data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "In the complex plane, the graph of $|z - 3| = 2|z + 3|$ intersects the graph of $|z| = k$ in exactly one point. Find all possible values of $k.$\n\nEnter all possible values, separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 28849,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graphs $y = 3(x-h)^2 + j$ and $y = 2(x-h)^2 + k$ have $y$-intercepts of $2013$ and $2014$, respectively, and each graph has two positive integer $x$-intercepts. Find $h$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 28850,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 28851,
"solution": "1. **Understanding the Display Format**: The digital watch displays time in a 12-hour format with AM and PM, showing hours and minutes. The hours can range from 01 to 12, and the minutes from 00 to 59.\n\n2. **Maximizing the Hour Digits**: \n - The hours are disp... |
deepscaler | [
{
"role": "user",
"content": "Given a set \\( A = \\{0, 1, 2, \\cdots, 9\\} \\), and a family of non-empty subsets \\( B_1, B_2, \\cdots, B_j \\) of \\( A \\), where for \\( i \\neq j \\), \\(\\left|B_i \\cap B_j\\right| \\leqslant 2\\), determine the maximum value of \\( k \\). Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "175"
} | {
"split": "train",
"index": 28852,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any real numbers \\( x \\) and \\( y \\), the operation is defined as\n\\[ x \\oplus y = x + 2y + 3. \\]\n\nGiven that real numbers \\( a \\) and \\( b \\) satisfy\n\\[ \\left(a^3 \\oplus a^2\\right) \\oplus a = a^3 \\oplus \\left(a^2 \\oplus a\\right) = b, \\]\n\nfind the ... | math | {
"style": "rule",
"ground_truth": "\\frac{21}{8}"
} | {
"split": "train",
"index": 28853,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest number of 3-cell L-shaped tiles that can be placed in an 8x8 square such that no more of these tiles can be placed in the square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 28854,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many whole numbers are between $\\sqrt{8}$ and $\\sqrt{80}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 28855,
"solution": "1. **Identify the range of numbers between $\\sqrt{8}$ and $\\sqrt{80}$:**\n - We start by approximating $\\sqrt{8}$ and $\\sqrt{80}$. We know that $\\sqrt{9} = 3$ and $\\sqrt{4} = 2$, and since $8$ is between $4$ and $9$, it follows that $\\sqrt{8}$ is between $... |
deepscaler | [
{
"role": "user",
"content": "Determine the value of $$2002+\\frac{1}{2}\\left(2001+\\frac{1}{2}\\left(2000+\\cdots+\\frac{1}{2}\\left(3+\\frac{1}{2} \\cdot 2\\right)\\right) \\cdots\\right)$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4002"
} | {
"split": "train",
"index": 28856,
"solution": "We can show by induction that $n+\\frac{1}{2}\\left([n-1]+\\frac{1}{2}\\left(\\cdots+\\frac{1}{2} \\cdot 2\\right) \\cdots\\right)=2(n-1)$. For $n=3$ we have $3+\\frac{1}{2} \\cdot 2=4$, giving the base case, and if the result holds for $n$, then $(n+1)+\\frac{1}{2... |
deepscaler | [
{
"role": "user",
"content": "Compute $\\dbinom{5}{3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 28857,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many pairs of nonzero integers $(c, d)$ with $-2015 \\leq c, d \\leq 2015$ do the equations $c x=d$ and $d x=c$ both have an integer solution? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8060"
} | {
"split": "train",
"index": 28858,
"solution": "We need both $c / d$ and $d / c$ to be integers, which is equivalent to $|c|=|d|$, or $d= \\pm c$. So there are 4030 ways to pick $c$ and 2 ways to pick $d$, for a total of 8060 pairs."
