data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "At the end of a game, each of the five members of a basketball team shake hands with each of the five members of the other team, and all of the players shake hands with the two referees. How many handshakes occur? Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 18649,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of rhombus $ABCD$ given that the radii of the circles circumscribed around triangles $ABD$ and $ACD$ are $12.5$ and $25$, respectively.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "400"
} | {
"split": "train",
"index": 18650,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any sequence of real numbers $A=\\{a_1, a_2, a_3, \\ldots\\}$, define $\\triangle A$ as the sequence $\\{a_2 - a_1, a_3 - a_2, a_4 - a_3, \\ldots\\}$, where the $n$-th term is $a_{n+1} - a_n$. Assume that all terms of the sequence $\\triangle (\\triangle A)$ are $1$ and $a_... | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"split": "train",
"index": 18651,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence $a_{1}, a_{2}, a_{3}, \\ldots$ of positive reals satisfies $a_{n+1}=\\sqrt{\\frac{1+a_{n}}{2}}$. Determine all $a_{1}$ such that $a_{i}=\\frac{\\sqrt{6}+\\sqrt{2}}{4}$ for some positive integer $i$. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}+\\sqrt{6}}{2}, \\frac{\\sqrt{3}}{2}, \\frac{1}{2}"
} | {
"split": "train",
"index": 18652,
"solution": "Clearly $a_{1}<1$, or else $1 \\leq a_{1} \\leq a_{2} \\leq a_{3} \\leq \\ldots$ We can therefore write $a_{1}=\\cos \\theta$ for some $0<\\theta<90^{\\circ}$. Note that $\\cos \\frac{\\theta}{2}=\\sqrt{\\frac{1+\\cos \\theta}{2}}$, and $\\cos 15^{\\circ}=$ $\\frac... |
deepscaler | [
{
"role": "user",
"content": "Simplify $(576)^\\frac{1}{4}(216)^\\frac{1}{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 18653,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice is thinking of a positive real number $x$, and Bob is thinking of a positive real number $y$. Given that $x^{\\sqrt{y}}=27$ and $(\\sqrt{x})^{y}=9$, compute $x y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16 \\sqrt[4]{3}"
} | {
"split": "train",
"index": 18654,
"solution": "Note that $$27^{\\sqrt{y}}=\\left(x^{\\sqrt{y}}\\right)^{\\sqrt{y}}=x^{y}=(\\sqrt{x})^{2 y}=81$$ so $\\sqrt{y}=4 / 3$ or $y=16 / 9$. It follows that $x^{4 / 3}=27$ or $x=9 \\sqrt[4]{3}$. The final answer is $9 \\sqrt[4]{3} \\cdot 16 / 9=16 \\sqrt[4]{3}$."
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ is a right isosceles triangle. Points $D$, $E$ and $F$ are the midpoints of the sides of the triangle. Point $G$ is the midpoint of segment $DF$ and point $H$ is the midpoint of segment $FE$. What is the ratio of the shaded area to the non-shaded area in triangle... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{11}"
} | {
"split": "train",
"index": 18655,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $p(x)$ is a function such that $p(x) + (x^5+3x^3+9x) = (7x^3+24x^2+25x+1)$. Express $p(x)$ as a polynomial with the degrees of the terms in decreasing order. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-x^5+4x^3+24x^2+16x+1"
} | {
"split": "train",
"index": 18656,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parametric equations of curve $C_1$ are $$\\begin{cases} x=2\\cos\\theta \\\\ y=\\sin\\theta\\end{cases}(\\theta \\text{ is the parameter}),$$ and the parametric equations of curve $C_2$ are $$\\begin{cases} x=-3+t \\\\ y= \\frac {3+3t}{4}\\end{cases}(t \\text{ is the... | math | {
"style": "rule",
"ground_truth": "\\frac {12-2 \\sqrt {13}}{5}"
} | {
"split": "train",
"index": 18657,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $x$: $0.05x + 0.07(30 + x) = 15.4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "110.8333"
} | {
"split": "train",
"index": 18658,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For some positive integer $n$, the number $110n^3$ has $110$ positive integer divisors, including $1$ and the number $110n^3$. How many positive integer divisors does the number $81n^4$ have?\n$\\textbf{(A) }110\\qquad\\textbf{(B) }191\\qquad\\textbf{(C) }261\\qquad\\textbf{(D)... | math | {
"style": "rule",
"ground_truth": "325"
} | {
"split": "train",
