data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "If triangle $PQR$ has sides of length $PQ = 7$, $PR = 8$, and $QR = 6$, then calculate\n\\[\n\\frac{\\cos \\frac{P - Q}{2}}{\\sin \\frac{R}{2}} - \\frac{\\sin \\frac{P - Q}{2}}{\\cos \\frac{R}{2}}.\n\\] Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{7}"
} | {
"split": "train",
"index": 4000,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $$f(x)= \\begin{cases} \\sqrt {x}+3, & x\\geq0 \\\\ ax+b, & x<0\\end{cases}$$ satisfies the condition: $y=f(x)$ is a monotonic function on $\\mathbb{R}$ and $f(a)=-f(b)=4$, then the value of $f(-1)$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 4001,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a$ is a constant such that $4x^2 - 12x + a$ is the square of a binomial, then what is $a$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 4002,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In order to test students' mastery of high school mathematics knowledge, two opaque boxes, Box A and Box B, are prepared. Box A contains 2 conceptual description questions and 2 calculation questions; Box B contains 2 conceptual description questions and 3 calculation questions... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{7}"
} | {
"split": "train",
"index": 4003,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x, y, z$ be real numbers such that $x + y + z = 2$, and $x \\ge -\\frac{2}{3}$, $y \\ge -1$, and $z \\ge -2$. Find the maximum value of \n\\[\\sqrt{3x + 2} + \\sqrt{3y + 4} + \\sqrt{3z + 7}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{57}"
} | {
"split": "train",
"index": 4004,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the 512 smallest positive integers written in base 8 use 5 or 6 (or both) as a digit? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "296"
} | {
"split": "train",
"index": 4005,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\frac{\\sin \\theta + \\cos \\theta}{\\sin \\theta - \\cos \\theta} = 2$, find the value of $\\frac{\\sin \\theta}{\\cos^{3} \\theta} + \\frac{\\cos \\theta}{\\sin^{3} \\theta}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{820}{27}"
} | {
"split": "train",
"index": 4006,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A and B play a repeated game. In each game, they take turns tossing a fair coin, and the player who first gets heads wins the game. In the first game, A tosses first. For subsequent games, the loser of the previous game starts first. What is the probability that A wins the 6th ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}\\left(1 - \\frac{1}{729}\\right)"
} | {
"split": "train",
"index": 4007,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express the product of $1001_2$ and $121_3$ in base 10. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 4008,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain intelligence station has four different passwords A, B, C, and D. Each week, one of the passwords is used, and the password for each week is equally likely to be randomly selected from the three passwords not used in the previous week. If password A is used in the fir... | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"split": "train",
"index": 4009,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( f(x) = \\max \\left| x^3 - a x^2 - b x - c \\right| \\) for \\( 1 \\leq x \\leq 3 \\), find the minimum value of \\( f(x) \\) as \\( a, b, \\) and \\( c \\) range over all real numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/4"
} | {
"split": "train",
"index": 4010,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the product of the values of $x$ that satisfy the equation $|4x|+3=35$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-64"
} | {
"split": "train",
"index": 4011,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the equation \\( x^{2} - a|x| + a^{2} - 3 = 0 \\) has a unique real solution, then \\( a = \\) ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\sqrt{3}"
} | {
"split": "train",
"index": 4012,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "4.\tFind the biggest positive integer $n$ , lesser thar $2012$ , that has the following property:\nIf $p$ is a prime divisor of $n$ , then $p^2 - 1$ is a divisor of $n$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1944"
} | {
"split": "train",
"index": 4013,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest value of $n$ for which the series \\[1\\cdot 3^1 + 2\\cdot 3^2 + 3\\cdot 3^3 + \\cdots + n\\cdot 3^n\\] exceeds $3^{2007}$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"split": "train",
