data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Rectangle $ABCD$ is the base of pyramid $PABCD$. If $AB = 8$, $BC = 4$, $\\overline{PA}\\perp \\overline{AB}$, $\\overline{PA}\\perp\\overline{AD}$, and $PA = 6$, then what is the volume of $PABCD$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 7500,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the midpoint of the segment with endpoints (7,-6) and (-3,4)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(2,-1)"
} | {
"split": "train",
"index": 7501,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest number, all of whose digits are either 5, 3, or 1, and whose digits add up to $15$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "555"
} | {
"split": "train",
"index": 7502,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $R$ if $\\sqrt[4]{R^3} = 64\\sqrt[16]{4}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "256 \\cdot 2^{1/6}"
} | {
"split": "train",
"index": 7503,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For all non-zero numbers $x$ and $y$ such that $x = 1/y$, $\\left(x-\\frac{1}{x}\\right)\\left(y+\\frac{1}{y}\\right)$ equals Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x^2-y^2"
} | {
"split": "train",
"index": 7504,
"solution": "1. **Substitute $y$ in terms of $x$:** Given $x = \\frac{1}{y}$, we can rewrite $y$ as $y = \\frac{1}{x}$. \n\n2. **Substitute and simplify the expression:** We start by substituting $y = \\frac{1}{x}$ into the expression $\\left(x-\\frac{1}{x}\\right)\\left(y+\\fra... |
deepscaler | [
{
"role": "user",
"content": "How many pairs of integers $a$ and $b$ are there such that $a$ and $b$ are between $1$ and $42$ and $a^9 = b^7 \\mod 43$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 7505,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the $111$th digit after the decimal point when $\\frac{33}{555}$ is expressed as a decimal? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 7506,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Célia wants to trade with Guilherme stickers from an album about Brazilian animals. Célia wants to trade four butterfly stickers, five shark stickers, three snake stickers, six parakeet stickers, and six monkey stickers. All of Guilherme's stickers are of spiders. They know tha... | math | {
"style": "rule",
"ground_truth": "171"
} | {
"split": "train",
"index": 7507,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest possible three-digit number that is divisible by 3 and divisible by 6? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "996"
} | {
"split": "train",
"index": 7508,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC,$ $AB=AC=25$ and $BC=23.$ Points $D,E,$ and $F$ are on sides $\\overline{AB},$ $\\overline{BC},$ and $\\overline{AC},$ respectively, such that $\\overline{DE}$ and $\\overline{EF}$ are parallel to $\\overline{AC}$ and $\\overline{AB},$ respectively. What is... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 7509,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two lines $l_{1}$: $(a+2)x+(a+3)y-5=0$ and $l_{2}$: $6x+(2a-1)y-5=0$ are parallel, then $a=$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\dfrac{5}{2}"
} | {
"split": "train",
"index": 7510,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A and B are playing a guessing game. First, A thinks of a number, denoted as $a$, then B guesses the number A was thinking of, denoting B's guess as $b$. Both $a$ and $b$ belong to the set $\\{0,1,2,…,9\\}$. The probability that $|a-b|\\leqslant 1$ is __________. Let's think st... | math | {
"style": "rule",
"ground_truth": "\\dfrac{7}{25}"
} | {
"split": "train",
"index": 7511,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain number quiz, the test score of a student with seat number $n$ ($n=1,2,3,4$) is denoted as $f(n)$. If $f(n) \\in \\{70,85,88,90,98,100\\}$ and it satisfies $f(1)<f(2) \\leq f(3)<f(4)$, then the total number of possible combinations of test scores for these 4 student... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 7512,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m$ be the smallest positive integer such that $m^2+(m+1)^2+\\cdots+(m+10)^2$ is the square of a positive integer $n$ . Find $m+n$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "95"
} | {
"split": "train",
"index": 7513,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The square quilt block shown is made from sixteen unit squares, where eight of these squares have been divided in half diagonally to form triangles. Each triangle is shaded. What fraction of the square quilt is shaded? Express your answer as a common fraction. Let's think step ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 7514,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Laura constructs a cone for an art project. The cone has a height of 15 inches and a circular base with a diameter of 8 inches. Laura needs to find the smallest cube-shaped box to transport her cone safely to the art gallery. What is the volume of this box, in cubic inches? Let... | math | {
