data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "One side of a rectangle (the width) was increased by 10%, and the other side (the length) by 20%. \na) Could the perimeter increase by more than 20% in this case?\nb) Find the ratio of the sides of the original rectangle if it is known that the perimeter of the new rectangle is... | math | {
"style": "rule",
"ground_truth": "1:4"
} | {
"split": "train",
"index": 2100,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For real numbers $a$ and $b$ , define $$ f(a,b) = \\sqrt{a^2+b^2+26a+86b+2018}. $$ Find the smallest possible value of the expression $$ f(a, b) + f (a,-b) + f(-a, b) + f (-a, -b). $$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4 \\sqrt{2018}"
} | {
"split": "train",
"index": 2101,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T_1$ be a triangle with side lengths $2011$, $2012$, and $2013$. For $n \\geq 1$, if $T_n = \\Delta ABC$ and $D, E$, and $F$ are the points of tangency of the incircle of $\\Delta ABC$ to the sides $AB$, $BC$, and $AC$, respectively, then $T_{n+1}$ is a triangle with side ... | math | {
"style": "rule",
"ground_truth": "\\frac{1509}{128}"
} | {
"split": "train",
"index": 2102,
"solution": "1. **Identify the side lengths of the initial triangle $T_1$:**\n Given $T_1$ has side lengths $2011$, $2012$, and $2013$.\n\n2. **Define the sequence of triangles $T_n$:**\n For each triangle $T_n = \\Delta ABC$, the incircle touches $AB$, $BC$, and $AC$ at poi... |
deepscaler | [
{
"role": "user",
"content": "Sierpinski's triangle is formed by taking a triangle, and drawing an upside down triangle inside each upright triangle that appears. A snake sees the fractal, but decides that the triangles need circles inside them. Therefore, she draws a circle inscribed in every upside down t... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{12}"
} | {
"split": "train",
"index": 2103,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $27^3 + 9(27^2) + 27(9^2) + 9^3$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "46656"
} | {
"split": "train",
"index": 2104,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the expression $8 \\times 10^{5}+4 \\times 10^{3}+9 \\times 10+5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "804095"
} | {
"split": "train",
"index": 2105,
"solution": "First, we write out the powers of 10 in full to obtain $8 \\times 100000+4 \\times 1000+9 \\times 10+5$. Simplifying, we obtain $800000+4000+90+5$ or 804095."
} |
deepscaler | [
{
"role": "user",
"content": "The product \\( 29 \\cdot 11 \\), and the numbers 1059, 1417, and 2312, are each divided by \\( d \\). If the remainder is always \\( r \\), where \\( d \\) is an integer greater than 1, what is \\( d - r \\) equal to? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 2106,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of positive integers $n \\le 1000$ that can be expressed in the form\n\\[\\lfloor x \\rfloor + \\lfloor 2x \\rfloor + \\lfloor 3x \\rfloor = n\\]for some real number $x.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "667"
} | {
"split": "train",
"index": 2107,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $$2\\sin^{2} \\frac {A+B}{2}+\\cos2C=1$$ \n(1) Find the magnitude of angle $C$; \n(2) If vector $$\\overrightarrow {m}=(3a，b)$$ and vector $$\... | math | {
"style": "rule",
"ground_truth": "\\sqrt {7}"
} | {
"split": "train",
"index": 2108,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 4 spheres in space with radii 2, 2, 3, and 3, respectively. Each sphere is externally tangent to the other 3 spheres. Additionally, there is a small sphere that is externally tangent to all 4 of these spheres. Find the radius of the small sphere. Let's think step by s... | math | {
"style": "rule",
"ground_truth": "6/11"
} | {
"split": "train",
"index": 2109,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rohan wants to cut a piece of string into nine pieces of equal length. He marks his cutting points on the string. Jai wants to cut the same piece of string into only eight pieces of equal length. He marks his cutting points on the string. Yuvraj then cuts the string at all the ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 2110,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the least three digit number that is equal to the sum of its digits plus twice the product of its digits. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "397"
} | {
"split": "train",
"index": 2111,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a, b, c$, and $d$ be positive real numbers such that\n\\[\n\\begin{array}{c@{\\hspace{3pt}}c@{\\hspace{3pt}}c@{\\hspace{3pt}}c@{\\hspace{3pt}}c}\na^2 + b^2 &=& c^2 + d^2 &=& 1458, \\\\\nac &=& bd &=& 1156.\n\\end{array}\n\\]\nIf $S = a + b + c + d$, compute the value of $\... | math | {
