data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "A construction team has 6 projects (A, B, C, D, E, F) that need to be completed separately. Project B must start after project A is completed, project C must start after project B is completed, and project D must immediately follow project C. Determine the number of different w... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 21049,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\tan (-3645^\\circ)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 21050,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all real values of $x$ that satisfy $\\frac{1}{x+1} + \\frac{3}{x+7} \\ge \\frac23.$ (Give your answer in interval notation.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-7, -4] \\cup (-1, 2]"
} | {
"split": "train",
"index": 21051,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest integer less than 150 for which the greatest common divisor of that integer and 18 is 6? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 21052,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose \\(AB = 1\\), and the slanted segments form an angle of \\(45^\\circ\\) with \\(AB\\). There are \\(n\\) vertices above \\(AB\\).\n\nWhat is the length of the broken line? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 21053,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m \\ge 5$ be an odd integer, and let $D(m)$ denote the number of quadruples $(a_1, a_2, a_3, a_4)$ of distinct integers with $1 \\le a_i \\le m$ for all $i$ such that $m$ divides $a_1+a_2+a_3+a_4$. There is a polynomial\n\\[q(x) = c_3x^3+c_2x^2+c_1x+c_0\\]such that $D(m) =... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 21054,
"solution": "We start by defining a transformation for each $a_i$:\n\\[ b_i = \\begin{cases} \na_i & \\text{if } 1 \\leq a_i \\leq \\frac{m-1}{2}, \\\\\na_i - m & \\text{if } \\frac{m-1}{2} + 1 \\leq a_i \\leq m - 1, \\\\\n0 & \\text{if } a_i = m \n\\end{cases} \\]\nThis transf... |
deepscaler | [
{
"role": "user",
"content": "Given \\(\\sin \\alpha + \\sin (\\alpha + \\beta) + \\cos (\\alpha + \\beta) = \\sqrt{3}\\), where \\(\\beta \\in \\left[\\frac{\\pi}{4}, \\pi\\right]\\), find the value of \\(\\beta\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 21055,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Derek and Julia are two of 64 players at a casual basketball tournament. The players split up into 8 teams of 8 players at random. Each team then randomly selects 2 captains among their players. What is the probability that both Derek and Julia are captains? Let's think step by... | math | {
"style": "rule",
"ground_truth": "5/84"
} | {
"split": "train",
"index": 21056,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A solid rectangular prism has dimensions 4 by 2 by 2. A 1 by 1 by 1 cube is cut out of the corner creating the new solid shown. What is the surface area of the new solid? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 21057,
"solution": "The original prism has four faces that are 4 by 2 rectangles, and two faces that are 2 by 2 rectangles. Thus, the surface area of the original prism is \\( 4(4 \\cdot 2)+2(2 \\cdot 2)=32+8=40 \\). When a 1 by 1 by cube is cut out, a 1 by 1 square is removed from ea... |
deepscaler | [
{
"role": "user",
"content": "If $x \\geq 0$, then $\\sqrt{x\\sqrt{x\\sqrt{x}}} =$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[8]{x^7}"
} | {
"split": "train",
"index": 21058,
"solution": "1. **Understanding the expression**: We start with the expression $\\sqrt{x\\sqrt{x\\sqrt{x}}}$. This involves nested square roots, which can be simplified using the property that $\\sqrt{y} = y^{\\frac{1}{2}}$ for all $y \\geq 0$.\n\n2. **Simplifying the innermost... |
deepscaler | [
{
"role": "user",
"content": "Elbert and Yaiza each draw 10 cards from a 20-card deck with cards numbered $1,2,3, \\ldots, 20$. Then, starting with the player with the card numbered 1, the players take turns placing down the lowest-numbered card from their hand that is greater than every card previously pla... | math | {
"style": "rule",
"ground_truth": "324"
} | {
"split": "train",
"index": 21059,
"solution": "Put each card in order and label them based on if Elbert or Yaiza got them. We will get a string of E's and Y's like EEYYYE ..., and consider the \"blocks\" of consecutive letters. It is not hard to see that only the first card of each block is played, and the numb... |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be the roots of the polynomial $x^2+2020x+c$ . Given that $\\frac{a}{b}+\\frac{b}{a}=98$ , compute $\\sqrt c$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "202"
} | {
"split": "train",
