A \"pass level game\" has the following rules: On the \\( n \\)-th level, a die is thrown \\( n \\) times. If the sum of the points that appear in these \\( n \\) throws is greater than \\( 2^{n} \\), then the player passes the level. \n\n1. What is the maximum number of levels... | math | {
"style": "rule",
"ground_truth": "\\frac{100}{243}"
} | {
"split": "train",
"index": 3464,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Vitya collects toy cars from the \"Retro\" series. The problem is that the total number of different models in the series is unknown — it's a big commercial secret. However, it is known that different cars are produced in equal quantities, so it can be assumed that all models a... | math | {
"style": "rule",
"ground_truth": "58"
} | {
"split": "train",
"index": 3465,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the nearest integer to $$100 \\sum_{n=1}^{\\infty} 3^{n} \\sin ^{3}\\left(\\frac{\\pi}{3^{n}}\\right)$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "236"
} | {
"split": "train",
"index": 3466,
"solution": "Note that we have $$\\sin 3 x=3 \\sin x-4 \\sin ^{3} x \\Longrightarrow \\sin ^{3} x=\\frac{1}{4}(3 \\sin x-\\sin 3 x)$$ which implies that $$\\frac{\\sin ^{3} x}{3 x}=\\frac{1}{4}\\left(\\frac{\\sin x}{x}-\\frac{\\sin 3 x}{3 x}\\right)$$ Substituting $x=\\frac{\\pi... |
deepscaler | [
{
"role": "user",
"content": "Given a sequence of natural numbers $\\left\\{x_{n}\\right\\}$ defined by:\n$$\nx_{1}=a, x_{2}=b, x_{n+2}=x_{n}+x_{n+1}, \\quad n=1,2,3,\\cdots\n$$\nIf an element of the sequence is 1000, what is the minimum possible value of $a+b$? Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 3467,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the polar equation of a circle is $\\rho=2\\cos \\theta$, the distance from the center of the circle to the line $\\rho\\sin \\theta+2\\rho\\cos \\theta=1$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac { \\sqrt {5}}{5}"
} | {
"split": "train",
"index": 3468,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A barrel with a height of 1.5 meters is completely filled with water and sealed with a lid. The mass of the water in the barrel is 1000 kg. A long, thin vertical tube with a cross-section of $1 \\mathrm{~cm}^{2}$ is inserted into the lid of the barrel and completely filled with... | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"split": "train",
"index": 3469,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many subsets $S$ of the set $\\{1,2, \\ldots, 10\\}$ satisfy the property that, for all $i \\in[1,9]$, either $i$ or $i+1$ (or both) is in $S$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 3470,
"solution": "We do casework on the number of $i$ 's not in $S$. Notice that these $i$ 's that are not in $S$ cannot be consecutive, otherwise there exists an index $i$ such that both $i$ and $i+1$ are both not in $S$. Hence if there are $k i$ 's not in $S$, we want to arrange $k... |
deepscaler | [
{
"role": "user",
"content": "The carbon dioxide emissions in a certain region reach a peak of a billion tons (a > 0) and then begin to decline. The relationship between the carbon dioxide emissions S (in billion tons) and time t (in years) satisfies the function S = a · b^t. If after 7 years, the carbon di... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 3471,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A shooter, in a shooting training session, has the probabilities of hitting the 10, 9, 8, and 7 rings as follows:\n\n0.21, 0.23, 0.25, 0.28, respectively. Calculate the probability that the shooter in a single shot:\n\n(1) Hits either the 10 or 9 ring; (2) Scores less than 7 ri... | math | {
"style": "rule",
"ground_truth": "0.03"
} | {
"split": "train",
"index": 3472,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) is continuous. For every real number \\( x \\), the equation \\( f(x) \\cdot f(f(x)) = 1 \\) holds. It is known that \\( f(1000) = 999 \\). Find \\( f(500) \\). Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{500}"
} | {
"split": "train",
"index": 3473,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The young man paid with $100 for a gift that cost $18 and received $79 in change from Mr. Wang. Mr. Wang then returned the counterfeit $100 bill to the neighbor. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "97"
} | {
"split": "train",
"index": 3474,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $a=3$, $\\angle C = \\frac{2\\pi}{3}$, and the area of $ABC$ is $\\frac{3\\sqrt{3}}{4}$. Find the lengths of sides $b$ and $c$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{13}"
} | {
"split": "train",
"index": 3475,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the ratio of the area of the shaded square to the area of the large square given that the large square is divided into a grid of $25$ smaller squares each of side $1$ unit? The shaded square consists of a pentagon formed by connecting the centers of adjacent half-square... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{10}"
} | {
"split": "train",
"index": 3476,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The older brother and the younger brother each bought several apples. The older brother said to the younger brother, \"If I give you one apple, we will have the same number of apples.\" The younger brother thought for a moment and said to the older brother, \"If I give you one ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 3477,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x)=m\\sin (πx+α)+n\\cos (πx+β)+8$, where $m$, $n$, $α$, $β$ are all real numbers. If $f(2000)=-2000$, find $f(2015)$ = \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"split": "train",
"index": 3478,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice and Bob play a game on a circle with 8 marked points. Alice places an apple beneath one of the points, then picks five of the other seven points and reveals that none of them are hiding the apple. Bob then drops a bomb on any of the points, and destroys the apple if he dr... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 3479,
"solution": "Let the points be $0, \\ldots, 7(\\bmod 8)$, and view Alice's reveal as revealing the three possible locations of the apple. If Alice always picks $0,2,4$ and puts the apple randomly at 0 or 4 , by symmetry Bob cannot achieve more than $\\frac{1}{2}$. Here's a proof... |