"index": 18659,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ denote the value of the sum\n\\[\\sum_{n = 1}^{9800} \\frac{1}{\\sqrt{n + \\sqrt{n^2 - 1}}}\\]\n$S$ can be expressed as $p + q \\sqrt{r}$, where $p, q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime. Determine $p + q + r$.\n Let's thi... | math | {
"style": "rule",
"ground_truth": "121"
} | {
"split": "train",
"index": 18660,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are a few integers \\( n \\) such that \\( n^{2}+n+1 \\) divides \\( n^{2013}+61 \\). Find the sum of the squares of these integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "62"
} | {
"split": "train",
"index": 18661,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cone is inscribed in a sphere such that the slant height of the cone is equal to the diameter of the base. Find the ratio of the total surface area of the cone to the surface area of the sphere. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9/16"
} | {
"split": "train",
"index": 18662,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the real solutions of $(2 x+1)(3 x+1)(5 x+1)(30 x+1)=10$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-4 \\pm \\sqrt{31}}{15}"
} | {
"split": "train",
"index": 18663,
"solution": "$(2 x+1)(3 x+1)(5 x+1)(30 x+1)=[(2 x+1)(30 x+1)][(3 x+1)(5 x+1)]=\\left(60 x^{2}+32 x+1\\right)\\left(15 x^{2}+8 x+1\\right)=(4 y+1)(y+1)=10$, where $y=15 x^{2}+8 x$. The quadratic equation in $y$ yields $y=1$ and $y=-\\frac{9}{4}$. For $y=1$, we have $15 x^{2}+8 x... |
deepscaler | [
{
"role": "user",
"content": "How many integers are solutions to the equation $$(x-2)^{(25-x^2)}=1?$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 18664,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The two squares shown share the same center $O_{}$ and have sides of length 1. The length of $\\overline{AB}$ is $43/99$ and the area of octagon $ABCDEFGH$ is $m/n,$ where $m_{}$ and $n_{}$ are relatively prime positive integers. Find $m+n.$\n[asy] //code taken from thread for ... | math | {
"style": "rule",
"ground_truth": "185"
} | {
"split": "train",
"index": 18665,
"solution": "Triangles $AOB$, $BOC$, $COD$, etc. are congruent by symmetry (you can prove it rigorously by using the power of a point to argue that exactly two chords of length $1$ in the circumcircle of the squares pass through $B$, etc.), and each area is $\\frac{\\frac{43}{9... |
deepscaler | [
{
"role": "user",
"content": "Four friends rent a cottage for a total of £300 for the weekend. The first friend pays half of the sum of the amounts paid by the other three friends. The second friend pays one third of the sum of the amounts paid by the other three friends. The third friend pays one quarter o... | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 18666,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $α \\in (0,π)$, if $\\sin α + \\cos α = \\frac{\\sqrt{3}}{3}$, find the value of $\\cos^2 α - \\sin^2 α$. 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": 18667,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify the expression $\\frac{15b^4 - 45b^3}{75b^2}$ when $b=2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{5}"
} | {
"split": "train",
"index": 18668,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the unit cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, points $E, F, G$ are the midpoints of edges $A A_{1}, C_{1} D_{1}$, and $D_{1} A_{1}$, respectively. Find the distance from point $B_{1}$ to the plane $E F G$. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 18669,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the four vertices A, B, C, D of the tetrahedron A-BCD are all on the surface of the sphere O, AC ⊥ the plane BCD, and AC = 2√2, BC = CD = 2, calculate the surface area of the sphere O. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16\\pi"
} | {
"split": "train",
"index": 18670,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer with exactly 16 positive divisors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 18671,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given three vertices of a rectangle are located at $(2, 5)$, $(2, -4)$ and $(10, 5)$. Calculate the area of the intersection of this rectangle with the region inside the graph of the equation $(x - 10)^2 + (y - 5)^2 = 16$. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"split": "train",
