data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "In a modified game, each of 5 players rolls a standard 6-sided die. The winner is the player who rolls the highest number. If there is a tie for the highest roll, those involved must roll again. This continues until only one person has the highest number. If Cecilia is one of t... | math | {
"style": "rule",
"ground_truth": "\\frac{256}{1555}"
} | {
"split": "train",
"index": 20249,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $AB = AC = 3.6$, point $D$ lies on $AB$ with $AD = 1.2$, and point $E$ lies on the extension of $AC$. The area of $\\triangle AED$ is equal to the area of $\\triangle ABC$. Find the length of $AE$. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "10.8"
} | {
"split": "train",
"index": 20250,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the point of tangency of the parabolas $y = x^2 + 15x + 32$ and $x = y^2 + 49y + 593.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(-7,-24)"
} | {
"split": "train",
"index": 20251,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( g : \\mathbb{R} \\to \\mathbb{R} \\) be a function such that\n\\[ g(g(x) - y) = 2g(x) + g(g(y) - g(-x)) + y \\] for all real numbers \\( x \\) and \\( y \\).\n\nLet \\( n \\) be the number of possible values of \\( g(2) \\), and let \\( s \\) be the sum of all possible ... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 20252,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a debate competition with four students participating, the rules are as follows: Each student must choose one question to answer from two given topics, Topic A and Topic B. For Topic A, answering correctly yields 100 points and answering incorrectly results in a loss of 100 ... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 20253,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A standard deck of 52 cards is randomly arranged. What is the probability that the top three cards are $\\spadesuit$, $\\heartsuit$, and $\\spadesuit$ in that sequence? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{78}{5100}"
} | {
"split": "train",
"index": 20254,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The value of $\\tan {75}^{{o}}$ is $\\dfrac{\\sqrt{6}+\\sqrt{2}}{4}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2+\\sqrt{3}"
} | {
"split": "train",
"index": 20255,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Wang Hong's father deposited 20,000 yuan in the bank for a fixed term of three years, with an annual interest rate of 3.33%. How much money, including the principal and interest, can Wang Hong's father withdraw at the end of the term? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "21998"
} | {
"split": "train",
"index": 20256,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$\\frac{x^{2}}{9} + \\frac{y^{2}}{7} = 1$, where $F_{1}$ and $F_{2}$ are the foci of the ellipse. Given that point $A$ lies on the ellipse and $\\angle AF_{1}F_{2} = 45^{\\circ}$, find the area of triangle $AF_{1}F_{2}$. Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{2}"
} | {
"split": "train",
"index": 20257,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $f(x)$ is an odd function defined on $\\mathbb{R}$, and for $x \\geqslant 0$, $f(x) = \\begin{cases} \\log_{\\frac{1}{2}}(x+1), & 0 \\leqslant x < 1 \\\\ 1-|x-3|, & x \\geqslant 1 \\end{cases}$, determine the sum of all zeros of the function $y = f(x) + \\frac{1}{2}$... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} - 1"
} | {
"split": "train",
"index": 20258,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ and $y$ be positive real numbers. Find the maximum value of\n\\[\\frac{(x + y)^2}{x^2 + y^2}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 20259,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Olya, after covering one-fifth of the way from home to school, realized that she forgot her notebook. If she does not return for it, she will reach school 6 minutes before the bell rings, but if she returns, she will be 2 minutes late. How much time (in minutes) does the journe... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 20260,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A flock of geese was flying over several lakes. On each lake, half of the geese and an additional half goose landed, while the rest continued flying. All the geese landed after seven lakes. How many geese were in the flock? Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "127"
} | {
"split": "train",
"index": 20261,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The list $p, q, r, s$ consists of four consecutive integers listed in increasing order. If $p + s = 109$, what is the value of $q + r$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "109"
} | {
"split": "train",
"index": 20262,
"solution": "Since $p, q, r, s$ is a list of consecutive integers in increasing order, then $q$ is 1 more than $p$ and $r$ is 1 less than $s$. This means that $q + r = (p + 1) + (s - 1) = p + s = 109$. Therefore, $q + r = 109$."