"index": 4014,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice and the Cheshire Cat play a game. At each step, Alice either (1) gives the cat a penny, which causes the cat to change the number of (magic) beans that Alice has from $n$ to $5n$ or (2) gives the cat a nickel, which causes the cat to give Alice another bean. Alice wins (a... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 4015,
"solution": "Consider the number of beans Alice has in base 5. Note that $2008=31013_{5}, 42=132_{5}$, and $100=400_{5}$. Now, suppose Alice has $d_{k} \\cdots d_{2} d_{1}$ beans when she wins; the conditions for winning mean that these digits must satisfy $d_{2} d_{1}=32, d_{k}... |
deepscaler | [
{
"role": "user",
"content": "Three non-collinear lattice points $A,B,C$ lie on the plane $1+3x+5y+7z=0$ . The minimal possible area of triangle $ABC$ can be expressed as $\\frac{\\sqrt{m}}{n}$ where $m,n$ are positive integers such that there does not exists a prime $p$ dividing $n$ with $p^2... | math | {
"style": "rule",
"ground_truth": "8302"
} | {
"split": "train",
"index": 4016,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the numbers $1, 2, 3, 4, 5, 6$ are randomly arranged in a row, represented as $a, b, c, d, e, f$, what is the probability that the number $a b c + d e f$ is odd? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/10"
} | {
"split": "train",
"index": 4017,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A given sequence $r_1, r_2, \\dots, r_n$ of distinct real numbers can be put in ascending order by means of one or more \"bubble passes\". A bubble pass through a given sequence consists of comparing the second term with the first term, and exchanging them if and only if the se... | math | {
"style": "rule",
"ground_truth": "931"
} | {
"split": "train",
"index": 4018,
"solution": "If any of $r_1, \\ldots, r_{19}$ is larger than $r_{20}$, one of these numbers will be compared with $r_{20}$ on the 19th step of the first bubble pass and $r_{20}$ will be moved back to the 19th position. Thus, $r_{20}$ must be the largest of the first 20 terms. In... |
deepscaler | [
{
"role": "user",
"content": "What multiple of 15 is closest to 3500? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3510"
} | {
"split": "train",
"index": 4019,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$\\sqrt{53+20\\sqrt{7}}$ can be written in the form $a+b\\sqrt{c}$, where $a,$ $b,$ and $c$ are integers and $c$ has no factors which is a perfect square of any positive integer other than 1. Find $a+b+c$. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 4020,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I planned to work 25 hours a week for 15 weeks to earn $3750$ for a vacation. However, due to a family emergency, I couldn't work for the first three weeks. How many hours per week must I work for the remaining weeks to still afford the vacation? Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "31.25"
} | {
"split": "train",
"index": 4021,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of all polynomials of the form $z^3 + az^2 + bz + c$, where $a$, $b$, and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $|z| = 20$ or $|z| = 13$. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "540"
} | {
"split": "train",
"index": 4022,
"solution": "There are two cases: either all the roots are real, or one is real and two are imaginary.\nCase 1: All roots are real. Then each of the roots is a member of the set $\\{-20, 20, -13, 13\\}$. It splits into three sub-cases: either no two are the same, exactly two are... |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC,$ $M$ is the midpoint of $\\overline{BC},$ $AB = 15,$ and $AC = 21.$ Let $E$ be on $\\overline{AC},$ and $F$ be on $\\overline{AB},$ and let $G$ be the intersection of $\\overline{EF}$ and $\\overline{AM}.$ If $AE = 3AF,$ then find $\\frac{EG}{GF}.$ Let's think... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{2}"
} | {
"split": "train",
"index": 4023,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(t)$ be the cubic polynomial for $t$ such that $\\cos 3x=f(\\cos x)$ holds for all real number $x$ .\r\nEvaluate\r\n\r\n\\[\\int_0^1 \\{f(t)\\}^2 \\sqrt{1-t^2}dt\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{8}"
} | {
"split": "train",