"index": 18672,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the period of $y = \\cos \\frac{x}{2}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4 \\pi"
} | {
"split": "train",
"index": 18673,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using the digits 1, 2, 3, 4, 5, how many even four-digit numbers less than 4000 can be formed if each digit can be used more than once? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 18674,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find a positive integer that is divisible by 14 and has a square root between 25 and 25.3. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "630"
} | {
"split": "train",
"index": 18675,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the slope of the line containing the midpoint of the segment with endpoints at (0, 0) and (2, 3) and the midpoint of the segment with endpoints at (5, 0) and (6, 3)? Express your answer in simplest form. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 18676,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be positive integers such that $2a - 9b + 18ab = 2018$ . Find $b - a$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "223"
} | {
"split": "train",
"index": 18677,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\sin 2x - \\cos \\left(2x+\\dfrac{\\pi}{6}\\right)$.\n\n(1) Find the value of $f\\left(\\dfrac{\\pi}{6}\\right)$.\n\n(2) Find the minimum positive period and the interval of monotonic increase of the function $f(x)$.\n\n(3) Find the maximum and minim... | math | {
"style": "rule",
"ground_truth": "-\\dfrac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 18678,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given complex numbers \\( z_{1}, z_{2}, z_{3} \\) such that \\( \\left|z_{1}\\right| \\leq 1 \\), \\( \\left|z_{2}\\right| \\leq 1 \\), and \\( \\left|2 z_{3}-\\left(z_{1}+z_{2}\\right)\\right| \\leq \\left|z_{1}-z_{2}\\right| \\). What is the maximum value of \\( \\left|z_{3}\... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 18679,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A mathematical contest had $3$ problems, each of which was given a score between $0$ and $7$ ( $0$ and $7$ included). It is known that, for any two contestants, there exists at most one problem in which they have obtained the same score (for example, there are no two c... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 18680,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Doug and Dave shared a pizza with $8$ equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was $8$ dollars, and there was an additional cost of $2$ dollars for putting anchovies on one half. Dave ate all the sli... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 18681,
"solution": "1. **Calculate the total cost of the pizza:** \n The pizza is plain and costs $8$ dollars. Adding anchovies to half of it costs an additional $2$ dollars. Therefore, the total cost of the pizza is:\n \\[\n 8 + 2 = 10 \\text{ dollars}\n \\]\n\n2. **Determin... |
deepscaler | [
{
"role": "user",
"content": "There are 8 seats in a row, and 3 people are sitting in the same row. If there are empty seats on both sides of each person, the number of different seating arrangements is \\_\\_\\_\\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 18682,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five consecutive two-digit positive integers, each less than 40, are not prime. What is the largest of these five integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 18683,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\sqrt{\\frac{2}{x} + 2} = \\frac{3}{2}$, solve for $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 18684,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express $\\frac{37}{80}$ as a terminating decimal. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.4625"
} | {
"split": "train",
"index": 18685,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular $2018$ -gon is inscribed in a circle. The numbers $1, 2, ..., 2018$ are arranged on the vertices of the $2018$ -gon, with each vertex having one number on it, such that the sum of any $2$ neighboring numbers ( $2$ numbers are neighboring if the vertices they ar... | math | {
"style": "rule",
"ground_truth": "2 \\times 1008!"