} |
deepscaler | [
{
"role": "user",
"content": "Which type of conic section is described by the equation \\[|y+5| = \\sqrt{(x-2)^2 + y^2}?\\]Enter \"C\" for circle, \"P\" for parabola, \"E\" for ellipse, \"H\" for hyperbola, and \"N\" for none of the above. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "\\text{(P)}"
} | {
"split": "train",
"index": 20263,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If P and Q are points on the line y = 1 - x and the curve y = -e^x, respectively, find the minimum value of |PQ|. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 20264,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $b$ is an odd multiple of 9, find the greatest common divisor of $8b^2 + 81b + 289$ and $4b + 17$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 20265,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a sequence $x_{n}$ such that $x_{1}=x_{2}=1, x_{3}=\\frac{2}{3}$. Suppose that $x_{n}=\\frac{x_{n-1}^{2} x_{n-2}}{2 x_{n-2}^{2}-x_{n-1} x_{n-3}}$ for all $n \\geq 4$. Find the least $n$ such that $x_{n} \\leq \\frac{1}{10^{6}}$. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 20266,
"solution": "The recursion simplifies to $\\frac{x_{n-1}}{x_{n}}+\\frac{x_{n-3}}{x_{n-2}}=2 \\frac{x_{n-2}}{x_{n-1}}$. So if we set $y_{n}=\\frac{x_{n-1}}{x_{n}}$ for $n \\geq 2$ then we have $y_{n}-y_{n-1}=y_{n-1}-y_{n-2}$ for $n \\geq 3$, which means that $\\left\\{y_{n}\\rig... |
deepscaler | [
{
"role": "user",
"content": "Remove all perfect squares from the sequence of positive integers $1, 2, 3, \\ldots$ to obtain a new sequence, and find the 2003rd term of this new sequence. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2048"
} | {
"split": "train",
"index": 20267,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $X \\sim N(5, 4)$, find $P(1 < X \\leq 7)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.9759"
} | {
"split": "train",
"index": 20268,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagram shows the miles traveled by bikers Alberto and Bjorn. After four hours about how many more miles has Alberto biked than Bjorn? [asy]\n/* AMC8 1999 #4 Problem */\ndraw((0,0)--(5,0)--(5,3.75)--(0,3.75)--cycle);\nfor(int x=0; x <= 5; ++x) {\n\nfor(real y=0; y <=3.75; y... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 20269,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nA secret facility is in the shape of a rectangle measuring $200 \\times 300$ meters. There is a guard at each of the four corners outside the facility. An intruder approached the perimeter of the secret facility from the outside, and all the guards ran towards the intruder by... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 20270,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two fuses, each of which burns for exactly one minute if lit from one end (but may burn non-uniformly), how can you measure 45 seconds using these fuses? (You can light the fuse from either of its two ends.) Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 20271,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right triangle $XYZ$, we have $\\angle Y = \\angle Z$ and $XY = 8\\sqrt{2}$. What is the area of $\\triangle XYZ$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 20272,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A long piece of paper $5$ cm wide is made into a roll for cash registers by wrapping it $600$ times around a cardboard tube of diameter $2$ cm, forming a roll $10$ cm in diameter. Approximate the length of the paper in meters. (Pretend the paper forms $600$ concentric circles w... | math | {
"style": "rule",
"ground_truth": "36\\pi"
} | {
"split": "train",
"index": 20273,
"solution": "1. **Understanding the Problem:**\n The paper is wrapped around a cardboard tube, forming concentric circles. Each wrap increases the diameter of the roll by the thickness of the paper, which is $5$ cm. The total number of wraps is $600$, and the final diameter o... |
deepscaler | [
{
"role": "user",
"content": "Let \\( g(x) = 10x + 5 \\). Find the sum of all \\( x \\) that satisfy the equation \\( g^{-1}(x) = g((3x)^{-2}) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 20274,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Of the 100 students in the science club, 65 take mathematics, 43 take physics and 10 students take both mathematics and physics. How many science club students take neither mathematics nor physics? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 20275,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A finite set $\\{a_1, a_2, ... a_k\\}$ of positive integers with $a_1 < a_2 < a_3 < ... < a_k$ is named *alternating* if $i+a$ for $i = 1, 2, 3, ..., k$ is even. The empty set is also considered to be alternating. The number of alternating subsets of $\\{1, 2, 3,..., ... | math | {