"index": 21060,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can 8 people be seated in a row of chairs if three of the people, John, Wilma, and Paul, refuse to sit in three consecutive seats? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36000"
} | {
"split": "train",
"index": 21061,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x,$ $y,$ and $z$ be positive real numbers satisfying the system of equations: \\begin{align*} \\sqrt{2x-xy} + \\sqrt{2y-xy} &= 1 \\\\ \\sqrt{2y-yz} + \\sqrt{2z-yz} &= \\sqrt2 \\\\ \\sqrt{2z-zx} + \\sqrt{2x-zx} &= \\sqrt3. \\end{align*} Then $\\left[ (1-x)(1-y)(1-z) \\right... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 21062,
"solution": "Let $1-x=a;1-y=b;1-z=c$, rewrite those equations\n$\\sqrt{(1-a)(1+b)}+\\sqrt{(1+a)(1-b)}=1$;\n$\\sqrt{(1-b)(1+c)}+\\sqrt{(1+b)(1-c)}=\\sqrt{2}$\n$\\sqrt{(1-a)(1+c)}+\\sqrt{(1-c)(1+a)}=\\sqrt{3}$\n square both sides, get three equations:\n$2ab-1=2\\sqrt{(1-a^2)(1-b^... |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $N$ with the following property: of the three numbers $N$, $N+1$, and $N+2$, one of them is divisible by $2^2$, one of them is divisible by $3^2$, one is divisible by $5^2$, and one is divisible by $7^2$. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "98"
} | {
"split": "train",
"index": 21063,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the total area of the region outside of an equilateral triangle but inside three circles each with radius 1, centered at the vertices of the triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2 \\pi-\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 21064,
"solution": "Let $A_{\\triangle}$ be the equilateral triangle. Let $A_{1}$ be the area of the region outside of the equilateral triangle but inside the second and third circles. Define $A_{2}, A_{3}$ analogously. We have $A_{1}=A_{2}=A_{3}=A_{k}=$ $$\\left(\\frac{1^{2} \\cdot \... |
deepscaler | [
{
"role": "user",
"content": "The sequences of real numbers $\\left\\{a_{i}\\right\\}_{i=1}^{\\infty}$ and $\\left\\{b_{i}\\right\\}_{i=1}^{\\infty}$ satisfy $a_{n+1}=\\left(a_{n-1}-1\\right)\\left(b_{n}+1\\right)$ and $b_{n+1}=a_{n} b_{n-1}-1$ for $n \\geq 2$, with $a_{1}=a_{2}=2015$ and $b_{1}=b_{2}=2013$... | math | {
"style": "rule",
"ground_truth": "1+\\frac{1}{2014 \\cdot 2015}"
} | {
"split": "train",
"index": 21065,
"solution": "First note that $a_{n}$ and $b_{n}$ are weakly increasing and tend to infinity. In particular, $a_{n}, b_{n} \\notin\\{0,-1,1\\}$ for all $n$. For $n \\geq 1$, we have $a_{n+3}=\\left(a_{n+1}-1\\right)\\left(b_{n+2}+1\\right)=\\left(a_{n+1}-1\\right)\\left(a_{n+1} ... |
deepscaler | [
{
"role": "user",
"content": "Among 51 consecutive odd numbers $1, 3, 5, \\cdots, 101$, select $\\mathrm{k}$ numbers such that their sum is 1949. What is the maximum value of $\\mathrm{k}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 21066,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ $c$ be real numbers such that $1 \\le a \\le b \\le c \\le 4.$ Find the minimum value of\n\\[(a - 1)^2 + \\left( \\frac{b}{a} - 1 \\right)^2 + \\left( \\frac{c}{b} - 1 \\right)^2 + \\left( \\frac{4}{c} - 1 \\right)^2.\\] Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "12 - 8 \\sqrt{2}"
} | {
"split": "train",
"index": 21067,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The $\\textit{arithmetic derivative}$ $D(n)$ of a positive integer $n$ is defined via the following rules:\n\n\n\n- $D(1) = 0$ ;\n- $D(p)=1$ for all primes $p$ ;\n- $D(ab)=D(a)b+aD(b)$ for all positive integers $a$ and $b$ .\n\nFind the sum of all positive integers ... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 21068,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest two-digit prime number such that reversing the digits of the number produces an even number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 21069,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $105, Dorothy paid $125, and Sammy paid $175. In order to share the costs equally, Tom gave Sammy $t$ dollars, and Dorothy gave Sammy $d$ dollars. What is $t-d$? Let's th... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 21070,
"solution": "1. **Calculate the total amount paid by all three:**\n Tom paid $105, Dorothy paid $125, and Sammy paid $175. Therefore, the total amount paid is:\n \\[\n 105 + 125 + 175 = 405\n \\]\n\n2. **Determine the amount each should have paid:**\n Since they agree... |
deepscaler | [
{
"role": "user",
"content": "Denote $\\mathbb{Z}_{>0}=\\{1,2,3,...\\}$ the set of all positive integers. Determine all functions $f:\\mathbb{Z}_{>0}\\rightarrow \\mathbb{Z}_{>0}$ such that, for each positive integer $n$,\n$\\hspace{1cm}i) \\sum_{k=1}^{n}f(k)$ is a perfect square, and\n$\\vspace{0.1cm}$\n$\... | math | {
"style": "rule",
"ground_truth": "f(n) = n^3"
} | {
"split": "train",
"index": 21071,
