data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "The value of $x$ at the intersection of $y=\\frac{8}{x^2+4}$ and $x+y=2$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 18049,
"solution": "1. **Substitute for $y$ in the second equation:** \n Given the equations $y = \\frac{8}{x^2 + 4}$ and $x + y = 2$, we can substitute $y$ from the second equation into the first:\n \\[\n x + y = 2 \\implies y = 2 - x.\n \\]\n Substituting $y = 2 - x$ into... |
deepscaler | [
{
"role": "user",
"content": "There are several soldiers forming a rectangular formation with exactly eight columns. If adding 120 people or removing 120 people from the formation can both form a square formation, how many soldiers are there in the original rectangular formation? Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "136"
} | {
"split": "train",
"index": 18050,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three students, A, B, and C, are playing badminton with the following rules:<br/>The player who loses two games in a row will be eliminated. Before the game, two players are randomly selected to play against each other, while the third player has a bye. The winner of each game ... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{16}"
} | {
"split": "train",
"index": 18051,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $\\mathcal{T}=A_0A_1A_2A_3$ is a tetrahedron with $\\angle A_1A_0A_3 = \\angle A_2A_0A_1 = \\angle A_3A_0A_2 = 90^\\circ$ , $A_0A_1=5, A_0A_2=12$ and $A_0A_3=9$ . A cube $A_0B_0C_0D_0E_0F_0G_0H_0$ with side length $s$ is inscribed inside $\\mathcal{T}$ with... | math | {
"style": "rule",
"ground_truth": "180/71"
} | {
"split": "train",
"index": 18052,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Our school's basketball team has won the national middle school basketball championship multiple times! In one competition, including our school's basketball team, 7 basketball teams need to be randomly divided into two groups (one group with 3 teams and the other with 4 teams)... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{7}"
} | {
"split": "train",
"index": 18053,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m \\ge 3$ be an integer and let $S = \\{3,4,5,\\ldots,m\\}$. Find the smallest value of $m$ such that for every partition of $S$ into two subsets, at least one of the subsets contains integers $a$, $b$, and $c$ (not necessarily distinct) such that $ab = c$.\n Let's think s... | math | {
"style": "rule",
"ground_truth": "243"
} | {
"split": "train",
"index": 18054,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The mean, median, and mode of the five numbers 12, 9, 11, 16, x are all equal. Find the value of x. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 18055,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $\\{a_n\\}$ is defined by \\[a_0 = 1,a_1 = 1, \\text{ and } a_n = a_{n - 1} + \\frac {a_{n - 1}^2}{a_{n - 2}}\\text{ for }n\\ge2.\\] The sequence $\\{b_n\\}$ is defined by \\[b_0 = 1,b_1 = 3, \\text{ and } b_n = b_{n - 1} + \\frac {b_{n - 1}^2}{b_{n - 2}}\\text{ fo... | math | {
"style": "rule",
"ground_truth": "561"
} | {
"split": "train",
"index": 18056,
"solution": "Rearranging the definitions, we have \\[\\frac{a_n}{a_{n-1}} = \\frac{a_{n-1}}{a_{n-2}} + 1,\\quad \\frac{b_n}{b_{n-1}} = \\frac{b_{n-1}}{b_{n-2}} + 1\\] from which it follows that $\\frac{a_n}{a_{n-1}} = 1+ \\frac{a_{n-1}}{a_{n-2}} = \\cdots = (n-1) + \\frac{a_{1}... |
deepscaler | [
{
"role": "user",
"content": "How many 9 step paths are there from $E$ to $G$ which pass through $F$?[asy]size(4cm,4cm);int w=6;int h=5;int i;pen p=fontsize(9);for (i=0; i<h; ++i){draw((0,i) -- (w-1,i));}for (i=0; i<w; ++i){draw((i, 0)--(i,h-1));}label(\"G\", (w-1,0), SE, p);label(\"E\", (0,h-1), NW, p);lab... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 18057,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $|\\overrightarrow{a}|=|\\overrightarrow{b}|=1$, $|\\overrightarrow{c}|=\\sqrt{2}$, and $\\overrightarrow{a}+\\overrightarrow{b}+\\overrightarrow{c}=\\overrightarrow{0}$, calculate $\\cos 〈\\overrightarrow{a}-\\overrightarrow{c}$,$\\overrightarrow{b}-\\overrightar... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"split": "train",
"index": 18058,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagram shows a square \\(PQRS\\). The arc \\(QS\\) is a quarter circle. The point \\(U\\) is the midpoint of \\(QR\\) and the point \\(T\\) lies on \\(SR\\). The line \\(TU\\) is a tangent to the arc \\(QS\\). What is the ratio of the length of \\(TR\\) to the length of \\... | math | {
"style": "rule",
"ground_truth": "4:3"
} | {
"split": "train",
"index": 18059,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\frac{2+3+4}{3}=\\frac{1990+1991+1992}{N}$, then $N=$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1991"