"style": "rule",
"ground_truth": "3375"
} | {
"split": "train",
"index": 7515,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$ we have $AB = 25$, $BC = 39$, and $AC=42$. Points $D$ and $E$ are on $AB$ and $AC$ respectively, with $AD = 19$ and $AE = 14$. What is the ratio of the area of triangle $ADE$ to the area of the quadrilateral $BCED$? Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "\\frac{19}{56}"
} | {
"split": "train",
"index": 7516,
"solution": "1. **Identify Similar Triangles and Use Ratios**:\n Given that $D$ and $E$ are on $AB$ and $AC$ respectively, with $AD = 19$ and $AE = 14$, and $AB = 25$ and $AC = 42$. We introduce point $F$ on $AC$ such that $DE \\parallel BF$. This implies $\\triangle ADE \\sim... |
deepscaler | [
{
"role": "user",
"content": "Quadrilateral $ABCD$ is a parallelogram. What is the degree measure of angle $A$? [asy]\nsize(100);\ndraw((0,0)--(5,2)--(6,7)--(1,5)--cycle);\ndraw((5,2)--(7.5,3));\ndraw(Arc((5,2),1,20,80));\nlabel(\"$D$\",(0,0),SW); label(\"$C$\",(5,2),SE); label(\"$B$\",(6,7),NE); label(\"$... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 7517,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\overline{CH}$ be an altitude of $\\triangle ABC$. Let $R\\,$ and $S\\,$ be the points where the circles inscribed in the triangles $ACH\\,$ and $BCH$ are tangent to $\\overline{CH}$. If $AB = 1995\\,$, $AC = 1994\\,$, and $BC = 1993\\,$, then $RS\\,$ can be expressed as ... | math | {
"style": "rule",
"ground_truth": "997"
} | {
"split": "train",
"index": 7518,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A, B, C$, and $D$ be points randomly selected independently and uniformly within the unit square. What is the probability that the six lines \\overline{A B}, \\overline{A C}, \\overline{A D}, \\overline{B C}, \\overline{B D}$, and \\overline{C D}$ all have positive slope? ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{24}"
} | {
"split": "train",
"index": 7519,
"solution": "Consider the sets of $x$-coordinates and $y$-coordinates of the points. In order to make 6 lines of positive slope, we must have smallest x -coordinate must be paired with the smallest y-coordinate, the second smallest together, and so forth. If we fix the order of ... |
deepscaler | [
{
"role": "user",
"content": "(1) Evaluate: $\\sin^2 120^\\circ + \\cos 180^\\circ + \\tan 45^\\circ - \\cos^2 (-330^\\circ) + \\sin (-210^\\circ)$; \n(2) Determine the monotonic intervals of the function $f(x) = \\left(\\frac{1}{3}\\right)^{\\sin x}$. Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 7520,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a right rectangular prism \\(B\\) with edge lengths \\(2,\\ 5,\\) and \\(6\\), including its interior. For any real \\(r \\geq 0\\), let \\(T(r)\\) be the set of points in 3D space within a distance \\(r\\) from some point in \\(B\\). The volume of \\(T(r)\\) is expres... | math | {
"style": "rule",
"ground_truth": "\\frac{8112}{240}"
} | {
"split": "train",
"index": 7521,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular piece of cardboard was cut along its diagonal. On one of the obtained pieces, two cuts were made parallel to the two shorter sides, at the midpoints of those sides. In the end, a rectangle with a perimeter of $129 \\mathrm{~cm}$ remained. The given drawing indicat... | math | {
"style": "rule",
"ground_truth": "258"
} | {
"split": "train",
"index": 7522,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse $C$: $\\frac{x^2}{a^2}+\\frac{y^2}{b^2}=1\\left(a \\gt b \\gt 0\\right)$ with the left and right foci $F_{1}$ and $F_{2}$ respectively. Let $F_{1}$ be the center of a circle with radius $|F_{1}F_{2}|$, intersecting the $y$-axis at points $A$ and $B$ (both point... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3} - 1"
} | {
"split": "train",
"index": 7523,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From a plywood circle with a diameter of 30 cm, two smaller circles with diameters of 20 cm and 10 cm are cut out. What is the diameter of the largest circle that can be cut from the remaining piece of plywood? Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 7524,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $a, b$, and $c$ are complex numbers satisfying $$\\begin{aligned} a^{2} & =b-c \\\\ b^{2} & =c-a, \\text { and } \\\\ c^{2} & =a-b \\end{aligned}$$ Compute all possible values of $a+b+c$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0, \\pm i \\sqrt{6}"
} | {
"split": "train",
"index": 7525,
"solution": "Summing the equations gives $a^{2}+b^{2}+c^{2}=0$ and summing $a$ times the first equation and etc. gives $a^{3}+b^{3}+c^{3}=0$. Let $a+b+c=k$. Then $a^{2}+b^{2}+c^{2}=0$ means $a b+b c+c a=k^{2} / 2$, and $a^{3}+b^{3}+c^{3}=0 \\Longrightarrow-3 a b c=a^{3}+b^{3}+c^... |