} | {
"split": "train",
"index": 18686,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$, $b$, $c$, $d$, and $e$ be distinct integers such that\n$(6-a)(6-b)(6-c)(6-d)(6-e)=45$\nWhat is $a+b+c+d+e$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 18687,
"solution": "1. **Identify the factors of 45**: We start by noting that the equation $(6-a)(6-b)(6-c)(6-d)(6-e)=45$ implies that the expressions $(6-a), (6-b), (6-c), (6-d), (6-e)$ are integers whose product is 45. We need to find distinct integers $a, b, c, d, e$ such that thi... |
deepscaler | [
{
"role": "user",
"content": "Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.303"
} | {
"split": "train",
"index": 18688,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of 6-digit even numbers that can be formed using the digits 1, 2, 3, 4, 5, 6 without repetition and ensuring that 1, 3, 5 are not adjacent to one another. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 18689,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.296"
} | {
"split": "train",
"index": 18690,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle $ABCD$, $DC = 2 \\cdot CB$ and points $E$ and $F$ lie on $\\overline{AB}$ so that $\\overline{ED}$ and $\\overline{FD}$ trisect $\\angle ADC$ as shown. What is the ratio of the area of $\\triangle DEF$ to the area of rectangle $ABCD$? Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{16}"
} | {
"split": "train",
"index": 18691,
"solution": "1. **Assigning Lengths**: Let the length of $AD$ be $x$. Since $DC = 2 \\cdot CB$ and $ABCD$ is a rectangle, $AB = 2x$ and $BC = x$. Therefore, the area of rectangle $ABCD$ is $AB \\times BC = 2x \\times x = 2x^2$.\n\n2. **Angle Analysis**: In rectangle $ABCD$, $\\... |
deepscaler | [
{
"role": "user",
"content": "I have five apples and ten oranges. If a fruit basket must contain at least one piece of fruit, how many kinds of fruit baskets can I make? (The apples are identical and the oranges are identical. A fruit basket consists of some number of pieces of fruit, and it doesn't matte... | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 18692,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Forty cards are placed into a box, each bearing a number 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10, with each number entered on four cards. Four cards are drawn from the box at random and without replacement. Let $p$ be the probability that all four cards bear the same number. Let $q$ b... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 18693,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let planes \\( \\alpha \\) and \\( \\beta \\) be parallel to each other. Four points are selected on plane \\( \\alpha \\) and five points are selected on plane \\( \\beta \\). What is the maximum number of planes that can be determined by these points? Let's think step by step... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 18694,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, with $AB=3$, $AC=4$, $BC=5$, let $I$ be the incenter of $\\triangle ABC$ and $P$ be a point inside $\\triangle IBC$ (including the boundary). If $\\overrightarrow{AP}=\\lambda \\overrightarrow{AB} + \\mu \\overrightarrow{AC}$ (where $\\lambda, \\mu \\in \\m... | math | {
"style": "rule",
"ground_truth": "7/12"
} | {
"split": "train",
"index": 18695,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circle $\\omega$ is inscribed in rhombus $H M_{1} M_{2} T$ so that $\\omega$ is tangent to $\\overline{H M_{1}}$ at $A, \\overline{M_{1} M_{2}}$ at $I, \\overline{M_{2} T}$ at $M$, and $\\overline{T H}$ at $E$. Given that the area of $H M_{1} M_{2} T$ is 1440 and the area of $E... | math | {
"style": "rule",
"ground_truth": "540"
} | {
"split": "train",
"index": 18696,
"solution": "First, from equal tangents, we know that $T E=T M$. As the sides of a rhombus are also equal, this gives from SAS similarity that $E M T \\sim T H M_{2}$. Further, the ratio of their areas is $\\frac{405}{1440 / 2}=\\frac{9}{16}$. This means that $T E=T M=\\frac{3}... |
deepscaler | [
{
"role": "user",
"content": "The absolute value of -9 is ; the reciprocal of -3 is . 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": 18697,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many pairs of positive integers \\( (m, n) \\) satisfy \\( m^2 \\cdot n < 30 \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 18698,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider numbers of the form $1a1$ , where $a$ is a digit. How many pairs of such numbers are there such that their sum is also a palindrome?\n\n*Note: A palindrome is a number which reads the same from left to right and from right to left. Examples: $353$ , $91719$ .* Let... | math | {
"style": "rule",
"ground_truth": "55"
} | {
"split": "train",