} |
deepscaler | [
{
"role": "user",
"content": "Find all possible values of $\\frac{d}{a}$ where $a^{2}-6 a d+8 d^{2}=0, a \\neq 0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}, \\frac{1}{4}"
} | {
"split": "train",
"index": 28859,
"solution": "Dividing $a^{2}-6 a d+8 d^{2}=0$ by $a^{2}$, we get $1-6 \\frac{d}{a}+8\\left(\\frac{d}{a}\\right)^{2}=0$. The roots of this quadratic are $\\frac{1}{2}, \\frac{1}{4}$."
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequences \\(\\{a_{n}\\}\\) and \\(\\{b_{n}\\}\\) with general terms \\(a_{n} = 2^{n}\\) and \\(b_{n} = 5n - 2\\), find the sum of all elements in the set \\(\\{a_{1}, a_{2}, \\cdots, a_{2019}\\} \\cap \\{b_{1}, b_{2}, \\cdots, b_{2019}\\}\\). Let's think step by step... | math | {
"style": "rule",
"ground_truth": "2184"
} | {
"split": "train",
"index": 28860,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Doug and Dave shared a pizza with $8$ equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was $8$ dollars, and there was an additional cost of $2$ dollars for putting anchovies on one half. Dave ate all the sli... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 28861,
"solution": "1. **Calculate the total cost of the pizza:** \n The pizza is plain and costs $8$ dollars. Adding anchovies to half of it costs an additional $2$ dollars. Therefore, the total cost of the pizza is:\n \\[\n 8 + 2 = 10 \\text{ dollars}\n \\]\n\n2. **Determin... |
deepscaler | [
{
"role": "user",
"content": "Find the measure of the angle\n\n$$\n\\delta=\\arccos \\left(\\left(\\sin 2905^{\\circ}+\\sin 2906^{\\circ}+\\cdots+\\sin 6505^{\\circ}\\right)^{\\cos } 2880^{\\circ}+\\cos 2881^{\\circ}+\\cdots+\\cos 6480^{\\circ}\\right)\n$$ Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 28862,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At a hypothetical school, there are three departments in the faculty of sciences: biology, physics and chemistry. Each department has three male and one female professor. A committee of six professors is to be formed containing three men and three women, and each department mus... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 28863,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five volunteers participate in community service for two days, Saturday and Sunday. Each day, two people are selected from the group to serve. Determine the number of ways to select exactly one person to serve for both days. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 28864,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the maximum possible value of \\[\\sin^2x_1+\\sin^2x_2+\\sin^2x_3+\\cdots+\\sin^2x_{2007},\\] where, for $1\\leq i\\leq 2007$ , $x_i$ is a nonnegative real number, and \\[x_1+x_2+x_3+\\cdots+x_{20... | math | {
"style": "rule",
"ground_truth": "2008"
} | {
"split": "train",
"index": 28865,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the maximum number of checkers that can be placed on a $6 \\times 6$ board so that no three checkers (more precisely, the centers of the cells they occupy) are collinear (in any direction)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 28866,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sequence $\\{a\\_n\\}$ is an arithmetic sequence with the sum of its first $n$ terms denoted as $S\\_n$. It is known that $a\\_1 + a\\_5 + a\\_9 = 27$. Find $a\\_5$ and $S\\_9$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 28867,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Walter wakes up at 6:30 a.m., catches the school bus at 7:30 a.m., has 7 classes that last 45 minutes each, enjoys a 30-minute lunch break, and spends an additional 3 hours at school for various activities. He takes the bus home and arrives back at 5:00 p.m. Calculate the total... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 28868,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A polynomial of degree four with leading coefficient 1 and integer coefficients has two real zeros, both of which are integers. Which of the following can also be a zero of the polynomial?\n\n(A) $\\frac{1 + i \\sqrt{11}}{2}$\n\n(B) $\\frac{1 + i}{2}$\n\n(C) $\\frac{1}{2} + i$\... | math | {
"style": "rule",
"ground_truth": "\\text{(A)}"
} | {
"split": "train",
"index": 28869,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sphere with center $O$ has radius $10$. A right triangle with sides $8, 15,$ and $17$ is situated in 3D space such that each side is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle?\n- **A)** $\\sqrt{84}$\n- **B)** $\\sqrt{85}$... | math | {