"style": "rule",
"ground_truth": "5702887"
} | {
"split": "train",
"index": 20276,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $ [\\sqrt{19992000}]$ where $ [x]$ is the greatest integer less than or equal to $ x$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4471"
} | {
"split": "train",
"index": 20277,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $\\lg 1000, \\lg \\left(1000 \\cos \\frac{\\pi}{3}\\right), \\lg \\left(1000 \\cos ^{2} \\frac{\\pi}{3}\\right), \\cdots, \\lg \\left(1000 \\cos ^{n-1} \\frac{\\pi}{3}\\right), \\cdots$, when the sum of the first $n$ terms is maximized, the value of $n$ is ( ). Let... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 20278,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the infinite series defined by the following progression:\n\\[2 + \\frac{1}{3} + \\frac{1}{9} + \\frac{1}{3^2} + \\frac{1}{9^2} + \\frac{1}{3^3} + \\frac{1}{9^3} + \\cdots\\]\nDetermine the limit of this series as it extends to infinity.\nA) $\\frac{1}{3}$\nB) $3$\nC) ... | math | {
"style": "rule",
"ground_truth": "\\frac{21}{8}"
} | {
"split": "train",
"index": 20279,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $PQ$ and $RS$ are two chords of a circle intersecting at a point $O$ . It is given that $PO=3 \\text{cm}$ and $SO=4 \\text{cm}$ . Moreover, the area of the triangle $POR$ is $7 \\text{cm}^2$ . Find the area of the triangle $QOS$ . Let's think step by st... | math | {
"style": "rule",
"ground_truth": "112/9"
} | {
"split": "train",
"index": 20280,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When the product $(3x+2y+1)(x+4y+5)$ is expanded, what is the sum of the coefficients of the terms which contain a nonzero power of $y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 20281,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If a line is perpendicular to a plane, then this line and the plane form a \"perpendicular line-plane pair\". In a cube, the number of \"perpendicular line-plane pairs\" formed by a line determined by two vertices and a plane containing four vertices is _________. Let's think s... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 20282,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A math class has fewer than $40$ students. When the students try to sit in rows of $7$, $3$ students sit in the last row. When the students try to sit in rows of $5$, $1$ student is left in the last row. How many students are in this class? Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 20283,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "You start with a single piece of chalk of length 1. Every second, you choose a piece of chalk that you have uniformly at random and break it in half. You continue this until you have 8 pieces of chalk. What is the probability that they all have length $\\frac{1}{8}$ ? Let's thi... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{63}"
} | {
"split": "train",
"index": 20284,
"solution": "There are 7! total ways to break the chalks. How many of these result in all having length $\\frac{1}{8}$ ? The first move gives you no choice. Then, among the remaining 6 moves, you must apply 3 breaks on the left side and 3 breaks on the right side, so there are ... |
deepscaler | [
{
"role": "user",
"content": "Determine the exact value of the series\n\\[\\frac{1}{3 + 1} + \\frac{2}{3^2 + 1} + \\frac{4}{3^4 + 1} + \\frac{8}{3^8 + 1} + \\frac{16}{3^{16} + 1} + \\dotsb.\\] 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": 20285,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $| \\overrightarrow{a}|=1$, $| \\overrightarrow{b}|= \\sqrt {2}$, and $\\overrightarrow{a} \\perp ( \\overrightarrow{a}- \\overrightarrow{b})$, find the angle between vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$. Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "\\frac {\\pi}{4}"
} | {
"split": "train",
"index": 20286,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a regular tetrahedron with edge length $2\\sqrt{6}$, the total length of the intersection between the sphere with center $O$ and radius $\\sqrt{3}$ and the surface of the tetrahedron is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8\\sqrt{2}\\pi"
} | {
"split": "train",