deepscaler | [
{
"role": "user",
"content": "In a rectangular parallelepiped $\\mathrm{ABCDA}_{1} \\mathrm{~B}_{1} \\mathrm{C}_{1} \\mathrm{D}_{1}$, with $\\mathrm{AB}=1$ cm, $\\mathrm{AD}=2$, and $\\mathrm{AA}_{1}=1$, find the minimum area of triangle $\\mathrm{PA}_{1} \\mathrm{C}$, where vertex $\\mathrm{P}$ lies on the... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 7526,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the equation \\(2 x^{3} + 24 x = 3 - 12 x^{2}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[3]{\\frac{19}{2}} - 2"
} | {
"split": "train",
"index": 7527,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest possible value of $k$ for which $3^{11}$ is expressible as the sum of $k$ consecutive positive integers.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "486"
} | {
"split": "train",
"index": 7528,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the circle $x^{2}-2x+y^{2}-2y+1=0$, find the cosine value of the angle between the two tangents drawn from the point $P(3,2)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 7529,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, \\(\\angle AFC = 90^\\circ\\), \\(D\\) is on \\(AC\\), \\(\\angle EDC = 90^\\circ\\), \\(CF = 21\\), \\(AF = 20\\), and \\(ED = 6\\). Determine the total area of quadrilateral \\(AFCE\\). Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "297"
} | {
"split": "train",
"index": 7530,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive odd integer $n$ such that the product $2^{1/7}2^{3/7}\\cdots2^{(2n+1)/7}$ is greater than $1000$? \n(In the product the denominators of the exponents are all sevens, and the numerators are the successive odd integers from $1$ to $2n+1$.) Let's thin... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 7531,
"solution": "1. **Expression Simplification**:\n The product given is $2^{1/7}2^{3/7}\\cdots2^{(2n+1)/7}$. We can combine the terms in the product to get a single exponent by adding all the exponents:\n \\[\n 2^{\\frac{1}{7} + \\frac{3}{7} + \\frac{5}{7} + \\cdots + \\frac... |
deepscaler | [
{
"role": "user",
"content": "Right triangle $ABC$ has one leg of length 9 cm, another leg of length 12 cm, and a right angle at $A$. A square has one side on the hypotenuse of triangle $ABC$ and a vertex on each of the two legs of triangle $ABC$. What is the length of one side of the square, in cm? Express... | math | {
"style": "rule",
"ground_truth": "\\frac{180}{37}"
} | {
"split": "train",
"index": 7532,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A subset of a student group is called an [i]ideal company[/i] if\n1) in this subset, all girls are liked by all young men,\n2) no one can be added to this subset without violating condition $1$.\nIn a certain group, $9$ female students and $15$ students study. Warden of the gro... | math | {
"style": "rule",
"ground_truth": "512"
} | {
"split": "train",
"index": 7533,
"solution": "\nTo solve this problem, we need to understand the concept of an \"ideal company\" as defined by the question. An ideal company is a subset of the student group where all female students in the subset are liked by all male students within the same subset, and no add... |
deepscaler | [
{
"role": "user",
"content": "Five women of different heights are standing in a line at a social gathering. Each woman decides to only shake hands with women taller than herself. How many handshakes take place? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 7534,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A charity sells $140$ benefit tickets for a total of $2001$. Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets? Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "$782"
} | {
"split": "train",
"index": 7535,
"solution": "\nLet's denote the number of full-price tickets as $f$ and the number of half-price tickets as $h$. Let $p$ be the price of a full-price ticket. Then, the price of a half-price ticket is $\\frac{p}{2}$. The total number of tickets sold is $140$, and the total revenu... |
deepscaler | [
{
"role": "user",
"content": "How many ordered pairs of integers $(x, y)$ satisfy the equation $x^{2020} + y^2 = 2y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 7536,
"solution": "\nWe start by analyzing the given equation:\n\\[ x^{2020} + y^2 = 2y. \\]\n\n#### Step 1: Rearrange the equation\nWe can rearrange the equation to isolate terms involving \\( y \\):\n\\[ y^2 - 2y + x^{2020} = 0. \\]\n\n#### Step 2: Complete the square\nCompleting th... |
deepscaler | [
{
"role": "user",
"content": "The volume of a regular triangular prism is $8$, the base edge length that minimizes the surface area of the prism is __________. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt[3]{4}"