"style": "rule",
"ground_truth": "\\sqrt{91}"
} | {
"split": "train",
"index": 28870,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alexio has 120 cards numbered 1-120, inclusive, and places them in a box. Alexio then chooses a card from the box at random. What is the probability that the number on the card he chooses is a multiple of 3, 4, or 7? Express your answer as a common fraction. Let's think step by... | math | {
"style": "rule",
"ground_truth": "\\dfrac{69}{120}"
} | {
"split": "train",
"index": 28871,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct sequences of four letters can be made from the letters in \"EXAMPLE\" if each letter can be used only once and each sequence must begin with X and not end with E? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 28872,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For positive integers $m, n$, let \\operatorname{gcd}(m, n) denote the largest positive integer that is a factor of both $m$ and $n$. Compute $$\\sum_{n=1}^{91} \\operatorname{gcd}(n, 91)$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "325"
} | {
"split": "train",
"index": 28873,
"solution": "Since $91=7 \\times 13$, we see that the possible values of \\operatorname{gcd}(n, 91) are 1, 7, 13, 91. For $1 \\leq n \\leq 91$, there is only one value of $n$ such that \\operatorname{gcd}(n, 91)=91. Then, we see that there are 12 values of $n$ for which \\opera... |
deepscaler | [
{
"role": "user",
"content": "What is the remainder when $4x^8 - x^7 + 5x^6 - 7x^4 + 3x^3 - 9$ is divided by $3x - 6$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1119"
} | {
"split": "train",
"index": 28874,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Professors Alpha, Beta, Gamma, and Delta choose their chairs so that each professor will be between two students. Given that there are 13 chairs in total, determine the number of ways these four professors can occupy their chairs. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "1680"
} | {
"split": "train",
"index": 28875,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the proposition \"$\\exists x\\in [-2,1], ax^{2}+2ax+3a > 1$\" is false, determine the maximum value of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 28876,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of $v_2$ when $x = 2$ for $f(x) = 3x^4 + x^3 + 2x^2 + x + 4$ using Horner's method. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 28877,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation $x^3 - 4x^2 + 5x - \\frac{19}{10} = 0$ has real roots $r,$ $s,$ and $t.$ Find the area of the triangle with sides $r,$ $s,$ and $t.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{5}"
} | {
"split": "train",
"index": 28878,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $w,$ $x,$ $y,$ and $z$ be nonnegative numbers whose sum is 100. Find the largest possible value of\n\\[wx + xy + yz.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2500"
} | {
"split": "train",
"index": 28879,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find a positive integer that is divisible by 21 and has a square root between 30 and 30.5. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "903"
} | {
"split": "train",
"index": 28880,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of the following expressions:\n\n1. $\\sqrt[4]{(3-\\pi )^{4}}+(0.008)\\;^{- \\frac {1}{3}}-(0.25)\\;^{ \\frac {1}{2}}×( \\frac {1}{ \\sqrt {2}})^{-4}$\n2. $\\log _{3} \\sqrt {27}-\\log _{3} \\sqrt {3}-\\lg 625-\\lg 4+\\ln (e^{2})- \\frac {4}{3}\\lg \\sqrt {8... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 28881,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)=|2\\{x\\}-1|$ where $\\{x\\}$ denotes the fractional part of $x$. The number $n$ is the smallest positive integer such that the equation \\[nf(xf(x))=x\\]has at least $2012$ real solutions. What is $n$?\n\nNote: the fractional part of $x$ is a real number $y=\\{x\\}$ ... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 28882,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the 17th FIFA World Cup, 35 teams participated, each with 23 players. How many players participated in total? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "805"
} | {
"split": "train",
"index": 28883,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each unit square of a 3-by-3 unit-square grid is to be colored either blue or red. For each square, either color is equally likely to be used. The probability of obtaining a grid that does not have a 2-by-2 red square is $\\frac {m}{n}$, where $m$ and $n$ are relatively prime p... | math | {