"index": 20287,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to put 7 balls in 2 boxes if the balls are distinguishable but the boxes are not? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 20288,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On each side of a 6 by 8 rectangle, construct an equilateral triangle with that side as one edge such that the interior of the triangle intersects the interior of the rectangle. What is the total area of all regions that are contained in exactly 3 of the 4 equilateral triangles... | math | {
"style": "rule",
"ground_truth": "\\frac{96 \\sqrt{3}-154}{\\sqrt{3}} \\text{ OR } \\frac{288-154 \\sqrt{3}}{3} \\text{ OR } 96-\\frac{154}{\\sqrt{3}} \\text{ OR } 96-\\frac{154 \\sqrt{3}}{3}"
} | {
"split": "train",
"index": 20289,
"solution": "Let the rectangle be $A B C D$ with $A B=8$ and $B C=6$. Let the four equilateral triangles be $A B P_{1}, B C P_{2}, C D P_{3}$, and $D A P_{4}$ (for convenience, call them the $P_{1^{-}}, P_{2^{-}}, P_{3^{-}}, P_{4^{-}}$triangles). Let $W=A P_{1} \\cap D P_{3}, X... |
deepscaler | [
{
"role": "user",
"content": "After learning about functions, the mathematics team of a high school first grade conducted a mathematical modeling activity. Through a survey of the sales of a certain product in a supermarket near the school, it was found that the relationship between the daily sales price P(... | math | {
"style": "rule",
"ground_truth": "121"
} | {
"split": "train",
"index": 20290,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the volume, in cubic units, of a cube whose surface area is 600 square units? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"split": "train",
"index": 20291,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sin(\\omega x+\\varphi)$ is monotonically increasing on the interval ($\\frac{π}{6}$,$\\frac{{2π}}{3}$), and the lines $x=\\frac{π}{6}$ and $x=\\frac{{2π}}{3}$ are the two symmetric axes of the graph of the function $y=f(x)$, determine the value of $f... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 20292,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the areas of three squares in the diagram, find the area of the triangle formed. The triangle shares one side with each of two squares and the hypotenuse with the third square.\n[asy]\n/* Modified AMC8-like Problem */\ndraw((0,0)--(10,0)--(10,10)--cycle);\ndraw((10,0)--(2... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 20293,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We placed 6 different dominoes in a closed chain on the table. The total number of points on the dominoes is $D$. What is the smallest possible value of $D$? (The number of points on each side of the dominoes ranges from 0 to 6, and the number of points must be the same on touc... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 20294,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that a triangle with integral sides is isosceles and has a perimeter of 12, find the area of the triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"split": "train",
"index": 20295,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and $5x^2+kx+12=0$ has at least one integer solution for $x$. What is $N$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "78"
} | {
"split": "train",
"index": 20296,
"solution": "1. **Identify the integer solution condition**: \n The quadratic equation given is:\n \\[\n 5x^2 + kx + 12 = 0\n \\]\n For this equation to have an integer solution, let's denote that integer by $-n$. Thus, the equation can be factored as:\n \\[\n (5... |
deepscaler | [
{
"role": "user",
"content": "Find the phase shift of the graph of \\( y = \\cos(5x - \\frac{\\pi}{2}) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{10}"
} | {
"split": "train",
"index": 20297,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain car beauty company launched a promotional activity to attract customers: for first-time customers, the charge is $200 per visit, and they can register as members. Members will receive corresponding discounts for each subsequent visit, as shown in the table below:\n\n|... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{15}"
} | {
"split": "train",
"index": 20298,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sides of triangle $CAB$ are in the ratio of $2:3:4$. Segment $BD$ is the angle bisector drawn to the shortest side, dividing it into segments $AD$ and $DC$. What is the length, in inches, of the longer subsegment of side $AC$ if the length of side $AC$ is $10$ inches? Expre... | math | {
"style": "rule",
"ground_truth": "\\frac {40}7"
} | {
"split": "train",