} | {
"split": "train",
"index": 18060,
"solution": "1. **Calculate the left-hand side of the equation:**\n \\[\n \\frac{2+3+4}{3} = \\frac{9}{3} = 3\n \\]\n\n2. **Set up the equation:**\n Given that\n \\[\n \\frac{2+3+4}{3} = \\frac{1990+1991+1992}{N}\n \\]\n substituting the value from step 1, we ha... |
deepscaler | [
{
"role": "user",
"content": "Compute $\\dbinom{25}{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 18061,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The three vertices of an inscribed triangle in a circle divide the circumference into three arcs of lengths $3$, $4$, and $5$. The area of this triangle is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{\\pi^{2}}(\\sqrt{3}+3)"
} | {
"split": "train",
"index": 18062,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Yesterday Han drove 1 hour longer than Ian at an average speed 5 miles per hour faster than Ian. Jan drove 2 hours longer than Ian at an average speed 10 miles per hour faster than Ian. Han drove 70 miles more than Ian. How many more miles did Jan drive than Ian? Let's think st... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 18063,
"solution": "1. **Define Variables:**\n Let $d$ be the distance Ian drove, $t$ be the time Ian drove, and $s$ be Ian's speed. Thus, Ian's equation is:\n \\[\n d = s \\cdot t\n \\]\n This is equation (1).\n\n2. **Set Up Han's Equation:**\n Han drove 1 hour longer tha... |
deepscaler | [
{
"role": "user",
"content": "One of the three cards had the number 18, another had the number 75, and the third had some two-digit number. If you sum all the distinct six-digit numbers that can be obtained by arranging these cards in a row, you get the number 2606058. What number is written on the third ca... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 18064,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six orange candies and four purple candies are available to create different flavors. A flavor is considered different if the percentage of orange candies is different. Combine some or all of these ten candies to determine how many unique flavors can be created based on their r... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 18065,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For $k\\ge 1$ , define $a_k=2^k$ . Let $$ S=\\sum_{k=1}^{\\infty}\\cos^{-1}\\left(\\frac{2a_k^2-6a_k+5}{\\sqrt{(a_k^2-4a_k+5)(4a_k^2-8a_k+5)}}\\right). $$ Compute $\\lfloor 100S\\rfloor$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "157"
} | {
"split": "train",
"index": 18066,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Assuming $a \\neq 3$, $b \\neq 4$, and $c \\neq 5$, what is the value in simplest form of the following expression?\n\\[\\frac{a-3}{5-c} \\cdot \\frac{b-4}{3-a} \\cdot \\frac{c-5}{4-b}\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 18067,
"solution": "1. **Identify the pattern in the expression**: We are given the expression:\n \\[\n \\frac{a-3}{5-c} \\cdot \\frac{b-4}{3-a} \\cdot \\frac{c-5}{4-b}\n \\]\n We notice that each fraction has a numerator and a denominator that are differences of variables and... |
deepscaler | [
{
"role": "user",
"content": "What is that largest positive integer $n$ for which $n^3+100$ is divisible by $n+10$?\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "890"
} | {
"split": "train",
"index": 18068,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a convex polygon, all its diagonals are drawn. These diagonals divide the polygon into several smaller polygons. What is the maximum number of sides that a polygon in the subdivision can have if the original polygon has:\n\na) 13 sides;\n\nb) 1950 sides? Let's think step by ... | math | {
"style": "rule",
"ground_truth": "1950"
} | {
"split": "train",
"index": 18069,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $m, n \\in \\mathbb{R}$, if the line $(m+1)x + (n+1)y - 2 = 0$ is tangent to the circle $x^2 + y^2 = 1$, find the maximum value of $m - n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 18070,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be square with 4 digits, such that all its digits are less than 6. If we add 1 to each digit the resulting number is another square. Find $n$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2025"
} | {
"split": "train",
"index": 18071,
"solution": "\nLet \\( n \\) be a number with four digits such that all digits are less than 6. We have \\( n = a^2 \\) where \\( a \\) is an integer. We need a transformation that, if we add 1 to each digit of \\( n \\), the result should be another perfect square.\n\nLet's de... |
deepscaler | [
{
"role": "user",