} | {
"split": "train",
"index": 7537,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A large rectangle measures 15 units by 20 units. One-quarter of this rectangle is shaded. If half of this quarter rectangle is shaded, what fraction of the large rectangle is shaded?\nA) $\\frac{1}{24}$\nB) $\\frac{1}{12}$\nC) $\\frac{1}{10}$\nD) $\\frac{1}{8}$\nE) $\\frac{1}{6... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 7538,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For what digit $d$ is the five-digit number $2345d$ a multiple of 9? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 7539,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Marisela is putting on a juggling show! She starts with $1$ ball, tossing it once per second. Lawrence tosses her another ball every five seconds, and she always tosses each ball that she has once per second. Compute the total number of tosses Marisela has made one minute aft... | math | {
"style": "rule",
"ground_truth": "390"
} | {
"split": "train",
"index": 7540,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a board game, I move on a linear track. For move 1, I stay still. For subsequent moves $n$ where $2 \\le n \\le 30$, I move forward two steps if $n$ is prime and three steps backward if $n$ is composite. How many steps in total will I need to make to return to my original st... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 7541,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N=\\overline{5 A B 37 C 2}$, where $A, B, C$ are digits between 0 and 9, inclusive, and $N$ is a 7-digit positive integer. If $N$ is divisible by 792, determine all possible ordered triples $(A, B, C)$. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "(0,5,5),(4,5,1),(6,4,9)"
} | {
"split": "train",
"index": 7542,
"solution": "First, note that $792=2^{3} \\times 3^{2} \\times 11$. So we get that $$\\begin{gathered} 8|N \\Rightarrow 8| \\overline{7 C 2} \\Rightarrow 8 \\mid 10 C+6 \\Rightarrow C=1,5,9 \\\\ 9|N \\Rightarrow 9| 5+A+B+3+7+C+2 \\Rightarrow A+B+C=1,10,19 \\\\ 11|N \\Rightarrow ... |
deepscaler | [
{
"role": "user",
"content": "Let \\(S\\) be the set of all nonzero real numbers. Let \\(f : S \\to S\\) be a function such that\n\\[f(x) + f(y) = cf(xyf(x + y))\\]\nfor all \\(x, y \\in S\\) such that \\(x + y \\neq 0\\) and for some nonzero constant \\(c\\). Determine all possible functions \\(f\\) that s... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"split": "train",
"index": 7543,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a $10 \\times 5$ grid, an ant starts from point $A$ and can only move right or up along the grid lines but is not allowed to pass through point $C$. How many different paths are there from point $A$ to point $B$? Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "1827"
} | {
"split": "train",
"index": 7544,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many two-digit numbers have digits whose sum is either a perfect square or a prime number up to 25? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 7545,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A semicircle with radius 2021 has diameter $AB$ and center $O$. Points $C$ and $D$ lie on the semicircle such that $\\angle AOC < \\angle AOD = 90^{\\circ}$. A circle of radius $r$ is inscribed in the sector bounded by $OA$ and $OC$ and is tangent to the semicircle at $E$. If $... | math | {
"style": "rule",
"ground_truth": "673"
} | {
"split": "train",
"index": 7546,
"solution": "We are given $$m \\angle EOC = m \\angle COD$$ and $$m \\angle AOC + m \\angle COD = 2m \\angle EOC + m \\angle COD = 90^{\\circ}$$ So $m \\angle EOC = 30^{\\circ}$ and $m \\angle AOC = 60^{\\circ}$. Letting the radius of the semicircle be $R$, we have $$(R-r) \\sin... |
deepscaler | [
{
"role": "user",
"content": "A triangular region is enclosed by the lines with equations $y = \\frac{1}{2} x + 3$, $y = -2x + 6$ and $y = 1$. What is the area of the triangular region? Express your answer as a decimal to the nearest hundredth. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "8.45"
} | {
"split": "train",
"index": 7547,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a list where each integer $n$ from 1 to 300 appears exactly $n$ times. What is the median of this number list? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "212"
} | {
"split": "train",
"index": 7548,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Dragoons take up \\(1 \\times 1\\) squares in the plane with sides parallel to the coordinate axes such that the interiors of the squares do not intersect. A dragoon can fire at another dragoon if the difference in the \\(x\\)-coordinates of their centers and the difference in ... | math | {
"style": "rule",
"ground_truth": "168"
} | {
"split": "train",