deepscaler | [
{
"role": "user",
"content": "A row consists of 10 chairs, but chair #5 is broken and cannot be used. Mary and James each sit in one of the available chairs, choosing their seats at random from the remaining chairs. What is the probability that they don't sit next to each other? Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"split": "train",
"index": 3480,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a \"prime date,\" both the month and the day are prime numbers. For example, Feb. 7 or 2/7 is a prime date. How many prime dates occurred in 2007? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 3481,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a quadrilateral $ABCD$ inscribed in a circle with side $AB$ extended beyond $B$ to point $E$, if $\\angle BAD=92^\\circ$ and $\\angle ADC=68^\\circ$, find $\\angle EBC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "68^\\circ"
} | {
"split": "train",
"index": 3482,
"solution": "1. **Identify Properties of Cyclic Quadrilateral**: Since $ABCD$ is a cyclic quadrilateral, by the Inscribed Angle Theorem, opposite angles in a cyclic quadrilateral sum to $180^\\circ$. Thus, we have:\n \\[\n \\angle ABC + \\angle ADC = 180^\\circ\n \\]\n\n2.... |
deepscaler | [
{
"role": "user",
"content": "A light flashes green every 3 seconds. Determine the number of times the light has flashed green after 671 seconds. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "154"
} | {
"split": "train",
"index": 3483,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, draw one at random. The probability of drawing a prime number is ____, and the probability of drawing a composite number is ____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"split": "train",
"index": 3484,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function \\( f(x) = \\cos x + \\log_2 x \\) for \\( x > 0 \\), if the positive real number \\( a \\) satisfies \\( f(a) = f(2a) \\), then find the value of \\( f(2a) - f(4a) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 3485,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An ordinary $6$-sided die has a number on each face from $1$ to $6$ (each number appears on one face). How many ways can I paint two faces of a die red, so that the numbers on the red faces don't add up to $7$? Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 3486,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The lengths of the sides of a triangle with positive area are $\\log_{2}9$, $\\log_{2}50$, and $\\log_{2}n$, where $n$ is a positive integer. Find the number of possible values for $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "445"
} | {
"split": "train",
"index": 3487,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the ratio of $b$ to $a$ is 3, then what is the value of $a$ when $b=12-5a$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 3488,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $\\{a\\_n\\}$ satisfies $(a_{n+1}-1)(1-a_{n})=a_{n}$, $a_{8}=2$, then $S_{2017}=$ _____ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {2017}{2}"
} | {
"split": "train",
"index": 3489,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence \\(\\left\\{a_{n}\\right\\}\\) satisfies: \\(a_1 = 1\\), and for each \\(n \\in \\mathbf{N}^{*}\\), \\(a_n\\) and \\(a_{n+1}\\) are the roots of the equation \\(x^2 + 3nx + b_n = 0\\). Find \\(\\sum_{k=1}^{20} b_k\\). Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "6385"
} | {
"split": "train",
"index": 3490,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $1 - 0.\\overline{123}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{292}{333}"
} | {
"split": "train",
"index": 3491,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using the digits 0, 1, 2, 3, 4, how many even three-digit numbers can be formed if each digit can be used more than once, and the number must be greater than 200? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 3492,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $m \\angle A= 60^\\circ$, $BC=12$ units, $\\overline{BD} \\perp \\overline{AC}$, $\\overline{CE} \\perp \\overline{AB}$ and $m \\angle DBC = 3m \\angle ECB$, the length of segment $EC$ can be expressed in the form $a(\\sqrt{b}+\\sqrt{c})$ units where $b$ and $c$ have... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 3493,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\alpha \\in (0, \\frac{\\pi}{2})$ and $\\beta \\in (0, \\frac{\\pi}{2})$, and $\\sin(2\\alpha + \\beta) = \\frac{3}{2} \\sin(\\beta)$, find the minimum value of $\\cos(\\beta)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{3}"
} | {
"split": "train",
"index": 3494,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A company is planning to increase the annual production of a product by implementing technical reforms in 2013. According to the survey, the product's annual production volume $x$ (in ten thousand units) and the technical reform investment $m$ (in million yuan, where $m \\ge 0$... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 3495,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify first, then evaluate: $(1-\\frac{1}{a+1})÷\\frac{a^2-2a+1}{a^2-1}$, where $a=-2$. 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": 3496,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $M=\\{1,2,x\\}$, we call the set $M$, where $1$, $2$, $x$ are elements of set $M$. The elements in the set have definiteness (such as $x$ must exist), distinctiveness (such as $x\\neq 1, x\\neq 2$), and unorderedness (i.e., changing the order of elements does not change t... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 3497,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define $E(a,b,c) = a \\times b^c$. What positive value of $r$ is the solution to the equation $E(r,r,3) = 625$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 3498,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We know that every natural number has factors. For a natural number $a$, we call the positive factors less than $a$ the proper factors of $a$. For example, the positive factors of $10$ are $1$, $2$, $5$, $10$, where $1$, $2$, and $5$ are the proper factors of $10$. The quotient... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 3499,
"solution": ""
} |
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