"index": 4024,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $\\frac{x^{2}}{a-3} + \\frac{y^{2}}{2-a} = 1$, with foci on the $y$-axis and a focal distance of $4$, determine the value of $a$. The options are:\nA) $\\frac{3}{2}$\nB) $5$\nC) $7$\nD) $\\frac{1}{2}$ Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 4025,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two concentric circles with radii of 19 and 29 units bound a shaded region. A third circle will be drawn with area equal to that of the shaded area. What must the radius of the third circle be? Express your answer in simplest radical form.\n\n[asy]\nfilldraw(circle((0,0),29),gr... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{30}"
} | {
"split": "train",
"index": 4026,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Grandpa is twice as strong as Grandma, Grandma is three times as strong as Granddaughter, Granddaughter is four times as strong as Doggie, Doggie is five times as strong as Cat, and Cat is six times as strong as Mouse. Grandpa, Grandma, Granddaughter, Doggie, and Cat together w... | math | {
"style": "rule",
"ground_truth": "1237"
} | {
"split": "train",
"index": 4027,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 7 volunteers to be arranged to participate in the promotional activities for the Shanghai World Expo on Saturday and Sunday, with 6 people participating each day and 3 people arranged for each day. There are __________ different arrangements possible (answer with a nu... | math | {
"style": "rule",
"ground_truth": "140"
} | {
"split": "train",
"index": 4028,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of the symbols $\\star$ and $*$ represents an operation in the set $\\{+,-,\\times,\\div\\}$, and $\\frac{12\\star 2}{9*3}=2$. What is the value of $\\frac{7\\star 3}{12*6}$? Express your answer as a common fraction. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{6}"
} | {
"split": "train",
"index": 4029,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be nonzero real numbers such that $\\tfrac{1}{3a}+\\tfrac{1}{b}=2011$ and $\\tfrac{1}{a}+\\tfrac{1}{3b}=1$ . What is the quotient when $a+b$ is divided by $ab$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1509"
} | {
"split": "train",
"index": 4030,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the future, MIT has attracted so many students that its buildings have become skyscrapers. Ben and Jerry decide to go ziplining together. Ben starts at the top of the Green Building, and ziplines to the bottom of the Stata Center. After waiting $a$ seconds, Jerry starts at t... | math | {
"style": "rule",
"ground_truth": "740"
} | {
"split": "train",
"index": 4031,
"solution": "Note that due to all the 3-4-5 triangles, we find $\\frac{x}{z}=\\frac{z}{y}=\\frac{4}{3}$, so $120=x+y=\\frac{25}{12} z$. Then, $$u=\\frac{5}{3} x=\\frac{20}{9} z=\\frac{16}{15} 120=128$$ while $$v=\\frac{5}{4} y=\\frac{15}{16} z=\\frac{9}{20} 120=54$$ Thus $u-v=74... |
deepscaler | [
{
"role": "user",
"content": "Centered at each lattice point in the coordinate plane are a circle radius $\\frac{1}{10}$ and a square with sides of length $\\frac{1}{5}$ whose sides are parallel to the coordinate axes. The line segment from $(0,0)$ to $(1001, 429)$ intersects $m$ of the squares and $n$ of t... | math | {
"style": "rule",
"ground_truth": "574"
} | {
"split": "train",
"index": 4032,
"solution": "This is mostly a clarification to Solution 1, but let's take the diagram for the origin to $(7,3)$. We have the origin circle and square intersected, then two squares, then the circle and square at $(7,3)$. If we take the circle and square at the origin out of the d... |
deepscaler | [
{
"role": "user",
"content": "If the system of equations \n\\[\n\\begin{align*}\n4x + y &= a, \\\\\n3x + 4y^2 &= 3a,\n\\end{align*}\n\\]\nhas a solution $(x,y)$ when $x=3$, compute $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9.75"
} | {
"split": "train",
"index": 4033,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right triangle $ABC$, $AB=9$, $BC=13$, and $\\angle B = 90^\\circ$. Points $D$ and $E$ are midpoints of $\\overline{AB}$ and $\\overline{AC}$ respectively; $\\overline{CD}$ and $\\overline{BE}$ intersect at point $X$. Compute the ratio of the area of quadrilateral $AEXD$ t... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 4034,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\tan 3825^\\circ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 4035,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the coefficient of the term containing $x^3$ in the expansion of ${(1+2x)}^{5}$. (The result should be represented as a number.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 4036,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of a triangle with side lengths 13, 14, and 14. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6.5\\sqrt{153.75}"
} | {
"split": "train",