"style": "rule",
"ground_truth": "77"
} | {
"split": "train",
"index": 2112,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is an integer $n > 1$. There are $n^2$ stations on a slope of a mountain, all at different altitudes. Each of two cable car companies, $A$ and $B$, operates $k$ cable cars; each cable car provides a transfer from one of the stations to a higher one (with no intermediate s... | math | {
"style": "rule",
"ground_truth": "n^2 - n + 1"
} | {
"split": "train",
"index": 2113,
"solution": "\nTo solve the problem involving cable car companies \\( A \\) and \\( B \\), we must determine the smallest integer \\( k \\) such that there are always two stations linked by both companies.\n\n### Understanding the Problem\n\n1. **Stations and Connections:**\n ... |
deepscaler | [
{
"role": "user",
"content": "Call an ordered triple $(a, b, c)$ of integers feral if $b -a, c - a$ and $c - b$ are all prime. \nFind the number of feral triples where $1 \\le a < b < c \\le 20$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 2114,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain year the price of gasoline rose by $20\\%$ during January, fell by $20\\%$ during February, rose by $25\\%$ during March, and fell by $x\\%$ during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the near... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 2115,
"solution": "1. **Assume the initial price**: Let's assume the initial price of gasoline at the beginning of January is $P_0 = 100$ dollars.\n\n2. **Price after January's increase**: The price increased by 20% in January. Therefore, the new price at the end of January is:\n \\... |
deepscaler | [
{
"role": "user",
"content": "Given $995 + 997 + 999 + 1001 + 1003 = 5100 - N$, determine $N$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 2116,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given four numbers $101010_{(2)}$, $111_{(5)}$, $32_{(8)}$, and $54_{(6)}$, the smallest among them is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32_{(8)}"
} | {
"split": "train",
"index": 2117,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequences where each term is a real number are denoted as $\\left\\{a_{n}\\right\\}$, with the sum of the first $n$ terms recorded as $S_{n}$. Given that $S_{10} = 10$ and $S_{30} = 70$, what is the value of $S_{40}$? Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 2118,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $1234 + 2341 + 3412 + 4123$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11110"
} | {
"split": "train",
"index": 2119,
"solution": "To solve the problem, we need to add the numbers $1234$, $2341$, $3412$, and $4123$. We can do this by aligning the numbers vertically and adding them column by column:\n\n\\[\n\\begin{array}{c}\n\\phantom{+}1234 \\\\\n+2341 \\\\\n+3412 \\\\\n+4123 \\\\\n\\hline\n\\... |
deepscaler | [
{
"role": "user",
"content": "In a group of nine people each person shakes hands with exactly two of the other people from the group. Let $N$ be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrange... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 2120,
"solution": "Given that each person shakes hands with two people, we can view all of these through graph theory as 'rings'. This will split it into four cases: Three rings of three, one ring of three and one ring of six, one ring of four and one ring of five, and one ring of nin... |
deepscaler | [
{
"role": "user",
"content": "Given the ratio of the legs of a right triangle is $3: 4$, determine the ratio of the corresponding segments of the hypotenuse created by dropping a perpendicular from the opposite vertex of the right angle onto the hypotenuse. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{9}"
} | {
"split": "train",
"index": 2121,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a sequence $\\{a_n\\}$ whose sum of the first $n$ terms $S_n = n^2 - 4n + 2$. Find the sum of the absolute values of the first ten terms: $|a_1| + |a_2| + \\cdots + |a_{10}|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "68"
} | {
"split": "train",
"index": 2122,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $l$ some line, that is not parallel to the coordinate axes. Find minimal $d$ that always exists point $A$ with integer coordinates, and distance from $A$ to $l$ is $\\leq d$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2\\sqrt{2}}"
} | {
"split": "train",
"index": 2123,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A$ and $B$ be points in space for which $A B=1$. Let $\\mathcal{R}$ be the region of points $P$ for which $A P \\leq 1$ and $B P \\leq 1$. Compute the largest possible side length of a cube contained within $\\mathcal{R}$. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}-1}{3}"