"solution": "\nWe are tasked with finding all functions \\( f: \\mathbb{Z}_{>0} \\rightarrow \\mathbb{Z}_{>0} \\) that satisfy the following conditions for each positive integer \\( n \\):\n\n1. \\( \\sum_{k=1}^{n} f(k) \\) is a perfect square.\n2. \\( f(n) \\) divides \\( n^3 ... |
deepscaler | [
{
"role": "user",
"content": "Given the sequence ${a_n}$ is an arithmetic sequence, with $a_1 \\geq 1$, $a_2 \\leq 5$, $a_5 \\geq 8$, let the sum of the first n terms of the sequence be $S_n$. The maximum value of $S_{15}$ is $M$, and the minimum value is $m$. Determine $M+m$. Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "600"
} | {
"split": "train",
"index": 21072,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{m}=(\\sin\\frac{x}{3}, -1)$, $\\overrightarrow{n}=(\\frac{\\sqrt{3}}{2}A, \\frac{1}{2}A\\cos\\frac{x}{3})$ ($A > 0$), the maximum value of the function $f(x)=\\overrightarrow{n}\\cdot\\overrightarrow{m}$ is $2$. \n$(1)$ Find the smallest positiv... | math | {
"style": "rule",
"ground_truth": "-\\frac{33}{65}"
} | {
"split": "train",
"index": 21073,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the equations $x^2+ax+b=0$ and $x^2+bx+a=0$ ($a<0$, $b<0$, $a \\neq b$) have one common root. Let the other two roots be $x_1$ and $x_2$.\n\n(1) Find the value of $x_1+x_2$.\n\n(2) Find the maximum value of $x_1x_2$. Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 21074,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of strictly increasing sequences of nonnegative integers with the following properties: - The first term is 0 and the last term is 12. In particular, the sequence has at least two terms. - Among any two consecutive terms, exactly one of them is even. Let's think... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 21075,
"solution": "For a natural number $n$, let $A_{n}$ be a set containing all sequences which satisfy the problem conditions but which 12 is replaced by $n$. Also, let $a_{n}$ be the size of $A_{n}$. We first consider $a_{1}$ and $a_{2}$. We get $a_{1}=1$, as the only sequence sat... |
deepscaler | [
{
"role": "user",
"content": "There are $64$ booths around a circular table and on each one there is a chip. The chips and the corresponding booths are numbered $1$ to $64$ in this order. At the center of the table there are $1996$ light bulbs which are all turned off. Every minute the chips move si... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 21076,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagram shows a square \\(PQRS\\) with sides of length 2. The point \\(T\\) is the midpoint of \\(RS\\), and \\(U\\) lies on \\(QR\\) so that \\(\\angle SPT = \\angle TPU\\). What is the length of \\(UR\\)? Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 21077,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a > 0$ and $b > 0$, if the inequality $\\frac{3}{a} + \\frac{1}{b} \\geq \\frac{m}{a + 3b}$ always holds true, find the maximum value of $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 21078,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jane and her brother each spin a spinner once. The spinner has six congruent sectors labeled from 1 to 6. If the non-negative difference of their numbers is less than 4, Jane wins. Otherwise, her brother wins. What is the probability that Jane wins? Express your answer as a com... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"split": "train",
"index": 21079,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( p \\) and \\( q \\) be positive integers such that\n\\[\n\\frac{6}{11} < \\frac{p}{q} < \\frac{5}{9}\n\\]\nand \\( q \\) is as small as possible. What is \\( p+q \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 21080,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Assume that savings banks offer the same interest rate as the inflation rate for a year to deposit holders. The government takes away $20 \\%$ of the interest as tax. By what percentage does the real value of government interest tax revenue decrease if the inflation rate drops ... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 21081,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\log_9 \\Big(\\log_4 (\\log_3 x) \\Big) = 1$, calculate the value of $x^{-2/3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3^{-174762.6667}"
} | {
"split": "train",
"index": 21082,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The region consisting of all points in three-dimensional space within $4$ units of line segment $\\overline{CD}$ has volume $544\\pi$. Calculate the length of $CD$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{86}{3}"
} | {
"split": "train",
"index": 21083,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p,$ $q,$ $r,$ $s$ be real numbers such that $p +q + r + s = 8$ and\n\\[pq + pr + ps + qr + qs + rs = 12.\\]Find the largest possible value of $s.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 + 3 \\sqrt{2}"