"index": 7549,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the triangle \\(A B C\\), angle \\(C\\) is a right angle, and \\(AC: AB = 3: 5\\). A circle with its center on the extension of leg \\(AC\\) beyond point \\(C\\) is tangent to the extension of hypotenuse \\(AB\\) beyond point \\(B\\) and intersects leg \\(BC\\) at point \\(P... | math | {
"style": "rule",
"ground_truth": "37/15"
} | {
"split": "train",
"index": 7550,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a sequence recursively by $t_1 = 20$, $t_2 = 21$, and\\[t_n = \\frac{5t_{n-1}+1}{25t_{n-2}}\\]for all $n \\ge 3$. Then $t_{2020}$ can be expressed as $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "626"
} | {
"split": "train",
"index": 7551,
"solution": "Let $t_n=\\frac{s_n}{5}$. Then, we have $s_n=\\frac{s_{n-1}+1}{s_{n-2}}$ where $s_1 = 100$ and $s_2 = 105$. By substitution, we find $s_3 = \\frac{53}{50}$, $s_4=\\frac{103}{105\\cdot50}$, $s_5=\\frac{101}{105}$, $s_6=100$, and $s_7=105$. So $s_n$ has a period of $5... |
deepscaler | [
{
"role": "user",
"content": "If 52 cards are dealt to 8 people as evenly as possible, how many people will end up with fewer than 7 cards? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 7552,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the greatest root of the equation $g(x) = 21x^4 - 20x^2 + 3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{21}}{7}"
} | {
"split": "train",
"index": 7553,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For what is the smallest $n$ such that there exist $n$ numbers within the interval $(-1, 1)$ whose sum is 0 and the sum of their squares is 42? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 7554,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cube with a side length of 1 meter was cut into smaller cubes with a side length of 1 centimeter and arranged in a straight line. What is the length of the resulting line? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10000"
} | {
"split": "train",
"index": 7555,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two plane vectors, the angle between them is $120^\\circ$, and $a=1$, $|b|=2$. If the plane vector $m$ satisfies $m\\cdot a=m\\cdot b=1$, then $|m|=$ ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{ \\sqrt{21}}{3}"
} | {
"split": "train",
"index": 7556,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $\\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}}$? 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": 7557,
"solution": "To solve the problem, we start by simplifying the expression \\[\\frac{2^{2014}+2^{2012}}{2^{2014}-2^{2012}}.\\]\n\n1. **Factor out common terms**:\n \\[\n \\frac{2^{2014} + 2^{2012}}{2^{2014} - 2^{2012}} = \\frac{2^{2012}(2^2 + 1)}{2^{2012}(2^2 - 1)}\n \\]\n ... |
deepscaler | [
{
"role": "user",
"content": "Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP > CP$. Let $O_{1}$ and $O_{2}$ be the circumcenters of triangles $ABP$ and $CDP$ respectively. Given that $AB = 12$ and $\\angle O_{1}PO_{2} = 120^{\\circ}$, then $AP = \\sqrt{a} + \\sqrt{b}$, where $a$ and $b$ are p... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 7558,
"solution": "Both $O_1$ and $O_2$ lie on the perpendicular bisector of $AB$.\nClaim: $O_1O_2=12$ and $O_1P=O_2P$.\nProof. Translate $O_1$ and $P$ $12$ units down, and let their images be $O_1'$ and $P'$, respectively. Note that $\\triangle ABP\\cong\\triangle DCP'$. Additionally... |
deepscaler | [
{
"role": "user",
"content": "Compute the prime factorization of \\(1007021035035021007001\\). (You should write your answer in the form \\(p_{1}^{e_{1}} p_{2}^{e_{2}} \\ldots p_{k}^{e_{k}}\\), where \\(p_{1}, \\ldots, p_{k}\\) are distinct prime numbers and \\(e_{1}, \\ldots, e_{k}\\) are positive integers... | math | {
"style": "rule",
"ground_truth": "7^7 * 11^7 * 13^7"
} | {
"split": "train",
"index": 7559,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three triangles. Inside triangle $ABC$, a random point $M$ is chosen. What is the probability that the area of one of the triangles $ABM$, $BCM$, or $CAM$ will be greater than the sum of the areas of the other two? Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "0.75"
} | {
"split": "train",
"index": 7560,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ that satisfies $3a_{n+1}+a_n=0$ and $a_2=-\\frac{4}{3}$, find the sum of the first $10$ terms of $\\{a_n\\}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3(1-3^{-10})"
} | {
"split": "train",
"index": 7561,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A merchant acquires goods at a discount of $30\\%$ of the list price and intends to sell them with a $25\\%$ profit margin after a $25\\%$ discount on the marked price. Determine the required percentage of the original list price that the goods should be marked. Let's think ste... | math | {