} | {
"split": "train",
"index": 2124,
"solution": "Let $h$ be the distance between the center of one sphere and the center of the opposite face of the cube. Let $x$ be the side length of the cube. Then we can draw a right triangle by connecting the center of the sphere, the center of the opposite face of the cube, a... |
deepscaler | [
{
"role": "user",
"content": "A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations? Let's think step b... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 2125,
"solution": "1. **Identify the fraction of pink and red flowers**: \n Given that six tenths (or $\\frac{6}{10}$) of the flowers are pink, we can simplify this fraction to $\\frac{3}{5}$. Consequently, the remaining flowers must be red, which is $\\frac{2}{5}$ of the total flow... |
deepscaler | [
{
"role": "user",
"content": "$a$,$b$,$c$ are the opposite sides of angles $A$,$B$,$C$ in $\\triangle ABC$. It is known that $\\sqrt{5}a\\sin B=b$. \n$(1)$ Find $\\sin A$; \n$(2)$ If $A$ is an obtuse angle, and $b=\\sqrt{5}$, $c=3$, find the perimeter of $\\triangle ABC$. Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{26} + \\sqrt{5} + 3"
} | {
"split": "train",
"index": 2126,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a round-robin chess tournament with $x$ players, two players dropped out after playing three matches each. The tournament ended with a total of 84 matches played. How many players were there initially? Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 2127,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of $r = \\cos \\theta$ is a circle. Find the smallest value of $t$ so that when $r = \\cos \\theta$ is plotted for $0 \\le \\theta \\le t,$ the resulting graph is the entire circle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\pi"
} | {
"split": "train",
"index": 2128,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to place four points in the plane such that the set of pairwise distances between the points consists of exactly 2 elements? (Two configurations are the same if one can be obtained from the other via rotation and scaling.) Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 2129,
"solution": "Let $A, B, C, D$ be the four points. There are 6 pairwise distances, so at least three of them must be equal. Case 1: There is no equilateral triangle. Then WLOG we have $A B=B C=C D=1$. - Subcase 1.1: $A D=1$ as well. Then $A C=B D \\neq 1$, so $A B C D$ is a squar... |
deepscaler | [
{
"role": "user",
"content": "How many points does one have to place on a unit square to guarantee that two of them are strictly less than 1/2 unit apart? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 2130,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the set $A = \\{x | x < 1 \\text{ or } x > 5\\}$, and $B = \\{x | a \\leq x \\leq b\\}$, and $A \\cup B = \\mathbb{R}$, $A \\cap B = \\{x | 5 < x \\leq 6\\}$, find the value of $2a - b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 2131,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a shooting competition, there are 8 clay targets arranged in 3 columns as shown in the diagram. A sharpshooter follows these rules to shoot down all the targets:\n1. First, select a column from which one target will be shot.\n2. Then, target the lowest remaining target in th... | math | {
"style": "rule",
"ground_truth": "560"
} | {
"split": "train",
"index": 2132,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Three people, A, B, and C, are applying to universities A, B, and C, respectively, where each person can only apply to one university, calculate the conditional probability $P\\left(A|B\\right)$. Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 2133,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The polynomial $ax^4 + bx^3 + 32x^2 - 16x + 6$ has a factor of $3x^2 - 2x + 1.$ Find the ordered pair $(a,b).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(18,-24)"
} | {
"split": "train",
"index": 2134,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The harmonic mean of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{12}{7}"
} | {
"split": "train",
"index": 2135,
"solution": "1. **Calculate the reciprocals of the numbers**: \n Given numbers are 1, 2, and 4. Their reciprocals are:\n \\[\n \\frac{1}{1}, \\frac{1}{2}, \\text{ and } \\frac{1}{4}\n \\]\n\n2. **Sum the reciprocals**:\n \\[\n \\frac{1}{1} + \\frac{1}{2} + \\frac{1}{... |
deepscaler | [
{
"role": "user",
"content": "Someone claims that the first five decimal digits of the square root of 51 are the same as the first five significant digits of the square root of 2. Verify this claim. Which rational approximating fraction can we derive from this observation for \\(\\sqrt{2}\\)? How many signi... | math | {