"style": "rule",
"ground_truth": "929"
} | {
"split": "train",
"index": 28884,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a right triangle with a right angle at $C.$ Two lines, one parallel to $AC$ and the other parallel to $BC,$ intersect on the hypotenuse $AB.$ The lines split the triangle into two triangles and a rectangle. The two triangles have areas $512$ and $32.$ ... | math | {
"style": "rule",
"ground_truth": "256"
} | {
"split": "train",
"index": 28885,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\frac{\\cos \\alpha}{1 + \\sin \\alpha} = \\sqrt{3}$, find the value of $\\frac{\\cos \\alpha}{\\sin \\alpha - 1}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 28886,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circle $C$ with radius 6 has diameter $\\overline{AB}$. Circle $D$ is internally tangent to circle $C$ at $A$. Circle $E$ is internally tangent to circle $C$, externally tangent to circle $D$, and tangent to $\\overline{AB}$. The radius of circle $D$ is twice the radius of circ... | math | {
"style": "rule",
"ground_truth": "186"
} | {
"split": "train",
"index": 28887,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the GCD (Greatest Common Divisor) of all the numbers of the form \\( n^{13} - n \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2730"
} | {
"split": "train",
"index": 28888,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When you simplify $\\sqrt[3]{24a^4b^6c^{11}}$, what is the sum of the exponents of the variables that are outside the radical? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 28889,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line is described by the equation $y-4=4(x-8)$. What is the sum of its $x$-intercept and $y$-intercept? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-21"
} | {
"split": "train",
"index": 28890,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Assume an even function $f(x)$ satisfies $f(x+6) = f(x) + f(3)$ for any $x \\in \\mathbb{R}$, and $f(x) = 5x$ when $x \\in (-3, -2)$. Calculate $f(201.2)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-16"
} | {
"split": "train",
"index": 28891,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On one particular Wednesday, Jack worked \\( t-2 \\) hours and earned \\( 3t-2 \\) dollars per hour. His coworker Bob worked 1.5 times more hours than Jack but earned \\( 2t-7 \\) dollars per hour less than Jack. After paying a fixed tax of $10 each, they both netted the same a... | math | {
"style": "rule",
"ground_truth": "\\frac{19}{3}"
} | {
"split": "train",
"index": 28892,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the set \\(T = \\{0,1,2,3,4,5,6\\}\\) and \\(M=\\left\\{\\frac{a_{1}}{7}+\\frac{a_{2}}{7^{2}}+\\frac{a_{3}}{7^{3}}+\\frac{a_{4}}{7^{4}}\\right\\}\\), where \\(a_{i} \\in \\mathbf{T}, i=\\{1,2,3,4\\}\\). Arrange the numbers in \\(M\\) in descending order. Determine the 2005t... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{7} + \\frac{1}{7^2} + \\frac{0}{7^3} + \\frac{4}{7^4}"
} | {
"split": "train",
"index": 28893,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify first, then evaluate: $(1-\\frac{2}{{m+1}})\\div \\frac{{{m^2}-2m+1}}{{{m^2}-m}}$, where $m=\\tan 60^{\\circ}-1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3-\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 28894,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "George has 10 different colors available to paint his room and he must choose exactly 2 of them. However, George prefers to include the color blue as one of his choices if possible. In how many ways can he choose the colors? Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 28895,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The curve $y = \\sin x$ cuts the line whose equation is $y = \\sin 70^\\circ$ into segments having the successive ratios\n\\[\\dots p : q : p : q \\dots\\]with $p < q.$ Compute the ordered pair of relatively prime positive integers $(p,q).$ Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "(1,8)"
} | {
"split": "train",