"content": "A palindrome is a number that reads the same forwards and backwards, such as 3003. How many positive four-digit integers are palindromes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 18072,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(\\mathcal{S}\\) be a set of 16 points in the plane, no three collinear. Let \\(\\chi(\\mathcal{S})\\) denote the number of ways to draw 8 line segments with endpoints in \\(\\mathcal{S}\\), such that no two drawn segments intersect, even at endpoints. Find the smallest p... | math | {
"style": "rule",
"ground_truth": "1430"
} | {
"split": "train",
"index": 18073,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three numbers, $a_1, a_2, a_3$, are drawn randomly and without replacement from the set $\\{1, 2, 3,\\ldots, 1000\\}$. Three other numbers, $b_1, b_2, b_3$, are then drawn randomly and without replacement from the remaining set of $997$ numbers. Let $p$ be the probability that,... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 18074,
"solution": "There is a total of $P(1000,6)$ possible ordered $6$-tuples $(a_1,a_2,a_3,b_1,b_2,b_3).$\n There are $C(1000,6)$ possible sets $\\{a_1,a_2,a_3,b_1,b_2,b_3\\}.$ We have five valid cases for the increasing order of these six elements:\n$aaabbb$\n$aababb$\n$aabbab$\n$... |
deepscaler | [
{
"role": "user",
"content": "For what base-6 digit $d$ is $2dd5_6$ divisible by the base 10 number 11? (Here $2dd5_6$ represents a base-6 number whose first digit is 2, whose last digit is 5, and whose middle two digits are both equal to $d$). Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 18075,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find\n\\[\\sin \\left( \\sin^{-1} \\frac{3}{5} + \\tan^{-1} 2 \\right).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11 \\sqrt{5}}{25}"
} | {
"split": "train",
"index": 18076,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ is isosceles with $AB=AC$ . The bisectors of angles $ABC$ and $ACB$ meet at $I$ . If the measure of angle $CIA$ is $130^\\circ$ , compute the measure of angle $CAB$ .\n\n*Proposed by Connor Gordon* Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 18077,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define $\\phi^{!}(n)$ as the product of all positive integers less than or equal to $n$ and relatively prime to $n$. Compute the remainder when $$ \\sum_{\\substack{2 \\leq n \\leq 50 \\\\ \\operatorname{gcd}(n, 50)=1}} \\phi^{!}(n) $$ is divided by 50 . Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 18078,
"solution": "First, $\\phi^{!}(n)$ is even for all odd $n$, so it vanishes modulo 2 . To compute the remainder modulo 25 , we first evaluate $\\phi^{!}(3)+\\phi^{!}(7)+\\phi^{!}(9) \\equiv 2+5 \\cdot 4+5 \\cdot 3 \\equiv 12$ $(\\bmod 25)$. Now, for $n \\geq 11$ the contribution... |
deepscaler | [
{
"role": "user",
"content": "How many different four-letter arrangements can be formed using the six letters $A, B, C, D, E$ and $F$, if the first letter must be $C$, one of the other letters must be $B$, and no letter can be used more than once in the arrangement? Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 18079,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\cos x$, where $x\\in[0,2\\pi]$, there are two distinct zero points $x\\_1$, $x\\_2$, and the equation $f(x)=m$ has two distinct real roots $x\\_3$, $x\\_4$. If these four numbers are arranged in ascending order to form an arithmetic sequence, the valu... | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 18080,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)= \\dfrac {4^{x}+1}{4^{x- \\frac {1}{2}}+1}$, find the sum of $f( \\dfrac {1}{2014})+f( \\dfrac {2}{2014})+…+f( \\dfrac {2013}{2014})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac {6039}{2}"
} | {
"split": "train",
"index": 18081,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square $ABCD$ has side length $13$, and points $E$ and $F$ are exterior to the square such that $BE=DF=5$ and $AE=CF=12$. Find $EF^{2}$. [asy]unitsize(0.2 cm); pair A, B, C, D, E, F; A = (0,13); B = (13,13); C = (13,0); D = (0,0); E = A + (12*12/13,5*12/13); F = D + (5*5/13,-5*... | math | {
"style": "rule",
"ground_truth": "578"
} | {
"split": "train",
"index": 18082,
"solution": "Drawing $EF$, it clearly passes through the center of $ABCD$. Letting this point be $P$, we note that $AEBP$ and $CFDP$ are congruent cyclic quadrilaterals, and that $AP=BP=CP=DP=\\frac{13}{\\sqrt{2}}.$ Now, from Ptolemy's, $13\\cdot EP=\\frac{13}{\\sqrt{2}}(12+5)\... |
deepscaler | [
{
"role": "user",
"content": "In a room containing $N$ people, $N > 3$, at least one person has not shaken hands with everyone else in the room.\nWhat is the maximum number of people in the room that could have shaken hands with everyone else? Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "N-1"
} | {
"split": "train",
"index": 18083,