"index": 4037,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a modified game similar to Deal or No Deal, participants choose a box at random from a set of 30 boxes, each containing one of the following values: \n\\begin{tabular}{|c|c|}\n\\hline\n\\$0.50 & \\$2,000 \\\\\n\\hline\n\\$2 & \\$10,000 \\\\\n\\hline\n\\$10 & \\$20,000 \\\\\n... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 4038,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = 4\\cos(\\omega x - \\frac{\\pi}{6})\\sin \\omega x - \\cos(2\\omega x + \\pi)$, where $\\omega > 0$.\n(I) Find the range of the function $y = f(x)$.\n(II) If $f(x)$ is an increasing function on the interval $[-\\frac{3\\pi}{2}, \\frac{\\pi}{2}]$, find... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 4039,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a checkered square with a side length of 2018, some cells are painted white and the rest are black. It is known that from this square, one can cut out a 10x10 square where all the cells are white, and a 10x10 square where all the cells are black. What is the smallest value f... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 4040,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest $2$-digit prime factor of the integer $n = {300\\choose 150}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "97"
} | {
"split": "train",
"index": 4041,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $(b_1,b_2,b_3,\\ldots,b_{10})$ be a permutation of $(1,2,3,\\ldots,10)$ for which\n$b_1>b_2>b_3>b_4 \\mathrm{\\ and \\ } b_4<b_5<b_6<b_7<b_8<b_9<b_{10}.$\nFind the number of such permutations. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 4042,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all real numbers $x$ for which $$\\lfloor\\lfloor\\cdots\\lfloor\\lfloor\\lfloor x\\rfloor+x\\rfloor+x\\rfloor \\cdots\\rfloor+x\\rfloor=2017 \\text { and }\\{\\{\\cdots\\{\\{\\{x\\}+x\\}+x\\} \\cdots\\}+x\\}=\\frac{1}{2017}$$ where there are $2017 x$ 's in both... | math | {
"style": "rule",
"ground_truth": "3025 \\frac{1}{2017}"
} | {
"split": "train",
"index": 4043,
"solution": "The two equations are equivalent to $2017\\lfloor x\\rfloor=2017$ and $\\{2017 x\\}=\\frac{1}{2017}$, respectively. The first equation reduces to $\\lfloor x\\rfloor=1$, so we must have $x=1+r$ for some real $r$ satisfying $0 \\leq r<1$. From the second equation, we... |
deepscaler | [
{
"role": "user",
"content": "Let $\\omega$ be a nonreal root of $x^3 = 1.$ Compute\n\\[(2 - \\omega + 2\\omega^2)^6 + (2 + \\omega - 2\\omega^2)^6.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "38908"
} | {
"split": "train",
"index": 4044,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be positive integers such that all but $2009$ positive integers are expressible in the form $ma + nb$ , where $m$ and $n$ are nonnegative integers. If $1776 $ is one of the numbers that is not expressible, \ffind $a + b$ . Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "133"
} | {
"split": "train",
"index": 4045,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers can be expressed in the form: $\\pm 1 \\pm 2 \\pm 3 \\pm 4 \\pm \\cdots \\pm 2018$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2037172"
} | {
"split": "train",
"index": 4046,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the final of the giraffe beauty contest, two giraffes, Tall and Spotted, reached the finals. There are 135 voters divided into 5 districts, with each district divided into 9 precincts, and each precinct having 3 voters. The voters in each precinct choose the winner by majori... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 4047,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an acute triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively. Given $a=4$, $b=5$, and the area of $\\triangle ABC$ is $5\\sqrt{3}$, find the values of $c$ and $\\sin A$. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{7}}{7}"
} | {
"split": "train",
"index": 4048,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can 7 people be seated in a row of chairs if two of the people, Wilma and Paul, refuse to sit next to each other? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3600"
} | {
"split": "train",