"style": "rule",
"ground_truth": "124\\%"
} | {
"split": "train",
"index": 7562,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Divide an $m$-by-$n$ rectangle into $m n$ nonoverlapping 1-by-1 squares. A polyomino of this rectangle is a subset of these unit squares such that for any two unit squares $S, T$ in the polyomino, either (1) $S$ and $T$ share an edge or (2) there exists a positive integer $n$ s... | math | {
"style": "rule",
"ground_truth": "470"
} | {
"split": "train",
"index": 7563,
"solution": "To span an $a \\times b$ rectangle, we need at least $a+b-1$ squares. Indeed, consider a square of the polyomino bordering the left edge of the rectangle and one bordering the right edge. There exists a path connecting these squares; suppose it runs through $c$ diff... |
deepscaler | [
{
"role": "user",
"content": "In the new clubroom, there were only chairs and a table. Each chair had four legs, and the table had three legs. Scouts came into the clubroom. Each sat on their own chair, two chairs remained unoccupied, and the total number of legs in the room was 101.\n\nDetermine how many c... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 7564,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle with a circumscribed and an inscribed square centered at the origin of a rectangular coordinate system with positive $x$ and $y$ axes is shown in each figure I to IV below.\n\nThe inequalities\n\\(|x|+|y| \\leq \\sqrt{2(x^{2}+y^{2})} \\leq 2\\mbox{Max}(|x|, |y|)\\)\nar... | math | {
"style": "rule",
"ground_truth": "II"
} | {
"split": "train",
"index": 7565,
"solution": "To solve this problem, we need to analyze the given inequalities and understand their geometric representations. We will consider each inequality separately and then find the figure that correctly represents all three inequalities simultaneously.\n\n1. **Understandi... |
deepscaler | [
{
"role": "user",
"content": "\nThere is a \\(4 \\times 4\\) square. Its cells are called neighboring if they share a common side. All cells are painted in two colors: red and blue. It turns out that each red cell has more red neighbors than blue ones, and each blue cell has an equal number of red and blue ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 7566,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a set of circles in the upper half-plane, all tangent to the $x$-axis. Begin with two circles of radii $50^2$ and $53^2$, externally tangent to each other, defined as Layer $L_0$. For each pair of consecutive circles in $\\bigcup_{j=0}^{k-1}L_j$, a new circle in Layer ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{1517}"
} | {
"split": "train",
"index": 7567,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence is defined by $a_0 = \\frac{1}{2}$ and $a_n = 1 + (a_{n - 1} - 1)^2.$ Compute\n\\[a_0 a_1 a_2 \\dotsm.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 7568,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Arrange 3 boys and 4 girls in a row. Calculate the number of different arrangements under the following conditions:\n(1) Person A and Person B must stand at the two ends; \n(2) All boys must stand together; \n(3) No two boys stand next to each other; \n(4) Exactly one person... | math | {
"style": "rule",
"ground_truth": "1200"
} | {
"split": "train",
"index": 7569,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a parallelepiped $A B C D A_1 B_1 C_1 D_1$, point $X$ is selected on the edge $A_1 D_1$, and point $Y$ is selected on the edge $B C$. It is known that $A_1 X = 5$, $B Y = 3$, and $B_1 C_1 = 14$. The plane $C_1 X Y$ intersects the ray $D A$ at point $Z$. Find $D Z$. Let's ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 7570,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $π < α < 2π$, $\\cos (α-9π)=- \\dfrac {3}{5}$, find the value of $\\cos (α- \\dfrac {11π}{2})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{4}{5}"
} | {
"split": "train",
"index": 7571,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A natural number of five digits is called *Ecuadorian*if it satisfies the following conditions: $\\bullet$ All its digits are different. $\\bullet$ The digit on the far left is equal to the sum of the other four digits. Example: $91350$ is an Ecuadorian number since $9 = ... | math | {
"style": "rule",
"ground_truth": "168"
} | {
"split": "train",
"index": 7572,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $6^6 \\cdot 3^6$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "34012224"
} | {
"split": "train",
"index": 7573,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rahul has ten cards face-down, which consist of five distinct pairs of matching cards. During each move of his game, Rahul chooses one card to turn face-up, looks at it, and then chooses another to turn face-up and looks at it. If the two face-up cards match, the game ends. If ... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 7574,