"solution": "To solve this problem, we need to determine the maximum number of people in a room of $N$ people who could have shaken hands with every other person, given that at least one person has not shaken hands with everyone else.\n\n1. **Understanding the Problem:**\n - ... |
deepscaler | [
{
"role": "user",
"content": "Given that $4:5 = 20 \\div \\_\\_\\_\\_\\_\\_ = \\frac{()}{20} = \\_\\_\\_\\_\\_\\_ \\%$, find the missing values. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 18084,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be a positive integer, and let Pushover be a game played by two players, standing squarely facing each other, pushing each other, where the first person to lose balance loses. At the HMPT, $2^{n+1}$ competitors, numbered 1 through $2^{n+1}$ clockwise, stand in a circle.... | math | {
"style": "rule",
"ground_truth": "\\frac{2^{n}-1}{8^{n}}"
} | {
"split": "train",
"index": 18085,
"solution": "At any point during this competition, we shall say that the situation is living if both players 1 and $2^{n}$ are still in the running. A living situation is far if those two players are diametrically opposite each other, and near otherwise, in which case (as one c... |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle ABC$ have sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively, and satisfy the equation $a\\sin B = \\sqrt{3}b\\cos A$. \n$(1)$ Find the measure of angle $A$. \n$(2)$ Choose one set of conditions from the following three sets to ensure t... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3} + 3\\sqrt{2}"
} | {
"split": "train",
"index": 18086,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ satisfying $a_1=1$, $|a_n-a_{n-1}|= \\frac {1}{2^n}$ $(n\\geqslant 2,n\\in\\mathbb{N})$, and the subsequence $\\{a_{2n-1}\\}$ is decreasing, while $\\{a_{2n}\\}$ is increasing, find the value of $5-6a_{10}$. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "\\frac {1}{512}"
} | {
"split": "train",
"index": 18087,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\sqrt{\\frac{3}{x} + 3} = \\frac{5}{3}$, solve for $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{27}{2}"
} | {
"split": "train",
"index": 18088,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the polar coordinate equation of circle C is ρ² + 2$\\sqrt {2}$ρsin(θ + $\\frac {π}{4}$) + 1 = 0, and the origin O of the rectangular coordinate system xOy coincides with the pole, and the positive semi-axis of the x-axis coincides with the polar axis. (1) Find the s... | math | {
"style": "rule",
"ground_truth": "\\frac {3}{2} + \\sqrt {2}"
} | {
"split": "train",
"index": 18089,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Pascal's Triangle starting with row 1 has the sum of elements in row $n$ given by $2^{n-1}$. What is the sum of the interior numbers of the ninth row, considering interior numbers are all except the first and last numbers in the row? Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "254"
} | {
"split": "train",
"index": 18090,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let n be the smallest positive integer such that n is divisible by 20, n^2 is a perfect square, and n^3 is a perfect fifth power. Find the value of n. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3200000"
} | {
"split": "train",
"index": 18091,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Say that an integer $A$ is yummy if there exist several consecutive integers, including $A$, that add up to 2014. What is the smallest yummy integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2013"
} | {
"split": "train",
"index": 18092,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate \\[\\frac 2{\\log_4{2000^6}} + \\frac 3{\\log_5{2000^6}},\\]giving your answer as a fraction in lowest terms. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\tfrac{1}{6}"
} | {
"split": "train",
"index": 18093,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 5 yellow pegs, 4 red pegs, 3 green pegs, 2 blue pegs, and 1 orange peg to be placed on a triangular peg board. In how many ways can the pegs be placed so that no (horizontal) row or (vertical) column contains two or more pegs of the same color? (Any two pegs of the s... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 18094,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a triangle such that $AB=6,BC=5,AC=7.$ Let the tangents to the circumcircle of $ABC$ at $B$ and $C$ meet at $X.$ Let $Z$ be a point on the circumcircle of $ABC.$ Let $Y$ be the foot of the perpendicular from $X$ to $CZ.$ Let $K$ be the inter... | math | {
"style": "rule",
"ground_truth": "147/10"
} | {
"split": "train",
"index": 18095,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kim earned scores of 86, 82, and 89 on her first three mathematics examinations. She is expected to increase her average score by at least 2 points with her fourth exam. What is the minimum score Kim must achieve on her fourth exam to meet this target? Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "94"