"index": 28896,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many Pythagorean triangles are there in which one of the legs is equal to 2013? (A Pythagorean triangle is a right triangle with integer sides. Identical triangles count as one.). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 28897,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a group of seven friends, the mean (average) age of three of the friends is 12 years and 3 months, and the mean age of the remaining four friends is 13 years and 5 months. In months, what is the mean age of all seven friends? Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "155"
} | {
"split": "train",
"index": 28898,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest possible value of the expression \\(\\frac{1}{a+\\frac{2010}{b+\\frac{1}{c}}}\\), where \\(a, b, c\\) are distinct non-zero digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/203"
} | {
"split": "train",
"index": 28899,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\\(ABCD\\) is a square with sides \\(8 \\, \\text{cm}\\). \\(M\\) is a point on \\(CB\\) such that \\(CM = 2 \\, \\text{cm}\\). If \\(N\\) is a variable point on the diagonal \\(DB\\), find the least value of \\(CN + MN\\). Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 28900,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand? Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "\\frac{47}{256}"
} | {
"split": "train",
"index": 28901,
"solution": "1. **Understanding the Problem:**\n We need to find the probability that no two adjacent people will stand after flipping their coins around a circular table with 8 people. Each person flips a fair coin, meaning each flip has a probability of $\\frac{1}{2}$ for h... |
deepscaler | [
{
"role": "user",
"content": "Consider sequences \\(a\\) of the form \\(a=\\left(a_{1}, a_{2}, \\ldots, a_{20}\\right)\\) such that each term \\(a_{i}\\) is either 0 or 1. For each such sequence \\(a\\), we can produce a sequence \\(b=\\left(b_{1}, b_{2}, \\ldots, b_{20}\\right)\\), where \\(b_{i}= \\begin{... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 28902,
"solution": "Let the two sequences be \\(b\\) and \\(\\hat{b}\\). Then, observe that given \\(a\\), if \\(b_{1}=\\hat{b}_{1}\\) and \\(b_{2}=\\hat{b}_{2}\\), then \\(b=\\hat{b}\\) (since \\(a\\) will uniquely determine the remaining elements in \\(b\\) and \\(\\hat{b}\\)). Thus... |
deepscaler | [
{
"role": "user",
"content": "Point \\( M \\) lies on the side of a regular hexagon with a side length of 10. Find the sum of the distances from point \\( M \\) to the lines containing the other sides of the hexagon. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30\\sqrt{3}"
} | {
"split": "train",
"index": 28903,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $m>n>1$ are positive integers such that there exist $n$ complex numbers $x_{1}, x_{2}, \\ldots, x_{n}$ for which - $x_{1}^{k}+x_{2}^{k}+\\cdots+x_{n}^{k}=1$ for $k=1,2, \\ldots, n-1$ - $x_{1}^{n}+x_{2}^{n}+\\cdots+x_{n}^{n}=2$; and - $x_{1}^{m}+x_{2}^{m}+\\cdots+x_{n}^{... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 28904,
"solution": "Let $S_{k}=\\sum_{j=1}^{n} x_{j}^{k}$, so $S_{1}=S_{2}=\\cdots=S_{n-1}=1, S_{n}=2$, and $S_{m}=4$. The first of these conditions gives that $x_{1}, \\ldots, x_{n}$ are the roots of $P(x)=x^{n}-x^{n-1}-c$ for some constant $c$. Then $x_{i}^{n}=x_{i}^{n-1}+c$, and th... |
deepscaler | [
{
"role": "user",
"content": "What is the minimum value of $z$ if $z=x^2+2y^2+6x-4y+22?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 28905,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square-shaped floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is 37, how many tiles cover the floor? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "361"
} | {
"split": "train",
"index": 28906,
"solution": "1. **Understanding the problem**: We are given a square floor covered with congruent square tiles, and the total number of tiles on the two diagonals is 37. We need to find the total number of tiles covering the floor.\n\n2. **Analyzing the diagonals**: The diagona... |
deepscaler | [
{
"role": "user",