"solution": "Label the 10 cards $a_{1}, a_{2}, \\ldots, a_{5}, b_{1}, b_{2}, \\ldots, b_{5}$ such that $a_{i}$ and $b_{i}$ match for $1 \\leq i \\leq 5$. First, we'll show that Rahul cannot always end the game in less than 4 moves, in particular, when he turns up his fifth card ... |
deepscaler | [
{
"role": "user",
"content": "The bottoms of two vertical poles are 12 feet apart and are on a region of flat ground. One pole is 6 feet tall and the other is 15 feet tall. How long, in feet, is a wire stretched from the top of one pole to the top of the other pole? Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 7575,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the matrix $\\mathbf{M}$ such that\n\\[\\mathbf{M} \\mathbf{v} = -5 \\mathbf{v}\\]for all vectors $\\mathbf{v}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -5 & 0 \\\\ 0 & -5 \\end{pmatrix}"
} | {
"split": "train",
"index": 7576,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Piravena must make a trip from $A$ to $B$, then from $B$ to $C$, then from $C$ to $A$. Each of these three parts of the trip is made entirely by bus or entirely by airplane. The cities form a right-angled triangle as shown, with $C$ a distance of 3000 km from $A$ and with $B$... | math | {
"style": "rule",
"ground_truth": "\\$1012.50"
} | {
"split": "train",
"index": 7577,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given points $P(-2,-3)$ and $Q(5, 3)$ in the $xy$-plane; point $R(2,m)$ is taken so that $PR+RQ$ is minimized. Determine the value of $m$.\nA) $\\frac{3}{5}$\nB) $\\frac{2}{5}$\nC) $\\frac{3}{7}$\nD) $\\frac{1}{5}$ Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{7}"
} | {
"split": "train",
"index": 7578,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The smallest sum one could get by adding three different numbers from the set $\\{ 7,25,-1,12,-3 \\}$ is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 7579,
"solution": "To find the smallest sum possible by adding three different numbers from the set $\\{7, 25, -1, 12, -3\\}$, we need to consider the smallest numbers in the set, as adding smaller numbers will result in a smaller sum.\n\n1. **Identify the smallest numbers in the set*... |
deepscaler | [
{
"role": "user",
"content": "For real numbers $t,$ consider the point of intersection of the triplet of lines $3x - 2y = 8t - 5$, $2x + 3y = 6t + 9$, and $x + y = 2t + 1$. All the plotted points lie on a line. Find the slope of this line. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{6}"
} | {
"split": "train",
"index": 7580,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There exist unique positive integers $x$ and $y$ that satisfy the equation $x^2 + 84x + 2008 = y^2$. Find $x + y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 7581,
"solution": "We see that $y^2 \\equiv x^2 + 4 \\pmod{6}$. By quadratic residues, we find that either $x \\equiv 0, 3 \\pmod{6}$. Also, $y^2 \\equiv (x+42)^2 + 244 \\equiv (x+2)^2 \\pmod{4}$, so $x \\equiv 0, 2 \\mod{4}$. Combining, we see that $x \\equiv 0 \\mod{6}$.\n Testing $... |
deepscaler | [
{
"role": "user",
"content": "Let \\(a\\), \\(b\\), \\(c\\) be distinct complex numbers such that\n\\[\n\\frac{a+1}{2 - b} = \\frac{b+1}{2 - c} = \\frac{c+1}{2 - a} = k.\n\\]\nFind the sum of all possible values of \\(k\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"split": "train",
"index": 7582,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many irreducible fractions with numerator 2015 exist that are less than \\( \\frac{1}{2015} \\) and greater than \\( \\frac{1}{2016} \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1440"
} | {
"split": "train",
"index": 7583,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a pyramid A-PBC, where PA is perpendicular to plane ABC, AB is perpendicular to AC, and BA=CA=2=2PA, calculate the height from the base PBC to the apex A. 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": 7584,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One hundred people were surveyed. Of these, $87$ indicated they liked Mozart and $70$ indicated they liked Bach. What is the minimum number of people surveyed who could have said they liked both Mozart and Bach? Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "57"
} | {
"split": "train",
"index": 7585,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $x > 0, y > 0$, and $2x + 8y - xy = 0$, find the minimum value of $xy$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 7586,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nThe base of pyramid \\( T ABCD \\) is an isosceles trapezoid \\( ABCD \\) with the length of the shorter base \\( BC \\) equal to \\( \\sqrt{3} \\). The ratio of the areas of the parts of the trapezoid \\( ABCD \\), divided by the median line, is \\( 5:7 \\). All the lateral ... | math | {