} | {
"split": "train",
"index": 18096,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a, b, c, d$ be real numbers such that $a^{2}+b^{2}+c^{2}+d^{2}=1$. Determine the minimum value of $(a-b)(b-c)(c-d)(d-a)$ and determine all values of $(a, b, c, d)$ such that the minimum value is achieved. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{8}"
} | {
"split": "train",
"index": 18097,
"solution": "The minimum value is $-\\frac{1}{8}$. There are eight equality cases in total. The first one is $$\\left(\\frac{1}{4}+\\frac{\\sqrt{3}}{4},-\\frac{1}{4}-\\frac{\\sqrt{3}}{4}, \\frac{1}{4}-\\frac{\\sqrt{3}}{4},-\\frac{1}{4}+\\frac{\\sqrt{3}}{4}\\right)$$ Cyclic shif... |
deepscaler | [
{
"role": "user",
"content": "An ant starts at the point \\((1,0)\\). Each minute, it walks from its current position to one of the four adjacent lattice points until it reaches a point \\((x, y)\\) with \\(|x|+|y| \\geq 2\\). What is the probability that the ant ends at the point \\((1,1)\\)? Let's think s... | math | {
"style": "rule",
"ground_truth": "7/24"
} | {
"split": "train",
"index": 18098,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of distinct arrangements in a row of all natural numbers from 1 to 10 such that the sum of any three consecutive numbers is divisible by 3. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1728"
} | {
"split": "train",
"index": 18099,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that an office at a school needs to arrange a duty roster from the 1st to the 6th day with six designated individuals participating, find the total number of different arrangements possible, given that person A and person B cannot be adjacent, and person C and person D al... | math | {
"style": "rule",
"ground_truth": "336"
} | {
"split": "train",
"index": 18100,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a$ and $b \\in R$, the function $f(x) = \\ln(x + 1) - 2$ is tangent to the line $y = ax + b - \\ln2$ at $x = -\\frac{1}{2}$. Let $g(x) = e^x + bx^2 + a$. If the inequality $m \\leqslant g(x) \\leqslant m^2 - 2$ holds true in the interval $[1, 2]$, determine the real... | math | {
"style": "rule",
"ground_truth": "e + 1"
} | {
"split": "train",
"index": 18101,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a $3 \\times 3$ chessboard, each square contains a Chinese knight with $\\frac{1}{2}$ probability. What is the probability that there are two Chinese knights that can attack each other? (In Chinese chess, a Chinese knight can attack any piece which is two squares away from i... | math | {
"style": "rule",
"ground_truth": "\\frac{79}{256}"
} | {
"split": "train",
"index": 18102,
"solution": "attack each other using PIE. First notice that in any setup with two knights attack each other, the center square must be empty. Also, for any pair of knights that attack each other, one must be in a corner, and the other at the center of a nonadjacent side. There ... |
deepscaler | [
{
"role": "user",
"content": "When we say that Ray is climbing up the stairs $m$ at a time, we mean that he starts on the floor (step $0$) then jumps to step $m$ and then to $2m$ and so on until the number of steps to the top is less than $m$. Ray climbs up a flight of stairs of $n$ steps in two ways. When ... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 18103,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using the method of base prime representation, where the place of each digit represents an exponent in the prime factorization (starting with the smallest prime on the right), convert the number $196$ into base prime. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "2002"
} | {
"split": "train",
"index": 18104,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the arithmetic sequence $\\{a_n\\}$, the common difference is $\\frac{1}{2}$, and $a_1+a_3+a_5+\\ldots+a_{99}=60$. Find the value of $a_2+a_4+a_6+\\ldots+a_{100}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "85"
} | {
"split": "train",
"index": 18105,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $x$, such that $3^7 \\cdot 3^x = 81$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 18106,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=(a\\sin x+b\\cos x)\\cdot e^{x}$ has an extremum at $x= \\frac {\\pi}{3}$, determine the value of $\\frac {a}{b}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2- \\sqrt {3}"
} | {
"split": "train",
"index": 18107,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a sphere inscribed in a right cone with the base radius of 10 cm and height of 40 cm. The radius of the inscribed sphere can be expressed as $b\\sqrt{d} - b$ cm. Determine the value of $b+d$. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "19.5"