"content": "The graphs of a function $f(x)=3x+b$ and its inverse function $f^{-1}(x)$ intersect at the point $(-3,a)$. Given that $b$ and $a$ are both integers, what is the value of $a$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 28907,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pentagonal prism is used as the base of a new pyramid. One of the seven faces of this pentagonal prism will be chosen as the base of the pyramid. Calculate the maximum value of the sum of the exterior faces, vertices, and edges of the resulting structure after this pyramid is... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 28908,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=-\\cos^2 x + \\sqrt{3}\\sin x\\sin\\left(x + \\frac{\\pi}{2}\\right)$, find the sum of the minimum and maximum values of $f(x)$ when $x \\in \\left[0, \\frac{\\pi}{2}\\right]$. 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": 28909,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square A has a perimeter of $24$ cm. Square B has an area equal to one-fourth the area of square A. What is the perimeter of square B?\n\n[asy]\ndraw((0,0)--(7,0));\ndraw((7,0)--(7,7));\ndraw((7,7)--(0,7));\ndraw((0,7)--(0,0));\ndraw((11,2)--(11,5));\ndraw((11,5)--(14,5));\nd... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 28910,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A farmer sold domestic rabbits. By the end of the market, he sold exactly one-tenth as many rabbits as the price per rabbit in forints. He then distributed the revenue between his two sons. Starting with the older son, the boys alternately received one-hundred forint bills, but... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 28911,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a variant of SHORT BINGO, a $5\\times5$ card is modified such that the middle square is marked as FREE, and the remaining 24 squares are populated with distinct numbers. The modification includes:\n- First column contains 5 distinct numbers from the set $1-15$.\n- Second col... | math | {
"style": "rule",
"ground_truth": "360360"
} | {
"split": "train",
"index": 28912,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the probability that when two numbers are randomly selected from the set {1, 2, 3, 4}, one number is twice the other? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 28913,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C D$ be a square of side length 5, and let $E$ be the midpoint of side $A B$. Let $P$ and $Q$ be the feet of perpendiculars from $B$ and $D$ to $C E$, respectively, and let $R$ be the foot of the perpendicular from $A$ to $D Q$. The segments $C E, B P, D Q$, and $A R$ ... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 28914,
"solution": "We have $D Q \\perp C E$ and $A R \\perp D Q$, so $A R \\| C E$. Thus, we can show that $\\triangle A R D \\cong \\triangle D Q C \\cong \\triangle C P B$, so the median of the areas of the five regions is equal to the area of one of the three triangles listed abov... |
deepscaler | [
{
"role": "user",
"content": "Find the greatest positive number $\\lambda$ such that for any real numbers $a$ and $b$, the inequality $\\lambda a^{2} b^{2}(a+b)^{2} \\leqslant\\left(a^{2}+ab+b^{2}\\right)^{3}$ holds. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{27}{4}"
} | {
"split": "train",
"index": 28915,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( n \\) be a natural number. Decompose \\( n \\) into sums of powers of \\( p \\) (where \\( p \\) is a positive integer greater than 1), in such a way that each power \\( p^k \\) appears at most \\( p^2 - 1 \\) times. Denote by \\( C(n, p) \\) the total number of such de... | math | {
"style": "rule",
"ground_truth": "118"
} | {
"split": "train",
"index": 28916,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle with side lengths in the ratio 2:3:4 is inscribed in a circle of radius 4. What is the area of the triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 28917,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The following diagram shows equilateral triangle $\\vartriangle ABC$ and three other triangles congruent to it. The other three triangles are obtained by sliding copies of $\\vartriangle ABC$ a distance $\\frac18 AB$ along a side of $\\vartriangle ABC$ in the direction... | math | {
"style": "rule",
"ground_truth": "768"
} | {
"split": "train",
"index": 28918,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the 512 smallest positive integers written in base 8 use 5 or 6 (or both) as a digit? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "296"
} | {
"split": "train",
"index": 28919,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Acme T-Shirt Company charges a $\\$50$ set-up fee plus $\\$9$ for each shirt printed. Beta T-shirt Company has no set up fee, but charges $\\$14$ per shirt. What is the minimum number of shirts for which a customer saves money by using Acme? Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 28920,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area of the smallest square that can enclose a circle with a radius of 5? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 28921,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An amoeba is placed in a puddle one day, and on that same day, it splits into two amoebas with a probability of 0.8. Each subsequent day, every amoeba in the puddle has a probability of 0.8 to split into two new amoebas. After one week, assuming no amoebas die, how many amoebas... | math | {