"index": 20299,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Which five-digit numbers are greater in quantity: those not divisible by 5 or those whose first and second digits from the left are not a five? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72000"
} | {
"split": "train",
"index": 20300,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For positive integers $n$, define $S_n$ to be the minimum value of the sum\n\\[\\sum_{k=1}^n \\sqrt{(2k-1)^2+a_k^2},\\]where $a_1,a_2,\\ldots,a_n$ are positive real numbers whose sum is $17$. Find the unique positive integer $n$ for which $S_n$ is also an integer. Let's think s... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 20301,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers satisfy $-4 \\le 2x+1 \\le 6$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 20302,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two numbers are such that their difference, their sum, and their product are to one another as $1:7:24$. The product of the two numbers is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 20303,
"solution": "1. Let the two numbers be $x$ and $y$. According to the problem, their difference, sum, and product are in the ratio $1:7:24$. This can be expressed as:\n \\[\n x - y : x + y : xy = 1 : 7 : 24\n \\]\n This implies:\n \\[\n x+y = 7(x-y) \\quad \\text{and... |
deepscaler | [
{
"role": "user",
"content": "The number obtained from the last two nonzero digits of $90!$ is equal to $n$. What is $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 20304,
"solution": "1. **Count the number of factors of 10 in $90!$:**\n The number of factors of 10 in $90!$ is determined by the number of factors of 5, as there are more factors of 2 than 5. We calculate this using the formula for the number of factors of a prime $p$ in $n!$:\n ... |
deepscaler | [
{
"role": "user",
"content": "The Chinese government actively responds to changes in gas emissions and has set a goal to reduce carbon emission intensity by 40% by 2020 compared to 2005. It is known that in 2005, China's carbon emission intensity was about 3 tons per 10,000 yuan, and each year thereafter, t... | math | {
"style": "rule",
"ground_truth": "2030"
} | {
"split": "train",
"index": 20305,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a similar game setup, there are 30 boxes, each containing one of the following values: \\begin{tabular}{|c|c|}\\hline\\$.01&\\$1,000\\\\\\hline\\$1&\\$5,000\\\\\\hline\\$5&\\$10,000\\\\\\hline\\$10&\\$25,000\\\\\\hline\\$25&\\$50,000\\\\\\hline\\$50&\\$75,000\\\\\\hline\\$75... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 20306,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the diagram below, find the positive difference in the $x$-coordinates when lines $l$ and $m$ reach $y = 15.$ [asy]\nimport cse5; import olympiad;\nsize(120);\nadd(grid(8,8));\ndraw((0,0)--(8,0),linewidth(1.2));\ndraw((0,0)--(0,8),linewidth(1.2));\nlabel(\"$x$\",(8,0),E);\... | math | {
"style": "rule",
"ground_truth": "39.5"
} | {
"split": "train",
"index": 20307,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many three-digit numbers are there in which any two adjacent digits differ by 3? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 20308,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The line $y=kx+b$ is tangent to the curve $y=x^2+\\frac{a}{x}+1$ at the point $(2,3)$. Find the value of $b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"split": "train",
"index": 20309,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let\n\\[f(x) = \\left\\{\n\\begin{array}{cl}\nx + 5 & \\text{if $x < 15$}, \\\\\n3x - 1 & \\text{if $x \\ge 15$}.\n\\end{array}\n\\right.\\]Find $f^{-1}(10) + f^{-1}(50).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 20310,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence of positive integers is defined by $a_{0}=1$ and $a_{n+1}=a_{n}^{2}+1$ for each $n \\geq 0$. Find $\\operatorname{gcd}(a_{999}, a_{2004})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "677"
} | {
"split": "train",
"index": 20311,
"solution": "If $d$ is the relevant greatest common divisor, then $a_{1000}=a_{999}^{2}+1 \\equiv 1=a_{0}(\\bmod d)$, which implies (by induction) that the sequence is periodic modulo $d$, with period 1000 . In particular, $a_{4} \\equiv a_{2004} \\equiv 0$. So $d$ must divide ... |
deepscaler | [
{
"role": "user",
"content": "The three sides of a right triangle form a geometric sequence. Determine the ratio of the length of the hypotenuse to the length of the shorter leg. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1+\\sqrt{5}}{2}"