} | {
"split": "train",
"index": 18108,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 18109,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rectangle $PQRS$ is inscribed in a semicircle with diameter $\\overline{GH}$, such that $PR=20$, and $PG=SH=12$. Determine the area of rectangle $PQRS$.\nA) $120\\sqrt{6}$\nB) $150\\sqrt{6}$\nC) $160\\sqrt{6}$\nD) $180\\sqrt{6}$\nE) $200\\sqrt{6}$ Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "160\\sqrt{6}"
} | {
"split": "train",
"index": 18110,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a circle with radius $4$, and there are numerous line segments of length $6$ that are tangent to the circle at their midpoints. Compute the area of the region consisting of all such line segments.\nA) $8\\pi$ \nB) $7\\pi$ \nC) $9\\pi$ \nD) $10\\pi$ Let's think step ... | math | {
"style": "rule",
"ground_truth": "9\\pi"
} | {
"split": "train",
"index": 18111,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Regular hexagon $ABCDEF$ is the base of right pyramid $\\allowbreak PABCDEF$. If $PAD$ is an equilateral triangle with side length 8, then what is the volume of the pyramid? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 18112,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A natural number greater than 1 is defined as nice if it is equal to the product of its distinct proper divisors. A number \\( n \\) is nice if:\n1. \\( n = pq \\), where \\( p \\) and \\( q \\) are distinct prime numbers.\n2. \\( n = p^3 \\), where \\( p \\) is a prime number.... | math | {
"style": "rule",
"ground_truth": "182"
} | {
"split": "train",
"index": 18113,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $a_1,$ $a_2,$ $a_3,$ $\\dots,$ let $S_n$ denote the sum of the first $n$ terms of the sequence.\n\nIf $a_1 = 1$ and\n\\[a_n = \\frac{2S_n^2}{2S_n - 1}\\]for all $n \\ge 2,$ then find $a_{100}.$ Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{39203}"
} | {
"split": "train",
"index": 18114,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a right prism with triangular bases, given the sum of the areas of three mutually adjacent faces (that is, of two lateral faces and one base) is 24, find the maximum volume of the prism.\n\n[asy]\nunitsize(1 cm);\n\npair A, B, C, D, E, F;\n\nA = (0,0);\nB = (3,-1);\nC = (-1,... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 18115,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of the numbers $a_1,$ $a_2,$ $\\dots,$ $a_{95}$ is $\\pm 1.$ Find the smallest possible positive value of\n\\[\\sum_{1 \\le i < j \\le 95} a_i a_j.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 18116,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the smallest 216 positive integers written in base 6 use the digit 3 or 4 (or both) as a digit? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "168"
} | {
"split": "train",
"index": 18117,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What real number is equal to the expression $3 + \\frac{5}{2 + \\frac{5}{3 + \\frac{5}{2 + \\cdots}}}$, where the $2$s and $3$s alternate? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"split": "train",
"index": 18118,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the number of units in the area of the circle with center at $P$ and passing through $Q$? Express your answer in terms of $\\pi$.\n\n[asy]\nsize(150); pair P = (-3,4), Q=(9,-3); string stringpair(pair p){return \"$(\"+string(p.x)+\", \"+string(p.y)+\"$)\";}\ndraw((-15,0... | math | {
"style": "rule",
"ground_truth": "193\\pi"
} | {
"split": "train",
"index": 18119,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the sequence of numbers defined recursively by $t_1=1$ and for $n>1$ by $t_n=1+t_{n/2}$ when $n$ is even and by $t_n=\\frac{1}{t_{n-1}}$ when $n$ is odd. Given that $t_n=\\frac{19}{87}$, find $n.$ Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "1905"
} | {
"split": "train",
"index": 18120,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The teacher wrote a positive number $x$ on the board and asked Kolya, Petya, and Vasya to raise this number to the 3rd, 4th, and 12th powers, respectively. It turned out that Kolya's number had at least 9 digits before the decimal point, and Petya's number had no more than 11 d... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 18121,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four planes divide space into $n$ parts at most. Calculate $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 18122,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the phase shift, vertical shift, and the maximum and minimum values of the graph of \\( y = 3\\sin\\left(3x - \\frac{\\pi}{4}\\right) + 1 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 18123,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the product of the numerator and the denominator when $0.\\overline{012}$ is expressed as a fraction in lowest terms? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1332"