"index": 4049,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A zoo has a menagerie containing four pairs of different animals, one male and one female for each. The zookeeper wishes to feed the animals in a specific pattern: each time he feeds a single animal, the next one he feeds must be a different gender. If he starts by feeding the ... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 4050,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The matrix\n\\[\\begin{pmatrix} \\frac{4}{29} & -\\frac{10}{29} \\\\ -\\frac{10}{29} & \\frac{25}{29} \\end{pmatrix}\\]corresponds to projecting onto a certain vector $\\begin{pmatrix} x \\\\ y \\end{pmatrix}.$ Find $\\frac{y}{x}.$ Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "-\\frac{5}{2}"
} | {
"split": "train",
"index": 4051,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that a child builds towers with $2$ red cubes, $3$ blue cubes, and $4$ green cubes, determine the number of different towers with a height of $8$ cubes that can be built, with one cube left out. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "1,260"
} | {
"split": "train",
"index": 4052,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The curvature of a polyhedron is $2 \\pi$ minus the sum of the face angles (the internal angles of the faces of the polyhedron). For example, a cube's face angle is $2 \\pi - 3 \\times \\frac{\\pi}{2} = \\frac{\\pi}{2}$, and its total curvature is $\\frac{\\pi}{2} \\times 8 = 4... | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"split": "train",
"index": 4053,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all triples $(a, b, c)$ of real numbers such that\n$$ a^2 + ab + c = 0, $$\n$$b^2 + bc + a = 0, $$\n$$c^2 + ca + b = 0.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(0, 0, 0)\\left(-\\frac{1}{2}, -\\frac{1}{2}, -\\frac{1}{2}\\right)"
} | {
"split": "train",
"index": 4054,
"solution": "\nTo find all triples \\((a, b, c)\\) of real numbers satisfying the system of equations:\n\n\\[\n\\begin{align*}\n1. \\quad & a^2 + ab + c = 0, \\\\\n2. \\quad & b^2 + bc + a = 0, \\\\\n3. \\quad & c^2 + ca + b = 0, \n\\end{align*}\n\\]\n\nwe proceed by considering... |
deepscaler | [
{
"role": "user",
"content": "Two circles of radius $s$ are externally tangent to each other and internally tangent to the ellipse $x^2 + 4y^2 = 8.$ Find $s.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\frac{3}{2}}"
} | {
"split": "train",
"index": 4055,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The value of $ 21!$ is $ 51{,}090{,}942{,}171{,}abc{,}440{,}000$ , where $ a$ , $ b$ , and $ c$ are digits. What is the value of $ 100a \\plus{} 10b \\plus{} c$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "709"
} | {
"split": "train",
"index": 4056,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle rests in the interior of the parabola with equation $y = x^2,$ so that it is tangent to the parabola at two points. How much higher is the center of the circle than the points of tangency? 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": 4057,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two different digits from 1 to 9 are chosen. One digit is placed in each box to complete the two 2-digit numbers shown. The result of subtracting the bottom number from the top number is calculated. How many of the possible results are positive? Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 4058,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\alpha$ and $\\beta$ be complex numbers such that $\\alpha + \\beta$ and $i(\\alpha - 2 \\beta)$ are both positive real numbers. If $\\beta = 3 + 2i,$ compute $\\alpha.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6 - 2i"
} | {
"split": "train",
"index": 4059,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that point P is on the curve \\( y = -x^{2} - 1 \\) and point Q is on the curve \\( x = 1 + y^{2} \\), find the minimum value of \\( |PQ| \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{2}}{4}"
} | {
"split": "train",
"index": 4060,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Many states use a sequence of three letters followed by a sequence of three digits as their standard license-plate pattern. Given that each three-letter three-digit arrangement is equally likely, the probability that such a license plate will contain at least one palindrome (a ... | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 4061,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Select two different natural numbers from the set $\\{1, 2, 3, ..., 8\\}$. What is the probability that the greatest common factor of these two numbers is one? 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{3}{4}"
} | {
"split": "train",