} | {
"split": "train",
"index": 20312,
"solution": "Let the shorter leg have length $\\ell$, and the common ratio of the geometric sequence be $r>1$. Then the length of the other leg is $\\ell r$, and the length of the hypotenuse is $\\ell r^{2}$. Hence, $$\\ell^{2}+(\\ell r)^{2}=\\left(\\ell r^{2}\\right)^{2} \\Lon... |
deepscaler | [
{
"role": "user",
"content": "How many ordered pairs $(a,b)$ such that $a$ is a positive real number and $b$ is an integer between $2$ and $200$, inclusive, satisfy the equation $(\\log_b a)^{2017}=\\log_b(a^{2017})?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "597"
} | {
"split": "train",
"index": 20313,
"solution": "1. **Rewrite the given equation using logarithmic identities:**\n The given equation is $(\\log_b a)^{2017} = \\log_b(a^{2017})$. By the power rule of logarithms, $\\log_b(a^{2017}) = 2017 \\log_b a$. Thus, the equation simplifies to:\n \\[\n (\\log_b a)^{201... |
deepscaler | [
{
"role": "user",
"content": "Given that \\( O \\) is the circumcenter of \\( \\triangle ABC \\), and the equation\n\\[\n\\overrightarrow{A O} \\cdot \\overrightarrow{B C} + 2 \\overrightarrow{B O} \\cdot \\overrightarrow{C A} + 3 \\overrightarrow{C O} \\cdot \\overrightarrow{A B} = 0,\n\\]\nfind the minimu... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 20314,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Nine hundred forty-three minus eighty-seven equals Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "856"
} | {
"split": "train",
"index": 20315,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the lines $l_{1}$: $x+\\left(m-3\\right)y+m=0$ and $l_{2}$: $mx-2y+4=0$. \n$(1)$ If line $l_{1}$ is perpendicular to line $l_{2}$, find the value of $m$. \n$(2)$ If line $l_{1}$ is parallel to line $l_{2}$, find the distance between $l_{1}$ and $l_{2}$. Let's think step... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{5}}{5}"
} | {
"split": "train",
"index": 20316,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $\\sin 9^\\circ \\sin 45^\\circ \\sin 69^\\circ \\sin 81^\\circ.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{0.6293 \\sqrt{2}}{4}"
} | {
"split": "train",
"index": 20317,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three boys played a \"Word\" game in which they each wrote down ten words. For each word a boy wrote, he scored three points if neither of the other boys had the same word; he scored one point if only one of the other boys had the same word. No points were awarded for words whi... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 20318,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A green chameleon always tells the truth, while a brown chameleon lies and immediately turns green after lying. In a group of 2019 chameleons (both green and brown), each chameleon, in turn, answered the question, \"How many of them are green right now?\" The answers were the n... | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"split": "train",
"index": 20319,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the set of all equations $x^3 + a_2x^2 + a_1x + a_0 = 0$, where $a_2$, $a_1$, $a_0$ are real constants and $|a_i| < 2$ for $i = 0,1,2$. Let $r$ be the largest positive real number which satisfies at least one of these equations. Then Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2} < r < 3"
} | {
"split": "train",
"index": 20320,
"solution": "To find the largest positive real number $r$ that satisfies at least one equation of the form $x^3 + a_2x^2 + a_1x + a_0 = 0$ with $|a_i| < 2$ for $i = 0,1,2$, we need to consider how to maximize $x$ under these constraints.\n\n1. **Expression Rearrangement**: \n ... |
deepscaler | [
{
"role": "user",
"content": "Using the numbers 0, 1, 2, 3, 4, 5 to form unique three-digit numbers, determine the total number of even numbers that can be formed. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 20321,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $XYZ$ is a right, isosceles triangle. Angle $X$ measures 45 degrees. What is the number of degrees in the measure of angle $Y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 20322,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When a bucket is two-thirds full of water, the bucket and water weigh $a$ kilograms. When the bucket is one-half full of water the total weight is $b$ kilograms. In terms of $a$ and $b$, what is the total weight in kilograms when the bucket is full of water? Let's think step by... | math | {
"style": "rule",