"index": 18699,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(P\\) and \\(P+2\\) be both prime numbers satisfying \\(P(P+2) \\leq 2007\\). If \\(S\\) represents the sum of such possible values of \\(P\\), find the value of \\(S\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "106"
} | {
"split": "train",
"index": 18700,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_1,$ $a_2,$ $\\dots,$ $a_{2018}$ be the roots of the polynomial\n\\[x^{2018} + x^{2017} + \\dots + x^2 + x - 1345 = 0.\\]Compute\n\\[\\sum_{n = 1}^{2018} \\frac{1}{1 - a_n}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3027"
} | {
"split": "train",
"index": 18701,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cylindrical can has a circumference of 24 inches and a height of 7 inches. A spiral strip is painted on the can such that it winds around the can precisely once, reaching from the bottom to the top. However, instead of reaching directly above where it started, it ends 3 inche... | math | {
"style": "rule",
"ground_truth": "\\sqrt{778}"
} | {
"split": "train",
"index": 18702,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many non-congruent triangles with only integer side lengths have a perimeter of 15 units? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 18703,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Following the concept of a healthy, low-carbon lifestyle, an increasing number of people are renting bicycles for cycling tours. A particular bicycle rental point charges no fee for rentals that do not exceed two hours, and for rentals that exceed two hours, the charging standa... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{2}"
} | {
"split": "train",
"index": 18704,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f\\left(x\\right)=4\\cos x\\sin \\left(x-\\frac{π}{3}\\right)+a$ with a maximum value of $2$.\n\n(1) Find the value of $a$ and the minimum positive period of the function $f\\left(x\\right)$;\n\n(2) In $\\triangle ABC$, if $A < B$, and $f\\left(A\\right)=f\\... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 18705,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number of distinct ordered pairs $(x,y)$ where $x$ and $y$ have positive integral values satisfying the equation $x^4y^4-10x^2y^2+9=0$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 18706,
"solution": "1. **Rewrite the given equation**: Start by rewriting the equation $x^4y^4 - 10x^2y^2 + 9 = 0$ in terms of a new variable $a = (xy)^2$. This gives us:\n \\[\n a^2 - 10a + 9 = 0\n \\]\n\n2. **Factorize the quadratic equation**: The quadratic equation in $a$ ca... |
deepscaler | [
{
"role": "user",
"content": "In the triangle \\( \\triangle ABC \\), if \\(\\sin^2 A + \\sin^2 B + \\sin^2 C = 2\\), calculate the maximum value of \\(\\cos A + \\cos B + 2 \\cos C\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 18707,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/19 of the original integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "950"
} | {
"split": "train",
"index": 18708,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cube is inscribed in a regular octahedron in such a way that its vertices lie on the edges of the octahedron. By what factor is the surface area of the octahedron greater than the surface area of the inscribed cube? Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 18709,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Integers $1, 2, 3, ... ,n$ , where $n > 2$ , are written on a board. Two numbers $m, k$ such that $1 < m < n, 1 < k < n$ are removed and the average of the remaining numbers is found to be $17$ . What is the maximum sum of the two removed numbers? Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 18710,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alex bakes a total of $24$ pies, and each pie is apple, blueberry, or cherry. The ratio of apple to blueberry to cherry pies is $1:4:3$. How many cherry pies did Alex bake? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 18711,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the parametric equations of the line $l$ are $\\left\\{{\\begin{array}{l}{x=1+\\frac{1}{2}t}\\\\{y=\\sqrt{3}+\\frac{{\\sqrt{3}}}{2}t}\\end{array}}\\right.$ (where $t$ is a parameter), establish a polar coordinate system with the origin as the pole and the non-negativ... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3} - 1}{2}"
} | {
"split": "train",
"index": 18712,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}=(2\\cos\\omega x,-2)$ and $\\overrightarrow{b}=(\\sqrt{3}\\sin\\omega x+\\cos\\omega x,1)$, where $\\omega\\ \\ \\gt 0$, and the function $f(x)=\\overrightarrow{a}\\cdot\\overrightarrow{b}+1$. The distance between two adjacent symmetric center... | math | {
"style": "rule",
"ground_truth": "\\frac{3-\\sqrt{3}}{4}"