"style": "rule",
"ground_truth": "\\frac{99}{70}"
} | {
"split": "train",
"index": 2136,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square and four circles, each with a radius of 8 inches, are arranged as in the previous problem. What is the area, in square inches, of the square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1024"
} | {
"split": "train",
"index": 2137,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Inside a square with side length 12, two congruent equilateral triangles are drawn such that each has one vertex touching two adjacent vertices of the square and they share one side. What is the side length of the largest square that can be inscribed in the space inside the squ... | math | {
"style": "rule",
"ground_truth": "12 - 4\\sqrt{3}"
} | {
"split": "train",
"index": 2138,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits ($\\spadesuit$, $\\heartsuit$, $\\diamondsuit$, and $\\clubsuit$), such that there is exactly one card for any given rank and suit. Two of the suits ($\\spadesuit$ and $\\... | math | {
"style": "rule",
"ground_truth": "\\dfrac{11}{850}"
} | {
"split": "train",
"index": 2139,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be the number of ways in which the letters in \"HMMTHMMTHMMTHMMTHMMTHMMT\" (\"HMMT\" repeated six times) can be rearranged so that each letter is adjacent to another copy of the same letter. For example, \"MMMMMMTTTTTTHHHHHHHHHHHH\" satisfies this property, but \"HMMMMM... | math | {
"style": "rule",
"ground_truth": "78556"
} | {
"split": "train",
"index": 2140,
"solution": "We first count the number of arrangements for which each block of consecutive identical letters has even size. Pair up the letters into 3 pairs of $H, 6$ pairs of $M$, and 3 pairs of $T$, then rearrange the pairs. There are $\\frac{12!}{6!3!3!}=18480$ ways to do thi... |
deepscaler | [
{
"role": "user",
"content": "Consider the polynomial $49x^3 - 105x^2 + 63x - 10 = 0$ whose roots are in arithmetic progression. Determine the difference between the largest and smallest roots.\nA) $\\frac{2}{7}$\nB) $\\frac{1}{7}$\nC) $\\frac{3\\sqrt{11}}{7}$\nD) $\\frac{2\\sqrt{11}}{7}$\nE) $\\frac{4\\sqr... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{11}}{7}"
} | {
"split": "train",
"index": 2141,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence \\( x_{n} \\), satisfying \\( (n+1) x_{n+1}=x_{n}+n \\), and \\( x_{1}=2 \\), find \\( x_{2009} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2009! + 1}{2009!}"
} | {
"split": "train",
"index": 2142,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The divisors of a natural number \\( n \\) (including \\( n \\) and 1) which has more than three divisors, are written in ascending order: \\( 1 = d_{1} < d_{2} < \\ldots < d_{k} = n \\). The differences \\( u_{1} = d_{2} - d_{1}, u_{2} = d_{3} - d_{2}, \\ldots, u_{k-1} = d_{k}... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 2143,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p$ be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of $5$ heads before one encounters a run of $2$ tails. Given that $p$ can be written in the form $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 2144,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 1000 lamps and 1000 switches, each switch simultaneously controls all lamps whose number is a multiple of the switch's number. Initially, all lamps are on. Now, if the switches numbered 2, 3, and 5 are flipped, how many lamps remain on? Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "499"
} | {
"split": "train",
"index": 2145,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are exactly 120 ways to color five cells in a $5 \\times 5$ grid such that exactly one cell in each row and each column is colored.\n\nThere are exactly 96 ways to color five cells in a $5 \\times 5$ grid without the corner cell, such that exactly one cell in each row and... | math | {
"style": "rule",
"ground_truth": "78"
} | {
"split": "train",
"index": 2146,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $N$ be the number of ordered pairs of nonempty sets $\\mathcal{A}$ and $\\mathcal{B}$ that have the following properties:\n$\\mathcal{A} \\cup \\mathcal{B} = \\{1,2,3,4,5,6,7,8,9,10,11,12\\}$,\n$\\mathcal{A} \\cap \\mathcal{B} = \\emptyset$,\nThe number of elements of $\\ma... | math | {
"style": "rule",
"ground_truth": "772"
} | {
"split": "train",
"index": 2147,
"solution": "Regardless of the size $n$ of $A$ (ignoring the case when $n = 6$), $n$ must not be in $A$ and $12 - n$ must be in $A$.\n There are $10$ remaining elements whose placements have yet to be determined. Note that the actual value of $n$ does not matter; there is always... |
deepscaler | [
{
"role": "user",