} | {
"split": "train",
"index": 21084,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Which of the following is closest in value to 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{50}"
} | {
"split": "train",
"index": 21085,
"solution": "We note that $7=\\sqrt{49}$ and that $\\sqrt{40}<\\sqrt{49}<\\sqrt{50}<\\sqrt{60}<\\sqrt{70}<\\sqrt{80}$. \n This means that $\\sqrt{40}$ or $\\sqrt{50}$ is the closest to 7 of the given choices. \n Since $\\sqrt{40} \\approx 6.32$ and $\\sqrt{50} \\approx 7.07$, t... |
deepscaler | [
{
"role": "user",
"content": "Points $A$ and $C$ lie on a circle centered at $O$, each of $\\overline{BA}$ and $\\overline{BC}$ are tangent to the circle, and $\\triangle ABC$ is equilateral. The circle intersects $\\overline{BO}$ at $D$. What is $\\frac{BD}{BO}$? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 21086,
"solution": "\n1. **Identify Key Properties of the Triangle and Circle:**\n Since $\\triangle ABC$ is equilateral, each angle in the triangle is $60^\\circ$. The lines $\\overline{BA}$ and $\\overline{BC}$ are tangents to the circle at points $A$ and $C$, respectively. This i... |
deepscaler | [
{
"role": "user",
"content": "Given that a tetrahedron $ABCD$ is inscribed in a sphere $O$, and $AD$ is the diameter of the sphere $O$. If triangles $\\triangle ABC$ and $\\triangle BCD$ are equilateral triangles with side length 1, calculate the volume of tetrahedron $ABCD$. Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{12}"
} | {
"split": "train",
"index": 21087,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two groups of numerical sequences, each group consisting of 15 arithmetic progressions containing 10 terms each. The first terms of the progressions in the first group are $1,2,3, \\ldots, 15$ and their differences are $2,4,6, \\ldots, 30$, respectively. The second group ... | math | {
"style": "rule",
"ground_truth": "160/151"
} | {
"split": "train",
"index": 21088,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the number of ordered quintuples of nonnegative integers $(a_{1}, a_{2}, a_{3}, a_{4}, a_{5})$ such that $0 \\leq a_{1}, a_{2}, a_{3}, a_{4}, a_{5} \\leq 7$ and 5 divides $2^{a_{1}}+2^{a_{2}}+2^{a_{3}}+2^{a_{4}}+2^{a_{5}}$. Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "6528"
} | {
"split": "train",
"index": 21089,
"solution": "Let $f(n)$ denote the number of $n$-tuples $(a_{1}, \\ldots, a_{n})$ such that $0 \\leq a_{1}, \\ldots, a_{n} \\leq 7$ and $5 \\mid 2^{a_{1}}+\\ldots+2^{a_{n}}$. To compute $f(n+1)$ from $f(n)$, we note that given any $n$-tuple $(a_{1}, \\ldots, a_{n})$ such that $... |
deepscaler | [
{
"role": "user",
"content": "A right triangular pyramid has a base edge length of $2$, and its three side edges are pairwise perpendicular. Calculate the volume of this pyramid. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{3}"
} | {
"split": "train",
"index": 21090,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $(a_n)$ satisfies $a_0=0$ and $a_{n + 1} = \\frac85a_n + \\frac65\\sqrt {4^n - a_n^2}$ for $n\\geq 0$. Find the greatest integer less than or equal to $a_{10}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "983"
} | {
"split": "train",
"index": 21091,
"solution": "We can now simply start to compute the values $b_i$ by hand:\n\\begin{align*} b_1 & = \\frac 35 \\\\ b_2 & = \\frac 45\\cdot \\frac 35 + \\frac 35 \\sqrt{1 - \\left(\\frac 35\\right)^2} = \\frac{24}{25} \\\\ b_3 & = \\frac 45\\cdot \\frac {24}{25} + \\frac 35 \\sqr... |
deepscaler | [
{
"role": "user",
"content": "At HappyTails Training Center, cats can learn to do three tricks: jump, fetch, and spin. Of the cats at the center:\n- 40 cats can jump\n- 25 cats can fetch\n- 30 cats can spin\n- 15 cats can jump and fetch\n- 10 cats can fetch and spin\n- 12 cats can jump and spin\n- 5 cats ca... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 21092,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $n$ is a real number, then the simultaneous system\n$nx+y = 1$\n$ny+z = 1$\n$x+nz = 1$\nhas no solution if and only if $n$ is equal to Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 21093,
"solution": "\nTo determine the values of $n$ for which the given system of equations has no solution, we analyze the system step-by-step.\n\n**Step 1: Add all three equations.**\n\\[\n\\begin{align*}\nnx + y &= 1, \\\\\nny + z &= 1, \\\\\nx + nz &= 1.\n\\end{align*}\n\\]\nAddi... |
deepscaler | [
{
"role": "user",
"content": "John has 12 marbles of different colors, including one red, one green, and one blue marble. In how many ways can he choose 4 marbles, if exactly one of the chosen marbles is red, green, or blue? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "252"
} | {
"split": "train",