} | {
"split": "train",
"index": 18124,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What three-digit integer is equal to the sum of the factorials of its digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "145"
} | {
"split": "train",
"index": 18125,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Regular octagon \\( CH I L D R E N \\) has area 1. Determine the area of quadrilateral \\( L I N E \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 18126,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Round 1278365.7422389 to the nearest hundred. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1278400"
} | {
"split": "train",
"index": 18127,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose we keep rolling a fair 2014-sided die (whose faces are labelled 1, 2, .., 2014) until we obtain a value less than or equal to the previous roll. Let $E$ be the expected number of times we roll the die. Find the nearest integer to $100 E$. Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "272"
} | {
"split": "train",
"index": 18128,
"solution": "Let $n=2014$. Let $p_{k}$ denote the probability the sequence has length at least $k$. We observe that $$p_{k}=\\frac{\\binom{n}{k}}{n^{k}}$$ since every sequence of $k$ rolls can be sorted in exactly one way. Now the answer is $$\\sum_{k \\geq 0} p_{k}=\\left(1+\\... |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer whose cube ends in $888$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "192"
} | {
"split": "train",
"index": 18129,
"solution": "Let $x^3 = 1000a + 888$. We factor an $8$ out of the right hand side, and we note that $x$ must be of the form $x = 2y$, where $y$ is a positive integer. Then, this becomes $y^3 = 125a + 111$. Taking mod $5$, $25$, and $125$, we get $y^3 \\equiv 1\\pmod 5$, $y^3 \\... |
deepscaler | [
{
"role": "user",
"content": "A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 7 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{10}"
} | {
"split": "train",
"index": 18130,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine $S$, the sum of all the real coefficients of the expansion of $(1+ix)^{2020}$, and find $\\log_2(S)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"split": "train",
"index": 18131,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A uniform tetrahedron has its four faces numbered with 1, 2, 3, and 4. It is randomly thrown twice, and the numbers on the bottom face of the tetrahedron are $x_1$ and $x_2$, respectively. Let $t = (x_{1}-3)^{2}+(x_{2}-3)^{2}$. \n(1) Calculate the probabilities of $t$ reaching... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{16}"
} | {
"split": "train",
"index": 18132,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two tour guides are leading six tourists. The guides decide to split up. Each tourist must choose one of the guides, but with the stipulation that each guide must take at least one tourist. How many different groupings of guides and tourists are possible? Let's think step by st... | math | {
"style": "rule",
"ground_truth": "62"
} | {
"split": "train",
"index": 18133,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A team of loggers was scheduled to harvest $216 \\mathrm{~m}^{3}$ of wood over several days. For the first three days, the team met the daily target set by the plan. Then, they harvested an additional $8 \\mathrm{~m}^{3}$ above the daily target each day. As a result, they harve... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 18134,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Zebadiah has 3 red shirts, 3 blue shirts, and 3 green shirts in a drawer. Without looking, he randomly pulls shirts from his drawer one at a time. What is the minimum number of shirts that Zebadiah has to pull out to guarantee that he has a set of shirts that includes either 3 ... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 18135,
"solution": "Zebadiah must remove at least 3 shirts. If he removes 3 shirts, he might remove 2 red shirts and 1 blue shirt. If he removes 4 shirts, he might remove 2 red shirts and 2 blue shirts. Therefore, if he removes fewer than 5 shirts, it is not guaranteed that he removes... |
deepscaler | [
{
"role": "user",
"content": "From the set \\(\\left\\{-3, -\\frac{5}{4}, -\\frac{1}{2}, 0, \\frac{1}{3}, 1, \\frac{4}{5}, 2\\right\\}\\), two numbers are drawn without replacement. Find the concept of the two numbers being the slopes of perpendicular lines. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "3/28"
} | {
"split": "train",