} | {
"split": "train",
"index": 18713,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many pairs of consecutive integers in the set $\\{1100, 1101, 1102, \\ldots, 2200\\}$ is no carrying required when the two integers are added? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1100"
} | {
"split": "train",
"index": 18714,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the arithmetic mean of all positive two-digit multiples of 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 18715,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the circle \\(\\Gamma: x^{2} + y^{2} = 1\\) with two intersection points with the \\(x\\)-axis as \\(A\\) and \\(B\\) (from left to right), \\(P\\) is a moving point on circle \\(\\Gamma\\). Line \\(l\\) passes through point \\(P\\) and is tangent to circle \\(\\Gamma\\).... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5} + 2"
} | {
"split": "train",
"index": 18716,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive divisors does the number $24$ have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 18717,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sasha records the numbers 1, 2, 3, 4, and 5 in some order, places arithmetic operation signs \"+\", \"-\", \"x\" and parentheses, and looks at the result of the expression obtained. For example, he can get the number 8 using the expression \\((4-3) \\times (2+5) + 1\\). Can he ... | math | {
"style": "rule",
"ground_truth": "123"
} | {
"split": "train",
"index": 18718,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$, $b$, $c$, and $d$ be positive integers with $a < 3b$, $b < 3c$, and $c < 4d$. Additionally, suppose $b + d = 200$. The largest possible value for $a$ is:\nA) 438\nB) 440\nC) 445\nD) 449\nE) 455 Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "449"
} | {
"split": "train",
"index": 18719,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\sqrt[3]{1+27} \\cdot \\sqrt[3]{1+\\sqrt[3]{27}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[3]{112}"
} | {
"split": "train",
"index": 18720,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set $\\{1, 2, 3,\\dots, 2010\\}$. What is the probability that $abc + ab + a$ is divisible by $3$? Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{27}"
} | {
"split": "train",
"index": 18721,
"solution": "1. **Understanding the Problem**: We need to find the probability that the expression $abc + ab + a$ is divisible by $3$ when $a$, $b$, and $c$ are chosen randomly from the set $\\{1, 2, 3, \\dots, 2010\\}$.\n\n2. **Divisibility Analysis**: We can rewrite the expre... |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, the equation of curve $C_{1}$ is $x^{2}+y^{2}-4x=0$. The parameter equation of curve $C_{2}$ is $\\left\\{\\begin{array}{l}x=\\cos\\beta\\\\ y=1+\\sin\\beta\\end{array}\\right.$ ($\\beta$ is the parameter). Establish a polar coordinate ... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5} + 2"
} | {
"split": "train",
"index": 18722,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( a_{1}, a_{2}, \\cdots, a_{2006} \\) be 2006 positive integers (they can be the same) such that \\( \\frac{a_{1}}{a_{2}}, \\frac{a_{2}}{a_{3}}, \\cdots, \\frac{a_{2005}}{a_{2006}} \\) are all different from each other. What is the minimum number of distinct numbers in \\... | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 18723,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ordered pairs of real numbers $(x, y)$ are there such that $x^2+y^2 = 200$ and\n\\[\\sqrt{(x-5)^2+(y-5)^2}+\\sqrt{(x+5)^2+(y+5)^2}\\]\nis an integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 18724,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine all the pairs $ (p , n )$ of a prime number $ p$ and a positive integer $ n$ for which $ \\frac{ n^p + 1 }{p^n + 1} $ is an integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(p,n)=(p,p),(2,4)"
} | {
"split": "train",
"index": 18725,
"solution": "\nTo solve the problem of finding all pairs \\((p, n)\\) of a prime number \\(p\\) and a positive integer \\(n\\) for which \\(\\frac{n^p + 1}{p^n + 1}\\) is an integer, we start by analyzing the expression:\n\n\\[\n\\frac{n^p + 1}{p^n + 1}.\n\\]\n\n**Step 1: Initi... |
deepscaler | [
{
"role": "user",
"content": "In a certain city, license plate numbers consist of 6 digits (ranging from 0 to 9), but it is required that any two plates differ by at least 2 digits (for example, license plates 038471 and 030471 cannot both be used). Determine the maximum number of different license plates t... | math | {
"style": "rule",
"ground_truth": "100000"
} | {
"split": "train",
"index": 18726,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $63 \\times 57$ in your head. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3591"
} | {
"split": "train",