"style": "rule",
"ground_truth": "0.875"
} | {
"split": "train",
"index": 7587,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integer divisors of $3003^{3003}$ are divisible by exactly 3003 positive integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 7588,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of the expression \\(\\frac{5 x^{2}-8 x y+5 y^{2}-10 x+14 y+55}{\\left(9-25 x^{2}+10 x y-y^{2}\\right)^{5 / 2}}\\). If necessary, round the answer to hundredths. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.19"
} | {
"split": "train",
"index": 7589,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the line $5x + 8y + c = 0$, the sum of the $x$- and $y$-intercepts is $26$. Find $c$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-80"
} | {
"split": "train",
"index": 7590,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the hyperbola $C$: $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 (a > 0, b > 0)$. Its right focus and one endpoint of its conjugate axis are $F$ and $A$, respectively. Let $P$ be a point on the left branch of the hyperbola $C$. If the minimum value of the perimeter of $\\tr... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{85}}{7}"
} | {
"split": "train",
"index": 7591,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ $c$ be integers such that\n\\[\\mathbf{A} = \\frac{1}{5} \\begin{pmatrix} -3 & a \\\\ b & c \\end{pmatrix}\\]and $\\mathbf{A}^2 = \\mathbf{I}.$ Find the largest possible value of $a + b + c.$ Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 7592,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(ABC\\) be a triangle with \\(AB=13, BC=14\\), and \\(CA=15\\). Pick points \\(Q\\) and \\(R\\) on \\(AC\\) and \\(AB\\) such that \\(\\angle CBQ=\\angle BCR=90^{\\circ}\\). There exist two points \\(P_{1} \\neq P_{2}\\) in the plane of \\(ABC\\) such that \\(\\triangle P... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 7593,
"solution": "Let \\(T\\) be the foot of the \\(A\\)-altitude of \\(ABC\\). Recall that \\(BT=5\\) and \\(CT=9\\). Let \\(T'\\) be the foot of the \\(P\\)-altitude of \\(PQR\\). Since \\(T'\\) is the midpoint of the possibilities for \\(P\\), the answer is \\(\\sum_{P} d(P, BC)=2... |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive prime that divides \\( n^2 + 5n + 23 \\) for some integer \\( n \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 7594,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all functions $ f : \\mathbb{R} \\rightarrow \\mathbb{R} $ such that\n\\[ f( xf(x) + 2y) = f(x^2)+f(y)+x+y-1 \\]\nholds for all $ x, y \\in \\mathbb{R}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = x + 1"
} | {
"split": "train",
"index": 7595,
"solution": "\nTo solve the functional equation\n\n\\[\nf(xf(x) + 2y) = f(x^2) + f(y) + x + y - 1\n\\]\n\nfor all real numbers \\( x \\) and \\( y \\), we need to find all possible functions \\( f : \\mathbb{R} \\rightarrow \\mathbb{R} \\) that satisfy this condition.\n\n**Step ... |
deepscaler | [
{
"role": "user",
"content": "(1901 + 1902 + 1903 + \\cdots + 1993) - (101 + 102 + 103 + \\cdots + 193) = Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "167400"
} | {
"split": "train",
"index": 7596,
"solution": "1. **Identify the pattern in the sequences**: \n The problem involves two sums of arithmetic sequences. The first sequence starts at 1901 and ends at 1993, and the second sequence starts at 101 and ends at 193.\n\n2. **Recognize the relationship between the sequen... |
deepscaler | [
{
"role": "user",
"content": "Given that the area of a rectangle is 192 and its length is 24, what is the perimeter of the rectangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 7597,
"solution": "Since the area of the rectangle is 192 and its length is 24, then its width is $192 \\div 24=8$. Therefore, its perimeter is $2 \\times 24 + 2 \\times 8 = 64$."
} |
deepscaler | [
{
"role": "user",
"content": "Seven cubes, whose volumes are $1$, $8$, $27$, $64$, $125$, $216$, and $343$ cubic units, are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top. Except for the bottom cube, the bottom face of each cube lies completely on top of the... | math | {
"style": "rule",
"ground_truth": "749"
} | {
"split": "train",
"index": 7598,
"solution": "1. **Identify the side lengths of the cubes**: The volumes of the cubes are given as $1$, $8$, $27$, $64$, $125$, $216$, and $343$ cubic units. The side lengths of these cubes are the cube roots of these volumes, which are $1$, $2$, $3$, $4$, $5$, $6$, and $7$ units... |
deepscaler | [
{
"role": "user",
"content": "\nHow many kings can be placed on an $8 \\times 8$ chessboard without putting each other in check? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 7599,
"solution": ""
} |
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