"style": "rule",
"ground_truth": "26.8435456"
} | {
"split": "train",
"index": 28922,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x$ and $y$ are positive real numbers such that $5x^2 + 10xy = x^3 + 2x^2 y,$ what is the value of $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 28923,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the value of $a + b$ if the points $(2,a,b),$ $(a,3,b),$ and $(a,b,4)$ are collinear. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 28924,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four consecutive positive integers have a product of 840. What is the largest of the four integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 28925,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $$f(x)= \\begin{cases} 2\\cos \\frac {\\pi }{3}x & x\\leq 2000 \\\\ x-100 & x>2000\\end{cases}$$, then $f[f(2010)]$ equals \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 28926,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A train has five carriages, each containing at least one passenger. Two passengers are said to be 'neighbours' if either they are in the same carriage or they are in adjacent carriages. Each passenger has exactly five or exactly ten neighbours. How many passengers are there on ... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 28927,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the first character can be chosen from 5 digits (3, 5, 6, 8, 9), and the third character from the left can be chosen from 4 letters (B, C, D), and the other 3 characters can be chosen from 3 digits (1, 3, 6, 9), and the last character from the left can be chosen from... | math | {
"style": "rule",
"ground_truth": "960"
} | {
"split": "train",
"index": 28928,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Country $X$ has $40\\%$ of the world's population and $60\\%$ of the world's wealth. Country $Y$ has $20\\%$ of the world's population but $30\\%$ of its wealth. Country $X$'s top $50\\%$ of the population owns $80\\%$ of the wealth, and the wealth in Country $Y$ is equally sha... | math | {
"style": "rule",
"ground_truth": "1.6"
} | {
"split": "train",
"index": 28929,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\sin 6^\\circ \\sin 42^\\circ \\sin 66^\\circ \\sin 78^\\circ.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"split": "train",
"index": 28930,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number \\(n\\) is a three-digit positive integer and is the product of the three factors \\(x\\), \\(y\\), and \\(5x+2y\\), where \\(x\\) and \\(y\\) are integers less than 10 and \\((5x+2y)\\) is a composite number. What is the largest possible value of \\(n\\) given these... | math | {
"style": "rule",
"ground_truth": "336"
} | {
"split": "train",
"index": 28931,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an $8 \\times 8$ grid filled with different natural numbers, where each cell contains only one number, if a cell's number is greater than the numbers in at least 6 other cells in its row and greater than the numbers in at least 6 other cells in its column, then this cell is ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 28932,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Use the six digits 0, 1, 2, 3, 4, 5 to form four-digit even numbers without repeating any digit. Calculate the total number of such numbers that can be formed. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "156"
} | {
"split": "train",
"index": 28933,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The expansion of $(x+1)^n$ has 3 consecutive terms with coefficients in the ratio $1:2:3$ that can be written in the form\\[{n\\choose k} : {n\\choose k+1} : {n \\choose k+2}\\]Find the sum of all possible values of $n+k$.\n Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 28934,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The coefficients of the polynomial\n\\[ a_{12} x^{12} + a_{11} x^{11} + \\dots + a_2 x^2 + a_1 x + a_0 = 0 \\]\nare all integers, and its roots $s_1, s_2, \\dots, s_{12}$ are all integers. Furthermore, the roots of the polynomial\n\\[ a_0 x^{12} + a_1 x^{11} + a_2 x^{10} + \\do... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 28935,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a math test, the scores of 6 students are as follows: 98, 88, 90, 92, 90, 94. The mode of this set of data is ______; the median is ______; the average is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "92"