"index": 18727,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $\\left(\\sqrt{4!\\cdot 3!}\\right)^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 18728,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A coin is tossed. If heads appear, point \\( P \\) moves +1 on the number line; if tails appear, point \\( P \\) does not move. The coin is tossed no more than 12 times, and if point \\( P \\) reaches coordinate +10, the coin is no longer tossed. In how many different ways can ... | math | {
"style": "rule",
"ground_truth": "66"
} | {
"split": "train",
"index": 18729,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular 2017-gon \\( A_1 A_2 \\cdots A_{2017} \\) is inscribed in a unit circle \\( O \\). If two different vertices \\( A_i \\) and \\( A_j \\) are chosen randomly, what is the probability that \\( \\overrightarrow{O A_i} \\cdot \\overrightarrow{O A_j} > \\frac{1}{2} \\)? Le... | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"split": "train",
"index": 18730,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From post office $A$, a car leaves heading towards post office $B$. After 20 minutes, a motorcyclist departs in pursuit of the car, traveling at a speed of 60 km/h. Upon catching up with the car, the motorcyclist delivers a package to the driver's cab and immediately turns back... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 18731,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression, and the common difference $d$ is a multiple of $5$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 18732,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nFrom point $A$ to point $B$, three cars depart at equal time intervals. They all arrive at $B$ simultaneously, and then they proceed to point $C$, which is 120 km away from $B$. The first car arrives at point $C$ one hour after the second car. After reaching point $C$, the th... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 18733,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The walls of a room are in the shape of a triangle $A B C$ with $\\angle A B C=90^{\\circ}, \\angle B A C=60^{\\circ}$, and $A B=6$. Chong stands at the midpoint of $B C$ and rolls a ball toward $A B$. Suppose that the ball bounces off $A B$, then $A C$, then returns exactly to... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{21}"
} | {
"split": "train",
"index": 18734,
"solution": "Let $C^{\\prime}$ be the reflection of $C$ across $A B$ and $B^{\\prime}$ be the reflection of $B$ across $A C^{\\prime}$; note that $B^{\\prime}, A, C$ are collinear by angle chasing. The image of the path under these reflections is just the line segment $M M^{\\p... |
deepscaler | [
{
"role": "user",
"content": "Let $a, b, c, d$ be real numbers. Suppose that all the roots of $z^4+az^3+bz^2+cz+d=0$ are complex numbers lying on a circle in the complex plane centered at $0+0i$ and having radius $1$. The sum of the reciprocals of the roots is necessarily Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "-a"
} | {
"split": "train",
"index": 18735,
"solution": "1. **Identify the roots and their properties**: Let the roots of the polynomial $z^4 + az^3 + bz^2 + cz + d = 0$ be $z_1, z_2, z_3, z_4$. Given that all roots lie on a circle centered at the origin with radius 1, we have $|z_1| = |z_2| = |z_3| = |z_4| = 1$.\n\n2. *... |
deepscaler | [
{
"role": "user",
"content": "A child lines up $2020^2$ pieces of bricks in a row, and then remove bricks whose positions are square numbers (i.e. the 1st, 4th, 9th, 16th, ... bricks). Then he lines up the remaining bricks again and remove those that are in a 'square position'. This process is repeated un... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 18736,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $z_1$, $z_2$, $z_3$, $\\dots$, $z_{12}$ be the 12 zeroes of the polynomial $z^{12} - 2^{36}$. For each $j$, let $w_j$ be one of $z_j$ or $iz_j$. Find the maximum possible value of the real part of\n\\[\\sum_{j = 1}^{12} w_j.\\] Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "16 + 16 \\sqrt{3}"
} | {
"split": "train",
"index": 18737,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Line segments \\( AB \\) and \\( CD \\) are situated between two parallel planes \\( \\alpha \\) and \\( \\beta \\). \\( AC \\subset \\alpha \\) and \\( BD \\subset \\beta \\). Given \\( AB \\perp \\alpha \\), \\( AC = BD = 5 \\), \\( AB = 12 \\), and \\( CD = 13 \\). Points \\... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3} \\sqrt{7}"
} | {
"split": "train",
"index": 18738,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $O$ is the circumcenter of $\\triangle ABC$, $AC \\perp BC$, $AC = 3$, and $\\angle ABC = \\frac{\\pi}{6}$, find the dot product of $\\overrightarrow{OC}$ and $\\overrightarrow{AB}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-9"