"index": 18136,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a hemispherical raw material. If this material is processed into a cube through cutting, the maximum value of the ratio of the volume of the obtained workpiece to the volume of the raw material is ______. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {6}}{3\\pi }"
} | {
"split": "train",
"index": 18137,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice, Bob, and Carol each start with $3. Every 20 seconds, a bell rings, at which time each of the players who has an odd amount of money must give $1 to one of the other two players randomly. What is the probability that after the bell has rung 100 times, each player has exac... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{13}"
} | {
"split": "train",
"index": 18138,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $E$ is the midpoint of side $\\overline{CD}$ in square $ABCD,$ and $\\overline{BE}$ meets diagonal $\\overline{AC}$ at $F.$ The area of quadrilateral $AFED$ is $45.$ What is the area of $ABCD?$ Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 18139,
"solution": "1. **Identify the Coordinates of Points in the Square:**\n We place square $ABCD$ in the Cartesian coordinate system with $D$ as the origin. Thus, we have:\n - $D = (0, 0)$\n - $C = (1, 0)$\n - $B = (1, 1)$\n - $A = (0, 1)$\n\n2. **Determine the Coordinat... |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a trapezoid of bases $AB$ and $CD$ . Let $O$ be the intersection point of the diagonals $AC$ and $BD$ . If the area of the triangle $ABC$ is $150$ and the area of the triangle $ACD$ is $120$ , calculate the area of the triangle $BCO$ . Let's thi... | math | {
"style": "rule",
"ground_truth": "\\frac{200}{3}"
} | {
"split": "train",
"index": 18140,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is known that there is a four-person team consisting of boys A and B and girls C and D participating in a quiz activity organized by a TV station. The activity consists of four rounds. Let the probabilities for the boys to pass rounds one to four be $\\frac{5}{6}, \\frac{4}{... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{15}"
} | {
"split": "train",
"index": 18141,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ denote the sum of all of the three digit positive integers with three distinct digits. Compute the remainder when $S$ is divided by $1000$.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "680"
} | {
"split": "train",
"index": 18142,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a,b,c$ are integers from the set of positive integers less than $7$ such that \\begin{align*}\nabc&\\equiv 1\\pmod 7,\\\\\n5c&\\equiv 2\\pmod 7,\\\\\n6b&\\equiv 3+b\\pmod 7,\n\\end{align*}then what is the remainder when $a+b+c$ is divided by $7$? Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 18143,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three-fourths of the parrots on Bird Island are green, and the remainder are blue. If there are 92 parrots total on Bird Island, how many of those parrots are blue? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 18144,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a function defined on the set of positive integers as follows:\n\\[ f(n) = \\begin{cases} \nn - 3, & \\text{if } n \\geq 1000 \\\\\nf[f(n + 7)], & \\text{if } n < 1000 \n\\end{cases} \\]\nWhat is the value of \\( f(90) \\)? Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "999"
} | {
"split": "train",
"index": 18145,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In solving a problem that reduces to a quadratic equation one student makes a mistake only in the constant term of the equation and obtains $8$ and $2$ for the roots. Another student makes a mistake only in the coefficient of the first degree term and find $-9$ and $-1$ for the... | math | {
"style": "rule",
"ground_truth": "x^2-10x+9=0"
} | {
"split": "train",
"index": 18146,
"solution": "1. **Identify the correct form of the quadratic equation**: Let the correct quadratic equation be $x^2 + bx + c = 0$. Here, $b$ is the coefficient of the $x$ term, and $c$ is the constant term.\n\n2. **Use the properties of roots**: For a quadratic equation $x^2 + ... |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of the function \\( f(x) = 3^x - 9^x \\) for real numbers \\( x \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 18147,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangular array of squares has one square in the first row, two in the second, and in general, $k$ squares in the $k$th row for $1 \\leq k \\leq 11.$ With the exception of the bottom row, each square rests on two squares in the row immediately below (illustrated in the given... | math | {
"style": "rule",
"ground_truth": "640"
} | {
"split": "train",
"index": 18148,
"solution": ""
} |
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