"index": 21094,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a\\star b = \\dfrac{\\sqrt{a+b}}{\\sqrt{a-b}}$. If $ x \\star 24 = 7$, find $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 21095,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mia sells four burritos and five empanadas for $\\$$4.00 and she sells six burritos and three empanadas for $\\$$4.50. Assuming a fixed price per item, what is the cost, in dollars, of five burritos and seven empanadas? Express your answer as a decimal to the nearest hundredth.... | math | {
"style": "rule",
"ground_truth": "5.25"
} | {
"split": "train",
"index": 21096,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the limit of the function:\n\n\\[\n\\lim _{x \\rightarrow \\frac{\\pi}{4}}(\\tan x)^{1 / \\cos \\left(\\frac{3 \\pi}{4}-x\\right)}\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "e^2"
} | {
"split": "train",
"index": 21097,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $X,$ $Y,$ and $Z$ be points such that $\\frac{XZ}{XY} = \\frac{ZY}{XY} = \\frac{1}{2}.$ If $Y = (1, 7)$, $Z = (-1, -7)$, then what is the sum of the coordinates of $X$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-24"
} | {
"split": "train",
"index": 21098,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What percentage of a seven-by-seven grid is shaded?\nIn a seven-by-seven grid, alternate squares are shaded starting with the top left square similar to a checkered pattern. However, an entire row (the fourth row from the top) and an entire column (the fourth column from the le... | math | {
"style": "rule",
"ground_truth": "73.47\\%"
} | {
"split": "train",
"index": 21099,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Shift the graph of the function $y=3\\sin (2x+ \\frac {\\pi}{6})$ to the graph of the function $y=3\\cos 2x$ and determine the horizontal shift units. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {\\pi}{6}"
} | {
"split": "train",
"index": 21100,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To obtain the graph of the function $y=\\sin\\left(2x+\\frac{\\pi}{3}\\right)$, find the transformation required to obtain the graph of the function $y=\\cos\\left(2x-\\frac{\\pi}{3}\\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left(\\frac{\\pi}{12}\\right)"
} | {
"split": "train",
"index": 21101,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a triangle $ABC$ with sides $a$, $b$, $c$, and area $S$ satisfying $S=a^{2}-(b-c)^{2}$, and $b+c=8$. \n$(1)$ Find $\\cos A$; \n$(2)$ Find the maximum value of $S$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{64}{17}"
} | {
"split": "train",
"index": 21102,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is $\\frac{1}{29}$ of the original integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "725"
} | {
"split": "train",
"index": 21103,
"solution": "Suppose the original number is $N = \\overline{a_na_{n-1}\\ldots a_1a_0},$ where the $a_i$ are digits and the first digit, $a_n,$ is nonzero. Then the number we create is $N_0 = \\overline{a_{n-1}\\ldots a_1a_0},$ so \\[N = 29N_0.\\] But $N$ is $N_0$ with the digit... |
deepscaler | [
{
"role": "user",
"content": "The sum of an infinite geometric series is $64$ times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio? 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": 21104,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A and B start from points A and B simultaneously, moving towards each other and meet at point C. If A starts 2 minutes earlier, then their meeting point is 42 meters away from point C. Given that A's speed is \\( a \\) meters per minute, B's speed is \\( b \\) meters per minute... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 21105,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each third-grade classroom at Pearl Creek Elementary has $18$ students and $2$ pet rabbits. How many more students than rabbits are there in all $4$ of the third-grade classrooms? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 21106,
"solution": "We are given that each third-grade classroom at Pearl Creek Elementary has $18$ students and $2$ pet rabbits. We need to find the total number of students and rabbits in all $4$ classrooms and then determine how many more students there are than rabbits.\n\n#### St... |
deepscaler | [
{
"role": "user",
"content": "What is the diameter of the circle inscribed in triangle $DEF$ if $DE = 13,$ $DF = 8,$ and $EF = 15$? Express your answer in simplest radical form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 21107,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the fall of 1996, a total of 800 students participated in an annual school clean-up day. The organizers of the event expect that in each of the years 1997, 1998, and 1999, participation will increase by 50% over the previous year. The number of participants the organizers ... | math | {