"content": "Side $\\overline{A B}$ of $\\triangle A B C$ is the diameter of a semicircle, as shown below. If $A B=3+\\sqrt{3}, B C=3 \\sqrt{2}$, and $A C=2 \\sqrt{3}$, then the area of the shaded region can be written as $\\frac{a+(b+c \\sqrt{d}) \\pi}{e}$, where $a, b, c, d, e$ are intege... | math | {
"style": "rule",
"ground_truth": "147938"
} | {
"split": "train",
"index": 2148,
"solution": "Drop an altitude to point $D$ on $\\overline{A B}$ from $C$ and let $x=A D$. Solving for $x$, we find $$\\begin{aligned} 12-x^{2}=18-(3+\\sqrt{3}-x)^{2} & \\Rightarrow 12=18-9-6 \\sqrt{3}-3+2(3+\\sqrt{3}) x-x^{2} \\\\ & \\Rightarrow 6+6 \\sqrt{3}=(6+2 \\sqrt{3}) x \... |
deepscaler | [
{
"role": "user",
"content": "Calculate the area of the smallest square that can completely contain a circle with a radius of 7 units. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "196"
} | {
"split": "train",
"index": 2149,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=a\\sin x - \\sqrt{3}\\cos x$, one of its graphs has an axis of symmetry at $x=-\\frac{\\pi}{6}$, and $f(x_1) - f(x_2) = -4$, calculate the minimum value of $|x_1+x_2|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"split": "train",
"index": 2150,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The centers of the three circles A, B, and C are collinear with the center of circle B lying between the centers of circles A and C. Circles A and C are both externally tangent to circle B, and the three circles share a common tangent line. Given that circle A has radius $12$ ... | math | {
"style": "rule",
"ground_truth": "147"
} | {
"split": "train",
"index": 2151,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The world is currently undergoing a major transformation that has not been seen in a century. China is facing new challenges. In order to enhance students' patriotism and cohesion, a certain high school organized a knowledge competition on \"China's national conditions and the ... | math | {
"style": "rule",
"ground_truth": "\\frac{23}{3}"
} | {
"split": "train",
"index": 2152,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a geometric sequence $a$, $a(a-1)$, $a(a-1)^{2}$, $...$, where $S_{n}$ represents the sum of its first $n$ terms.\n1) Find the range of real values for $a$ and the expression for $S_{n}$;\n2) Does there exist a real value of $a$ such that $S_{1}$, $S_{3}$, $S_{2}$ form an... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 2153,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, four circles of radius 4 units intersect at the origin. What is the number of square units in the area of the shaded region? Express your answer in terms of $\\pi$. [asy]\nimport olympiad; import geometry; size(100); defaultpen(linewidth(0.8));\nfill(Arc((1,0),1... | math | {
"style": "rule",
"ground_truth": "32\\pi-64"
} | {
"split": "train",
"index": 2154,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the function $f(x) = \\sqrt{3}\\sin\\omega x - 2\\sin^2\\left(\\frac{\\omega x}{2}\\right)$ ($\\omega > 0$) has a minimum positive period of $3\\pi$,\n(I) Find the maximum and minimum values of the function $f(x)$ on the interval $[-\\pi, \\frac{3\\pi}{4}]$;\n(II) In... | math | {
"style": "rule",
"ground_truth": "\\frac{12 + 5\\sqrt{3}}{26}"
} | {
"split": "train",
"index": 2155,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)$, for any $x \\in \\mathbb{R}$, it satisfies $f(x+6) + f(x) = 0$, and the graph of $y=f(x-1)$ is symmetric about the point $(1,0)$. If $f(2) = 4$, find the value of $f(2014)$. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 2156,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A city adopts a lottery system for \"price-limited housing,\" where winning families can randomly draw a house number from the available housing in a designated community. It is known that two friendly families, Family A and Family B, have both won the lottery and decided to go... | math | {
"style": "rule",
"ground_truth": "\\dfrac{3}{5}"
} | {
"split": "train",
"index": 2157,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a set of points \\( M \\) on a plane and seven different circles \\( C_{1}, C_{2}, \\cdots, C_{7} \\). Circle \\( C_{7} \\) passes through exactly 7 points in \\( M \\), circle \\( C_{6} \\) passes through exactly 6 points in \\( M \\), and so on, with circle \\( C_{1}... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 2158,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of all trees planted at a five-foot distance from each other on a rectangular plot of land, the sides of which are 120 feet and 70 feet. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "375"
} | {
"split": "train",