"index": 4062,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A phone fully charges in 1 hour 20 minutes on fast charge and in 4 hours on regular charge. Fedya initially put the completely discharged phone on regular charge, then switched to fast charge once he found the proper adapter, completing the charging process. Determine the total... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 4063,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Schools A and B are having a sports competition with three events. In each event, the winner gets 10 points and the loser gets 0 points, with no draws. The school with the highest total score after the three events wins the championship. It is known that the probabilities of sc... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 4064,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle \\(A B C\\) is considered. Point \\(F\\) is the midpoint of side \\(A B\\). Point \\(S\\) lies on the ray \\(A C\\) such that \\(C S = 2 A C\\). In what ratio does the line \\(S F\\) divide side \\(B C\\)? Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "2:3"
} | {
"split": "train",
"index": 4065,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x) = 2\\sin x (\\sin x + \\cos x)$, where $x \\in \\mathbb{R}$.\n\n(Ⅰ) Find the intervals of monotonic increase for the function $f(x)$.\n\n(Ⅱ) If $f\\left( \\frac{a}{2} \\right) = 1 + \\frac{3\\sqrt{2}}{5}$, and $\\frac{3\\pi}{4} < a < \\frac{5\\pi}{4}$, find the valu... | math | {
"style": "rule",
"ground_truth": "-\\frac{7\\sqrt{2}}{10}"
} | {
"split": "train",
"index": 4066,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From a deck of cards, 5 spades, 4 clubs, and 6 hearts, totaling 15 cards, are drawn. If drawing $m$ cards such that all three suits are present is a certain event, then the minimum value of $m$ is \\_\\_\\_\\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 4067,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of unordered triples of distinct points in the $4 \\times 4 \\times 4$ lattice grid $\\{0,1,2,3\\}^{3}$ that are collinear in $\\mathbb{R}^{3}$ (i.e. there exists a line passing through the three points). Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "376"
} | {
"split": "train",
"index": 4068,
"solution": "Define a main plane to be one of the $x y, y z, z x$ planes. Define a space diagonal to be a set of collinear points not parallel to a main plane. We classify the lines as follows: (a) Lines parallel to two axes (i.e. orthogonal to a main plane). Notice that given a... |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a parallelogram with $\\angle{ABC}=120^\\circ$, $AB=16$ and $BC=10$. Extend $\\overline{CD}$ through $D$ to $E$ so that $DE=4$. If $\\overline{BE}$ intersects $\\overline{AD}$ at $F$, then $FD$ is closest to Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 4069,
"solution": "1. **Identify Key Properties of Parallelogram**: In parallelogram $ABCD$, since $\\angle ABC = 120^\\circ$, we know that $\\angle ADC = 120^\\circ$ as well because opposite angles in a parallelogram are equal.\n\n2. **Extend Line $CD$ to $E$**: Given that $DE = 4$, ... |
deepscaler | [
{
"role": "user",
"content": "A sphere with volume $V$ is inside a closed right triangular prism $ABC-A_{1}B_{1}C_{1}$, where $AB \\perp BC$, $AB=6$, $BC=8$, and $AA_{1}=3$. Find the maximum value of $V$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9\\pi}{2}"
} | {
"split": "train",
"index": 4070,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six natural numbers (with possible repetitions) are written on the faces of a cube, such that the numbers on adjacent faces differ by more than 1. What is the smallest possible sum of these six numbers? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 4071,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that it is currently between 4:00 and 5:00 o'clock, and eight minutes from now, the minute hand of a clock will be exactly opposite to the position where the hour hand was six minutes ago, determine the exact time now. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "4:45\\frac{3}{11}"
} | {
"split": "train",
"index": 4072,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular 15-gon has $L$ lines of symmetry, and the smallest positive angle for which it has rotational symmetry is $R$ degrees. What is $L+R$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 4073,
"solution": "1. **Determine the Lines of Symmetry, $L$:**\n A regular polygon with an odd number of sides, such as a regular 15-gon, has lines of symmetry that pass through each vertex and bisect the opposite side. This is because each line of symmetry aligns a vertex with the... |
deepscaler | [
{
"role": "user",