"style": "rule",
"ground_truth": "2700"
} | {
"split": "train",
"index": 21108,
"solution": "1. **Initial Participation**: In 1996, the number of participants was 800.\n\n2. **Annual Increase**: Each year, the number of participants increases by 50%. This means that each year, the number of participants is multiplied by $1.5$ (since $100\\% + 50\\% = 150\\... |
deepscaler | [
{
"role": "user",
"content": "A convex quadrilateral is determined by the points of intersection of the curves \\( x^{4}+y^{4}=100 \\) and \\( x y=4 \\); determine its area. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{17}"
} | {
"split": "train",
"index": 21109,
"solution": "By symmetry, the quadrilateral is a rectangle having \\( x=y \\) and \\( x=-y \\) as axes of symmetry. Let \\( (a, b) \\) with \\( a>b>0 \\) be one of the vertices. Then the desired area is \\( (\\sqrt{2}(a-b)) \\cdot(\\sqrt{2}(a+b))=2\\left(a^{2}-b^{2}\\right)=2 \... |
deepscaler | [
{
"role": "user",
"content": "Let $T$ be a positive integer whose only digits are 0s and 1s. If $X = T \\div 18$ and $X$ is an integer, what is the smallest possible value of $X$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6172839500"
} | {
"split": "train",
"index": 21110,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that 6 rational numbers are placed sequentially on a given circle, now select any 3 adjacent numbers $a$, $b$, $c$ in a clockwise direction, satisfying $a = |b - c|$. It is also known that the total sum of all the placed numbers is 1. What are the values of these 6 number... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 21111,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all real numbers $x$ which satisfy \\[\\frac{x-1}{x-3} \\ge 2.\\](Give your answer in interval notation.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(3, 5]"
} | {
"split": "train",
"index": 21112,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pyramid-like stack with a rectangular base containing $6$ apples by $9$ apples is constructed, with each apple above the first level fitting into a pocket formed by four apples below, until no more apples can be fit in a new layer. Determine the total number of apples in the ... | math | {
"style": "rule",
"ground_truth": "154"
} | {
"split": "train",
"index": 21113,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the product of all prime numbers between 1 and 20. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9699690"
} | {
"split": "train",
"index": 21114,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many numbers between 100 and 999 (inclusive) have digits that form an arithmetic progression when read from left to right?\n\nA sequence of three numbers \\( a, b, c \\) is said to form an arithmetic progression if \\( a + c = 2b \\). \n\nA correct numerical answer without ... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 21115,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum number of white dominoes that can be cut from the board shown on the left. A domino is a $1 \\times 2$ rectangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 21116,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose the edge length of a regular tetrahedron $ABC D$ is 1 meter. A bug starts at point $A$ and moves according to the following rule: at each vertex, it chooses one of the three edges connected to this vertex with equal probability and crawls along this edge to the next ver... | math | {
"style": "rule",
"ground_truth": "7/27"
} | {
"split": "train",
"index": 21117,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the functions \\( f(x) \\) and \\( g(x) \\) are defined on \\( \\mathbf{R} \\), and \\( f(x-y)=f(x)g(y)-g(x)f(y) \\), with \\( f(-2)=f(1) \\neq 0 \\), what is \\( g(1) + g(-1) \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 21118,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On an island, there are only knights, who always tell the truth, and liars, who always lie. There are at least two knights and at least two liars. One day, each islander pointed to each of the others in turn and said either \"You are a knight!\" or \"You are a liar!\". The phra... | math | {
"style": "rule",
"ground_truth": "526"
} | {
"split": "train",
"index": 21119,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Katie writes a different positive integer on the top face of each of the fourteen cubes in the pyramid shown. The sum of the nine integers written on the cubes in the bottom layer is 50. The integer written on each of the cubes in the middle and top layers of the pyramid is equ... | math | {
"style": "rule",
"ground_truth": "118"
} | {
"split": "train",