} | {
"split": "train",
"index": 18739,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagram shows a quadrilateral \\(PQRS\\) made from two similar right-angled triangles, \\(PQR\\) and \\(PRS\\). The length of \\(PQ\\) is 3, the length of \\(QR\\) is 4, and \\(\\angle PRQ = \\angle PSR\\).\nWhat is the perimeter of \\(PQRS\\)? Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 18740,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many 11 step paths are there from point A to point D, which pass through points B and C in that order? Assume the grid layout permits only right and down steps, where B is 2 right and 2 down steps from A, and C is 1 right and 3 down steps from B, and finally, D is 3 right a... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 18741,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $\\frac{11! - 10!}{9!}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 18742,
"solution": "1. Start by simplifying the expression given in the problem:\n \\[\n \\frac{11! - 10!}{9!}\n \\]\n Here, $n!$ denotes the factorial of $n$, which is the product of all positive integers up to $n$.\n\n2. Factor out $10!$ from the numerator:\n \\[\n \\fra... |
deepscaler | [
{
"role": "user",
"content": "A dress originally priced at $80$ dollars was put on sale for $25\\%$ off. If $10\\%$ tax was added to the sale price, then the total selling price (in dollars) of the dress was Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 18743,
"solution": "1. **Calculate the Sale Price After Discount:**\n The original price of the dress is $80$ dollars. The discount offered is $25\\%$. The amount of discount in dollars can be calculated as:\n \\[\n \\text{Discount} = 25\\% \\times 80 = 0.25 \\times 80 = 20 \\te... |
deepscaler | [
{
"role": "user",
"content": "Let $x=\\frac{4}{(\\sqrt{5}+1)(\\sqrt[4]{5}+1)(\\sqrt[8]{5}+1)(\\sqrt[16]{5}+1)}.$ Find $(x+1)^{48}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 18744,
"solution": "Like Solution $2$, let $z=\\sqrt[16]{5}$ Then, the expression becomes\n$x=\\frac{4}{(z+1)(z^2+1)(z^4+1)(z^8+1)}$ Now, multiplying by the conjugate of each binomial in the denominator, we obtain...\n$x=\\frac{4(z-1)(z^2-1)(z^4-1)(z^8-1)}{(z^2-1)(z^4-1)(z^8-1)(z^{16}... |
deepscaler | [
{
"role": "user",
"content": "In a certain country, there are 100 senators, each of whom has 4 aides. These senators and aides serve on various committees. A committee may consist either of 5 senators, of 4 senators and 4 aides, or of 2 senators and 12 aides. Every senator serves on 5 committees, and every ... | math | {
"style": "rule",
"ground_truth": "160"
} | {
"split": "train",
"index": 18745,
"solution": "If each senator gets a point for every committee on which she serves, and every aide gets $1 / 4$ point for every committee on which he serves, then the 100 senators get 500 points altogether, and the 400 aides get 300 points altogether, for a total of 800 points. ... |
deepscaler | [
{
"role": "user",
"content": "Let $S_1, S_2, \\ldots, S_{100}$ be finite sets of integers whose intersection is not empty. For each non-empty $T \\subseteq \\{S_1, S_2, \\ldots, S_{100}\\},$ the size of the intersection of the sets in $T$ is a multiple of the number of sets in $T$. What is the least possibl... | math | {
"style": "rule",
"ground_truth": "50 \\cdot \\binom{100}{50}"
} | {
"split": "train",
"index": 18746,
"solution": "\nLet \\( S_1, S_2, \\ldots, S_{100} \\) be finite sets of integers such that their intersection is not empty. For every non-empty subset \\( T \\) of \\( \\{S_1, S_2, \\ldots, S_{100}\\} \\), the size of the intersection of the sets in \\( T \\) is a multiple of t... |
deepscaler | [
{
"role": "user",
"content": "A set of six edges of a regular octahedron is called Hamiltonian cycle if the edges in some order constitute a single continuous loop that visits each vertex exactly once. How many ways are there to partition the twelve edges into two Hamiltonian cycles? Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 18747,
"solution": "Call the octahedron $A B C D E F$, where $A, B$, and $C$ are opposite $D, E$, and $F$, respectively. Note that each Hamiltonian cycle can be described in terms of the order it visits vertices in exactly 12 different ways. Conversely, listing the six vertices in som... |
deepscaler | [
{
"role": "user",
"content": "On the first day, 1 bee brings back 5 companions. On the second day, 6 bees (1 from the original + 5 brought back on the first day) fly out, each bringing back 5 companions. Determine the total number of bees in the hive after the 6th day. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "46656"
} | {
"split": "train",
"index": 18748,
"solution": ""
} |
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