"content": "The sum of two numbers is $45$. Their difference is $3$. What is the lesser of the two numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 4074,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a square, points A and B are midpoints of two adjacent sides. A line segment is drawn from point A to the opposite vertex of the side that does not contain B, forming a triangle. What fraction of the interior of the square is shaded, if the triangle is left unshaded?\n\n[asy... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 4075,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of the expression\n\n$$\n\\frac{\\left(3^{4}+4\\right) \\cdot\\left(7^{4}+4\\right) \\cdot\\left(11^{4}+4\\right) \\cdot \\ldots \\cdot\\left(2015^{4}+4\\right) \\cdot\\left(2019^{4}+4\\right)}{\\left(1^{4}+4\\right) \\cdot\\left(5^{4}+4\\right) \\cdot\\left... | math | {
"style": "rule",
"ground_truth": "4080401"
} | {
"split": "train",
"index": 4076,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square is inscribed in another square such that its vertices lie on the sides of the first square, and its sides form angles of $60^{\\circ}$ with the sides of the first square. What fraction of the area of the given square is the area of the inscribed square? Let's think ste... | math | {
"style": "rule",
"ground_truth": "4 - 2\\sqrt{3}"
} | {
"split": "train",
"index": 4077,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I am preparing for a French exam, which includes recalling vocabulary. There are 800 words in total on the exam, and the exam grade is based on the percentage of these words that I recall correctly. If I randomly guess the words I do not learn, my guessing has a 5% chance of be... | math | {
"style": "rule",
"ground_truth": "716"
} | {
"split": "train",
"index": 4078,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When three numbers are added two at a time the sums are 29, 46, and 53. What is the sum of all three numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 4079,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different positive, four-digit integers can be formed using the digits 2, 2, 9 and 9? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 4080,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A geometric sequence $(a_n)$ has $a_1=\\sin x$, $a_2=\\cos x$, and $a_3= \\tan x$ for some real number $x$. For what value of $n$ does $a_n=1+\\cos x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 4081,
"solution": "1. **Identify the common ratio**: Given a geometric sequence $(a_n)$ with $a_1 = \\sin x$, $a_2 = \\cos x$, and $a_3 = \\tan x$, we know that the common ratio $r$ satisfies $a_2 = r a_1$ and $a_3 = r a_2$. Thus, we have:\n \\[\n \\cos x = r \\sin x \\quad \\text... |
deepscaler | [
{
"role": "user",
"content": "On a grid, there are three locations $A, B, C$, with each small square having a side length of 100 meters. If roads are to be built along the grid lines to connect the three locations, what is the minimum total length of the roads in meters? Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"split": "train",
"index": 4082,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Which terms must be removed from the sum\n$\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{6}+\\frac{1}{8}+\\frac{1}{10}+\\frac{1}{12}$\nif the sum of the remaining terms is to equal $1$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8} \\text{ and } \\frac{1}{10}"
} | {
"split": "train",
"index": 4083,
"solution": "1. **Calculate the total sum of all terms**: We start by finding the sum of all the terms given in the problem:\n \\[\n \\frac{1}{2} + \\frac{1}{4} + \\frac{1}{6} + \\frac{1}{8} + \\frac{1}{10} + \\frac{1}{12}\n \\]\n To add these fractions, we need a common... |
deepscaler | [
{
"role": "user",
"content": "Using the digits $0$, $1$, $2$, $3$, $4$ to form a four-digit number without repeating any digit, determine the total number of four-digit numbers less than $2340$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 4084,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\alpha$ and $\\beta$ be real numbers. Find the minimum value of\n\\[(3 \\cos \\alpha + 4 \\sin \\beta - 7)^2 + (3 \\sin \\alpha + 4 \\cos \\beta - 12)^2.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 4085,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow {a}=( \\sqrt {3}\\sin x, m+\\cos x)$ and $\\overrightarrow {b}=(\\cos x, -m+\\cos x)$, and a function $f(x)= \\overrightarrow {a}\\cdot \\overrightarrow {b}$\n(1) Find the analytical expression of function $f(x)$;\n(2) When $x\\in[- \\frac {\\pi}... | math | {
"style": "rule",
"ground_truth": "\\frac {\\pi}{6}"
} | {
"split": "train",