"index": 21120,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nAlong a straight alley at equal intervals, there are 400 lampposts, numbered sequentially from 1 to 400. At the same time, Alla and Boris started walking towards each other from opposite ends of the alley with different constant speeds (Alla started from the first lamppost an... | math | {
"style": "rule",
"ground_truth": "163"
} | {
"split": "train",
"index": 21121,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $DEF$ has side lengths $DE = 15$, $EF = 36$, and $FD = 39$. Rectangle $WXYZ$ has vertex $W$ on $\\overline{DE}$, vertex $X$ on $\\overline{DF}$, and vertices $Y$ and $Z$ on $\\overline{EF}$. In terms of the side length $WX = \\omega$, the area of $WXYZ$ can be expresse... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 21122,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integer multiples of $1001$ can be expressed in the form $10^{j} - 10^{i}$, where $i$ and $j$ are integers and $0\\leq i < j \\leq 99$?\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "784"
} | {
"split": "train",
"index": 21123,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given real numbers \\( x, y \\in (1,+\\infty) \\) such that \\( xy - 2x - y + 1 = 0 \\), find the minimum value of \\( \\frac{3}{2} x^{2} + y^{2} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 21124,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\n\\sum_{n = 1}^\\infty \\frac{1}{n(n + 3)}.\n\\] 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": 21125,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "After Euclid High School's last basketball game, it was determined that $\\frac{1}{4}$ of the team's points were scored by Alexa and $\\frac{2}{7}$ were scored by Brittany. Chelsea scored $15$ points. None of the other $7$ team members scored more than $2$ points. What was the ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 21126,
"solution": "Let $x$ be the total number of points scored by the team. According to the problem, the points can be broken down as follows:\n- Alexa scored $\\frac{1}{4}x$ points.\n- Brittany scored $\\frac{2}{7}x$ points.\n- Chelsea scored $15$ points.\n- The other $7$ team mem... |
deepscaler | [
{
"role": "user",
"content": "In how many ways can four married couples sit around a circular table such that no man sits next to his wife? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1488"
} | {
"split": "train",
"index": 21127,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The positive integers $N$ and $N^2$ both end in the same sequence of four digits $abcd$ when written in base $10$, where digit a is not zero. Find the three-digit number $abc$.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "937"
} | {
"split": "train",
"index": 21128,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x)=\\dfrac{x-3}{x-4}$, then for what value of $x$ is $f^{-1}(x)$ undefined? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 21129,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive multiple of $32$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 21130,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The polynomial $-5x^2-xy^4+2^6xy+3$ has terms, and the degree of this polynomial is . The coefficient of the highest degree term is . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 21131,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circles with centers of $(2,2)$ and $(17,10)$ are both tangent to the $x$-axis. What is the distance between the closest points of the two circles? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 21132,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Logan is constructing a scaled model of his town. The city's water tower stands 40 meters high, and the top portion is a sphere that holds 100,000 liters of water. Logan's miniature water tower holds 0.1 liters. How tall, in meters, should Logan make his tower? Let's think step... | math | {
"style": "rule",
"ground_truth": "0.4"
} | {
"split": "train",
"index": 21133,
"solution": "1. **Identify the ratio of volumes between the actual water tower and the miniature model**: \n The actual water tower holds 100,000 liters, and Logan's miniature holds 0.1 liters. The ratio of the volumes is:\n \\[\n \\frac{100000 \\text{ liters}}{0.1 \\text... |
deepscaler | [
{
"role": "user",
"content": "Given that $|\\vec{a}|=4$, and $\\vec{e}$ is a unit vector. When the angle between $\\vec{a}$ and $\\vec{e}$ is $\\frac{2\\pi}{3}$, the projection of $\\vec{a} + \\vec{e}$ on $\\vec{a} - \\vec{e}$ is ______. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "\\frac{5\\sqrt{21}}{7}"
} | {
"split": "train",