"index": 2159,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the median from vertex $A$ is perpendicular to the median from vertex $B$. The lengths of sides $AC$ and $BC$ are 6 and 7 respectively. What is the length of side $AB$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{17}"
} | {
"split": "train",
"index": 2160,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The region shown is bounded by the arcs of circles having radius 4 units, having a central angle measure of 60 degrees and intersecting at points of tangency. The area of the region can be expressed in the form $a\\sqrt{b}+c\\pi$ square units, where $\\sqrt{b}$ is a radical in ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 2161,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many cubic feet are in three cubic yards? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 2162,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Count how many 8-digit numbers there are that contain exactly four nines as digits. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "433755"
} | {
"split": "train",
"index": 2163,
"solution": "There are $\\binom{8}{4} \\cdot 9^{4}$ sequences of 8 numbers with exactly four nines. A sequence of digits of length 8 is not an 8-digit number, however, if and only if the first digit is zero. There are $\\binom{7}{4} 9^{3}$ 8-digit sequences that are not 8-digit ... |
deepscaler | [
{
"role": "user",
"content": "\nIn a rectangular parallelepiped $ABCDEFGH$, the edge lengths are given as $AB = 30$, $AD = 32$, and $AA_1 = 20$. Point $E$ is marked at the midpoint of edge $A_1B_1$, and point $F$ is marked at the midpoint of edge $B_1C_1$. Find the distance between the lines $AE$ and $BF$. ... | math | {
"style": "rule",
"ground_truth": "19.2"
} | {
"split": "train",
"index": 2164,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle having radius $r_{1}$ centered at point $N$ is tangent to a circle of radius $r_{2}$ centered at $M$. Let $l$ and $j$ be the two common external tangent lines to the two circles. A circle centered at $P$ with radius $r_{2}$ is externally tangent to circle $N$ at the po... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 2165,
"solution": "Suppose the lines are parallel. Draw the other tangent line to $N$ and $P$ - since $M$ and $P$ have the same radius, it is tangent to all three circles. Let $j$ and $k$ meet circle $N$ at $A$ and $B$, respectively. Then by symmetry we see that $\\angle A N M=\\angle... |
deepscaler | [
{
"role": "user",
"content": "Ninety-four bricks, each measuring $4''\\times10''\\times19'',$ are to be stacked one on top of another to form a tower 94 bricks tall. Each brick can be oriented so it contributes $4''\\,$ or $10''\\,$ or $19''\\,$ to the total height of the tower. How many different tower hei... | math | {
"style": "rule",
"ground_truth": "465"
} | {
"split": "train",
"index": 2166,
"solution": "Using bricks of dimensions $4''\\times10''\\times19''$ is comparable to using bricks of dimensions $0''\\times6''\\times15''$ which is comparable to using bricks of dimensions $0''\\times2''\\times5''$. Using 5 bricks of height $2''$ can be replaced by using 2 brick... |
deepscaler | [
{
"role": "user",
"content": "Given that $f(n) = \\left\\{\\begin{matrix}\\log_{4}{n}, &\\text{if }\\log_{4}{n}\\text{ is rational,}\\\\ 0, &\\text{otherwise,}\\end{matrix}\\right.$, evaluate the sum $\\sum_{n = 1}^{1023}{f(n)}$. Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "22.5"
} | {
"split": "train",
"index": 2167,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a balance scale, $3$ green balls balance $6$ blue balls, $2$ yellow balls balance $5$ blue balls, and $6$ blue balls balance $4$ white balls. How many blue balls are needed to balance $4$ green, $2$ yellow and $2$ white balls? Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 2168,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Seven balls are numbered 1 through 7 and placed in a bowl. Josh will randomly choose a ball from the bowl, look at its number, and then put it back into the bowl. Then Josh will again randomly choose a ball from the bowl and look at its number. What is the probability that the ... | math | {
"style": "rule",
"ground_truth": "\\frac{6}{49}"
} | {
"split": "train",
"index": 2169,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Stock investor Li Jin bought shares of a certain company last Saturday for $27 per share. The table below shows the price changes of the stock within the week.\n\n| Day of the Week | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday |\n|-----------------|--------|----... | math | {
"style": "rule",
"ground_truth": "24.5"
} | {
"split": "train",
"index": 2170,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It costs 2.5 cents to copy a page. How many pages can you copy for $\\$20$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "800"