"index": 4086,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( x, y, z > 0 \\) and \\( x + y + z = 1 \\), find the maximum value of \n$$\nf(x, y, z) = \\sum \\frac{x(2y - z)}{1 + x + 3y}.\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/7"
} | {
"split": "train",
"index": 4087,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest negative root of the equation\n\n$$\n4 \\sin (3 x) + 13 \\cos (3 x) = 8 \\sin (x) + 11 \\cos (x)\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-0.1651"
} | {
"split": "train",
"index": 4088,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A company has 45 male employees and 15 female employees. A 4-person research and development team was formed using stratified sampling. \n(1) Calculate the probability of an employee being selected and the number of male and female employees in the research and development tea... | math | {
"style": "rule",
"ground_truth": "\\frac {1}{2}"
} | {
"split": "train",
"index": 4089,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "George walks $1$ mile to school. He leaves home at the same time each day, walks at a steady speed of $3$ miles per hour, and arrives just as school begins. Today he was distracted by the pleasant weather and walked the first $\\frac{1}{2}$ mile at a speed of only $2$ miles per... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 4090,
"solution": "1. **Calculate the normal time to get to school**: George walks 1 mile to school at a speed of 3 miles per hour. The time taken to walk this distance is calculated by the formula:\n \\[\n \\text{Time} = \\frac{\\text{Distance}}{\\text{Speed}} = \\frac{1 \\text{ ... |
deepscaler | [
{
"role": "user",
"content": "Calculate the integer nearest to $500\\sum_{n=4}^{10005}\\frac{1}{n^2-9}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "174"
} | {
"split": "train",
"index": 4091,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For the \"Skillful Hands\" club, Anton needs to cut several identical pieces of wire (the length of each piece is an integer number of centimeters). Initially, Anton took a piece of wire 10 meters long and was able to cut only 9 required pieces from it. Then Anton took a piece ... | math | {
"style": "rule",
"ground_truth": "111"
} | {
"split": "train",
"index": 4092,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate:<br/>$(1)-9+5-\\left(-12\\right)+\\left(-3\\right)$;<br/>Calculate:<br/>$(2)-(+1.5)-(-4\\frac{1}{4})+3.75-(-8\\frac{1}{2})$;<br/>$(3)$Read the following solution process and answer the question:<br/>Calculate:$\\left(-15\\right)\\div (-\\frac{1}{2}×\\frac{25}{3}$）$÷\\... | math | {
"style": "rule",
"ground_truth": "\\frac{108}{5}"
} | {
"split": "train",
"index": 4093,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $x-y=1$ and $x^3-y^3=2$, find the values of $x^4+y^4$ and $x^5-y^5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{29}{9}"
} | {
"split": "train",
"index": 4094,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\Delta ABC$, $\\angle A$ satisfies the condition $\\sqrt{3}\\sin A+\\cos A=1$, and $AB=2$, $BC=2\\sqrt{3}$. Determine the area $S=$_______ of $\\Delta ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 4095,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be positive real numbers such that $a + 2b = 1.$ Find the minimum value of\n\\[\\frac{1}{a} + \\frac{2}{b}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 4096,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sqrt{3}\\sin{2x}-\\cos{2x}$. \n$(1)$ Find the smallest positive period and the interval of monotonic increase of the function $f(x)$. \n$(2)$ In triangle $ABC$, where the internal angles $A$, $B$, $C$ are opposite to sides $a$, $b$, $c$ respectively... | math | {
"style": "rule",
"ground_truth": "\\sqrt{7}"
} | {
"split": "train",
"index": 4097,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Miki extracts 12 ounces of juice from 4 pears and 6 ounces of juice from 3 oranges. Determine the percentage of pear juice in a blend using 8 pears and 6 oranges. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "66.67\\%"
} | {
"split": "train",
"index": 4098,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\omega$ be a circle with radius $1$ . Equilateral triangle $\\vartriangle ABC$ is tangent to $\\omega$ at the midpoint of side $BC$ and $\\omega$ lies outside $\\vartriangle ABC$ . If line $AB$ is tangent to $\\omega$ , compute the side length of $\\v... | math | {
"style": "rule",
"ground_truth": "\\frac{2 \\sqrt{3}}{3}"
} | {
"split": "train",
"index": 4099,
"solution": ""
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.