} | {
"split": "train",
"index": 28936,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $7$ students, among whom $A_{1}$, $A_{2}$, $A_{3}$ excel in mathematics, $B_{1}$, $B_{2}$ excel in physics, and $C_{1}$, $C_{2}$ excel in chemistry. One student with excellent performance in mathematics, one in physics, and one in chemistry will be selected to form a ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"split": "train",
"index": 28937,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = x^3 - 3x$,\n\n(Ⅰ) Find the intervals of monotonicity for $f(x)$;\n\n(Ⅱ) Find the maximum and minimum values of $f(x)$ in the interval $[-3,2]$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-18"
} | {
"split": "train",
"index": 28938,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the positive divisors of 3240 are multiples of 3? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 28939,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $p$ and $q$ are both real numbers, and $\\sin \\alpha$ and $\\cos \\alpha$ are the two real roots of the equation $x^{2}+px+q=0$ with respect to $x$. Find the minimum value of $p+q$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 28940,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An eight-sided die, with faces numbered from 1 to 8, is tossed three times. Given that the sum of the first two tosses equals the third, calculate the probability that at least one \"2\" is tossed. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{28}"
} | {
"split": "train",
"index": 28941,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the function $f(x)$ is an even function on $(-\\infty, +\\infty)$, and $f(5+x) = f(5-x)$, if $f(x)$ only equals $0$ at $f(1)=0$ within the interval $[0,5]$, determine the number of zeros of $f(x)$ in the interval $[-2012, 2012]$. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "806"
} | {
"split": "train",
"index": 28942,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a shooter who has a probability of $\\frac{3}{4}$ of hitting target A with a single shot and a probability of $\\frac{2}{3}$ of hitting target B with each of two shots, determine the probability that the shooter hits exactly one of the three shots. Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{36}"
} | {
"split": "train",
"index": 28943,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $h(x)$, with terms in order of decreasing degree, if \\[9x^3-3x+1+h(x)=3x^2-5x+3.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-9x^3+3x^2-2x+2"
} | {
"split": "train",
"index": 28944,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\left(\\dfrac{-1+i\\sqrt{3}}{2}\\right)^6+\\left(\\dfrac{-1-i\\sqrt{3}}{2}\\right)^6.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 28945,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I am dining at a Mexican restaurant with a friend who is vegan and allergic to nuts. The restaurant menu lists 8 dishes that are vegan. These vegan options constitute one-fourth of the entire menu. However, 5 of these vegan dishes contain nuts. What fraction of the menu can my ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{32}"
} | {
"split": "train",
"index": 28946,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 24 four-digit whole numbers that use each of the four digits 2, 4, 5 and 7 exactly once. Only one of these four-digit numbers is a multiple of another one. What is it? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7425"
} | {
"split": "train",
"index": 28947,
"solution": "To solve this problem, we need to determine which of the given four-digit numbers is a multiple of another four-digit number formed using the digits 2, 4, 5, and 7 exactly once.\n\n1. **Identify the Range of Numbers**: The numbers formed by the digits 2, 4, 5, and ... |
deepscaler | [
{
"role": "user",
"content": "Determine the smallest natural number $n =>2$ with the property:\n For every positive integers $a_1, a_2,. . . , a_n$ the product of all differences $a_j-a_i$ , \n $1 <=i <j <=n$ , is divisible by 2001. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 28948,
"solution": ""
} |
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