} | {
"split": "train",
"index": 2171,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest $n$ for which it is possible to construct two bi-infinite sequences $A$ and $B$ such that any subsequence of $B$ of length $n$ is contained in $A$, $A$ has a period of 1995, and $B$ does not have this property (is either non-periodic or has a period of a dif... | math | {
"style": "rule",
"ground_truth": "1995"
} | {
"split": "train",
"index": 2172,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Seven test scores have a mean of $85$, a median of $88$, and a mode of $90$. Calculate the sum of the three lowest test scores. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "237"
} | {
"split": "train",
"index": 2173,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively in $\\triangle ABC$, and $\\sin A$, $\\sin B$, and $\\sin C$ form a geometric sequence. When $B$ takes the maximum value, the maximum value of $\\sin A + \\sin C$ is _____. Let's think ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 2174,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a ten-digit number. From left to right:\n- Its first digit indicates how many zeros are in the number.\n- Its second digit indicates how many ones are in the number.\n- Its third digit indicates how many twos are in the number.\n- $\\cdots \\cdots$\n- Its tenth digit i... | math | {
"style": "rule",
"ground_truth": "6210001000"
} | {
"split": "train",
"index": 2175,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 100 positive integers written on a board. At each step, Alex composes 50 fractions using each number written on the board exactly once, brings these fractions to their irreducible form, and then replaces the 100 numbers on the board with the new numerators and denomin... | math | {
"style": "rule",
"ground_truth": "99"
} | {
"split": "train",
"index": 2176,
"solution": "\nTo solve this problem, we aim to find the smallest positive integer \\( n \\) such that after \\( n \\) steps, the 100 numbers on the board are all pairwise coprime regardless of their initial values.\n\n### Key Observations\n\n1. **Irreducible Fractions**: At eac... |
deepscaler | [
{
"role": "user",
"content": "A round-robin tennis tournament consists of each player playing every other player exactly once. How many matches will be held during an 8-person round-robin tennis tournament? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 2177,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ellen wants to color some of the cells of a $4 \\times 4$ grid. She wants to do this so that each colored cell shares at least one side with an uncolored cell and each uncolored cell shares at least one side with a colored cell. What is the largest number of cells she can color... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 2178,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In response to the call for rural revitalization, Xiao Jiao, a college graduate who has successfully started a business in another place, resolutely returned to her hometown to become a new farmer and established a fruit and vegetable ecological planting base. Recently, in orde... | math | {
"style": "rule",
"ground_truth": "500"
} | {
"split": "train",
"index": 2179,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express the quotient and remainder of $3232_5 \\div 21_5$ in base $5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "130_5 \\, R2_5"
} | {
"split": "train",
"index": 2180,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $A_k = \\frac {k(k - 1)}2\\cos\\frac {k(k - 1)\\pi}2,$ find $|A_{19} + A_{20} + \\cdots + A_{98}|.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 2181,
"solution": "Though the problem may appear to be quite daunting, it is actually not that difficult. $\\frac {k(k-1)}2$ always evaluates to an integer (triangular number), and the cosine of $n\\pi$ where $n \\in \\mathbb{Z}$ is 1 if $n$ is even and -1 if $n$ is odd. $\\frac {k(k-... |
deepscaler | [
{
"role": "user",
"content": "The first $20$ numbers of an arrangement are shown below. What would be the value of the $40^{\\mathrm{th}}$ number if the arrangement were continued?\n\n$\\bullet$ Row 1: $2,$ $2$\n\n$\\bullet$ Row 2: $4,$ $4,$ $4,$ $4$\n\n$\\bullet$ Row 3: $6,$ $6,$ $6,$ $6,$ $6,$ $6$\n\n$\\b... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 2182,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kacey is handing out candy for Halloween. She has only $15$ candies left when a ghost, a goblin, and a vampire arrive at her door. She wants to give each trick-or-treater at least one candy, but she does not want to give any two the same number of candies. How many ways can s... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 2183,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The value of $[2 - 3(2 - 3)^{-1}]^{-1}$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"split": "train",