"index": 21134,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A monkey in Zoo becomes lucky if he eats three different fruits. What is the largest number of monkeys one can make lucky, by having $20$ oranges, $30$ bananas, $40$ peaches and $50$ tangerines? Justify your answer. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 21135,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sin (\\omega x+ \\frac {\\pi}{3})$ ($\\omega > 0$), if $f( \\frac {\\pi}{6})=f( \\frac {\\pi}{3})$ and $f(x)$ has a minimum value but no maximum value in the interval $( \\frac {\\pi}{6}, \\frac {\\pi}{3})$, determine the value of $\\omega$. Let's thi... | math | {
"style": "rule",
"ground_truth": "\\frac {14}{3}"
} | {
"split": "train",
"index": 21136,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $1-\\frac{4}{x}+\\frac{4}{x^2}=0$, then $\\frac{2}{x}$ equals Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 21137,
"solution": "1. Start with the given equation:\n \\[\n 1 - \\frac{4}{x} + \\frac{4}{x^2} = 0\n \\]\n\n2. Multiply each term by \\(x^2\\) to clear the denominators:\n \\[\n x^2 \\cdot 1 - 4x \\cdot x + 4 = x^2 - 4x + 4 = 0\n \\]\n\n3. Factor the quadratic equation:\n... |
deepscaler | [
{
"role": "user",
"content": "Solve for $r$: $r = \\displaystyle\\frac{\\sqrt{5^2+12^2}}{\\sqrt{16+9}}$. Express as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{5}"
} | {
"split": "train",
"index": 21138,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a $4 \\times 5$ grid, place 5 crosses such that each row and each column contains at least one cross. How many ways can this be done? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 21139,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct arrangements of the letters in the word \"balloon\" are there, considering the repeated 'l' and 'o'? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1260"
} | {
"split": "train",
"index": 21140,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $C$: $\\frac{{x}^{2}}{3}+\\frac{{y}^{2}}{{b}^{2}}=1\\left(b \\gt 0\\right)$ with the right focus $F$, two lines passing through $F$ are tangent to the circle $O$: $x^{2}+y^{2}=r^{2}(r \\gt 0)$ at points $A$ and $B$, forming a right triangle $\\triangle ABF$. It... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 21141,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the regular octagon $ABCDEFGH$ with its center at $J$, and each of the vertices and the center associated with the digits 1 through 9, with each digit used once, such that the sums of the numbers on the lines $AJE$, $BJF$, $CJG$, and $DJH$ are equal, determine the number ... | math | {
"style": "rule",
"ground_truth": "1152"
} | {
"split": "train",
"index": 21142,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a regular quadrangular pyramid \\( S-ABCD \\) with a side edge length of 4, and \\( \\angle ASB = 30^\\circ \\). Points \\( E \\), \\( F \\), and \\( G \\) are taken on side edges \\( SB \\), \\( SC \\), and \\( SD \\), respectively. Find the minimum value of \\( AE + EF ... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"split": "train",
"index": 21143,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangular array of numbers has a first row consisting of the odd integers $1,3,5,\\ldots,99$ in increasing order. Each row below the first has one fewer entry than the row above it, and the bottom row has a single entry. Each entry in any row after the top row equals the sum... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 21144,
"solution": "The result above is fairly intuitive if we write out several rows and then divide all numbers in row $r$ by $2^{r-1}$ (we can do this because dividing by a power of 2 doesn't affect divisibility by $67$). The second row will be $2, 4, 6, \\cdots , 98$, the third ro... |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{M} = \\begin{pmatrix} 2 & 0 \\\\ 1 & -3 \\end{pmatrix}.$ Find constants $a$ and $b$ so that\n\\[\\mathbf{M}^{-1} = a \\mathbf{M} + b \\mathbf{I}.\\]Enter the ordered pair $(a,b).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left( \\frac{1}{6}, \\frac{1}{6} \\right)"
} | {
"split": "train",
"index": 21145,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that vehicles are not allowed to turn back at a crossroads, calculate the total number of driving routes. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 21146,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), side \\(AB\\) is 21, the bisector \\(BD\\) is \\(8 \\sqrt{7}\\), and \\(DC\\) is 8. Find the perimeter of the triangle \\(ABC\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 21147,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integ... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 21148,
"solution": "\nTo solve this problem, we need to ensure that the average after each score is entered is an integer. This means that the sum of the scores entered at each step must be divisible by the number of scores entered up to that step.\n\nGiven scores in ascending order: ... |
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