"index": 2184,
"solution": "To solve the expression $[2 - 3(2 - 3)^{-1}]^{-1}$, we start by simplifying the innermost expression first.\n\n1. **Simplify $(2 - 3)^{-1}$:**\n \\[\n (2 - 3)^{-1} = (-1)^{-1} = -1\n \\]\n\n2. **Substitute back into the expression:**\n \\[\n 2 - 3(-1) = 2 ... |
deepscaler | [
{
"role": "user",
"content": "Regular octagon $ABCDEFGH$ is divided into eight smaller equilateral triangles, such as $\\triangle ABJ$, where $J$ is the center of the octagon. By connecting every second vertex starting from $A$, we obtain a larger equilateral triangle $\\triangle ACE$. Compute the ratio of ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 2185,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The value of $\\log_5{3125}$ is between which consecutive integers $c$ and $d$? Find $c+d$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 2186,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When two fair dice are rolled, the probability that the sum of the numbers facing up is $5$ equals \\_\\_\\_\\_\\_\\_． Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"split": "train",
"index": 2187,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number $6545$ can be written as a product of a pair of positive two-digit numbers. What is the sum of this pair of numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "162"
} | {
"split": "train",
"index": 2188,
"solution": "To solve the problem, we first need to find the prime factorization of the number $6545$.\n\n1. **Prime Factorization of $6545$:**\n \\[\n 6545 = 5 \\times 1309\n \\]\n We need to check if $1309$ is prime or can be further factored. Testing divisibility by s... |
deepscaler | [
{
"role": "user",
"content": "From a large sheet of aluminum, triangular sheets (with each cell side equal to 1) are cut with vertices at marked points. What is the minimum area of the triangle that can be obtained? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 2189,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a trapezoid, the smaller base is 1 decimeter, and the angles adjacent to it are $135^{\\circ}$. The angle between the diagonals, opposite to the base, is $150^{\\circ}$. Find the area of the trapezoid. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"split": "train",
"index": 2190,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $(2^8+4^5)(2^3-(-2)^3)^{10}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1342177280"
} | {
"split": "train",
"index": 2191,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the expansion of $(1+\\frac{a}{x}){{(2x-\\frac{1}{x})}^{5}}$, find the constant term. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 2192,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many 10-digit numbers are there whose digits are all 1, 2, or 3 and in which adjacent digits differ by 1? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 2193,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a fair coin and a fair 6-sided die. The die begins with the number 1 face up. A *step* starts with a toss of the coin: if the coin comes out heads, we roll the die; otherwise (if the coin comes out tails), we do nothing else in this step. After 5 such steps, what is... | math | {
"style": "rule",
"ground_truth": "37/192"
} | {
"split": "train",
"index": 2194,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the set $A={m+2,2m^2+m}$, find the value of $m$ if $3\\in A$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"split": "train",
"index": 2195,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all possible three-digit numbers that can be obtained by removing three digits from the number 112277. Sum them and write the result as the answer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1159"
} | {
"split": "train",
"index": 2196,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Seven thousand twenty-two can be written as Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7022"
} | {
"split": "train",
"index": 2197,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $\\frac{2^2 \\cdot 2^{-3}}{2^3 \\cdot 2^{-2}}$? 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": 2198,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A boss plans a business meeting at Starbucks with the two engineers below him. However, he fails to set a time, and all three arrive at Starbucks at a random time between 2:00 and 4:00 p.m. When the boss shows up, if both engineers are not already there, he storms out and cance... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{24}"
} | {
"split": "train",
"index": 2199,
"solution": ""
} |
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