"ground_truth": "3a - 2b"
} | {
"split": "train",
"index": 20323,
"solution": "\nLet's denote:\n- $x$ as the weight of the empty bucket.\n- $y$ as the weight of the water when the bucket is full.\n\nFrom the problem, we have two equations based on the given conditions:\n1. When the bucket is two-thirds full, the total weight is $a$ kilograms:... |
deepscaler | [
{
"role": "user",
"content": "Three distinct numbers are selected simultaneously and at random from the set $\\{1, 2, 3, 4, 6\\}$. What is the probability that at least one number divides another among the selected numbers? Express your answer as a common fraction. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{10}"
} | {
"split": "train",
"index": 20324,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that the roots of $x^3+3x^2+4x-11=0$ are $a$, $b$, and $c$, and that the roots of $x^3+rx^2+sx+t=0$ are $a+b$, $b+c$, and $c+a$. Find $t$.\n ~ pi_is_3.14 Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 20325,
"solution": "By Vieta's formulas on the polynomial $P(x) = x^3+3x^2+4x-11 = (x-a)(x-b)(x-c) = 0$, we have $a + b + c = s = -3$, $ab + bc + ca = 4$, and $abc = 11$. Then\n$t = -(a+b)(b+c)(c+a) = -(s-a)(s-b)(s-c) = -(-3-a)(-3-b)(-3-c)$\n This is just the definition for $-P(-3) = ... |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $\\angle C=90^\\circ$, $AC=6$ and $BC=8$. Points $D$ and $E$ are on $\\overline{AB}$ and $\\overline{BC}$, respectively, and $\\angle BED=90^\\circ$. If $DE=4$, then what is the length of $BD$? [asy]\nimport olympiad; import geometry; size(150); defaultpen(li... | math | {
"style": "rule",
"ground_truth": "\\frac{20}{3}"
} | {
"split": "train",
"index": 20326,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $(a_n)$ satisfies $a_0=0$ and $a_{n + 1} = \\frac{8}{5}a_n + \\frac{6}{5}\\sqrt{4^n - a_n^2}$ for $n \\geq 0$. Find $a_{10}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{24576}{25}"
} | {
"split": "train",
"index": 20327,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ratio of the areas of a square and a circle is $\\frac{250}{196}$. After rationalizing the denominator, the ratio of the side length of the square to the radius of the circle can be expressed in the simplified form $\\frac{a\\sqrt{b}}{c}$ where $a$, $b$, and $c$ are integer... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 20328,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $\\frac{x^{2}}{4}+\\frac{y^{2}}{3}=1$ with left and right foci denoted as $F_{1}$ and $F_{2}$, respectively. Draw a line $l$ passing through the right focus that intersects the ellipse at points $P$ and $Q$. What is the maximum area of the inscribed circle of ... | math | {
"style": "rule",
"ground_truth": "\\frac{9\\pi}{16}"
} | {
"split": "train",
"index": 20329,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=ax^{3}-4x+4$, where $a\\in\\mathbb{R}$, $f′(x)$ is the derivative of $f(x)$, and $f′(1)=-3$.\n\n(1) Find the value of $a$;\n\n(2) Find the extreme values of the function $f(x)$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "-\\frac{4}{3}"
} | {
"split": "train",
"index": 20330,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The factors of $x^4+64$ are: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(x^2-4x+8)(x^2+4x+8)"
} | {
"split": "train",
"index": 20331,
"solution": "1. **Identify the Expression**: We start with the expression \\(x^4 + 64\\).\n\n2. **Recognize the Form**: Notice that \\(64\\) can be rewritten as \\(2^6\\), so we have:\n \\[\n x^4 + 64 = x^4 + 2^6\n \\]\n\n3. **Apply the Sophie Germain Identity**: The Soph... |
deepscaler | [
{
"role": "user",
"content": "\nGrandpa is twice as strong as Grandma, Grandma is three times as strong as Granddaughter, Granddaughter is four times as strong as Dog, Dog is five times as strong as Cat, Cat is six times as strong as Mouse. Grandpa, Grandma, Granddaughter, Dog, and Cat together with Mouse c... | math | {
"style": "rule",
"ground_truth": "1237"
} | {
"split": "train",
"index": 20332,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, the parametric equations of curve $C_{1}$ are $\\left\\{{\\begin{array}{l}{x=1+t,}\\\\{y=\\sqrt{3}t}\\end{array}}\\right.$ (where $t$ is the parameter), and the parametric equations of curve $C_{2}$ are $\\left\\{{\\begin{array}{l}{x=\\... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 20333,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $X$ is a discrete random variable, $P(X=x_{1})= \\frac {2}{3},P(X=x_{2})= \\frac {1}{3}$, and $x_{1} < x_{2}$, it is also known that $EX= \\frac {4}{9}$, $DX=2$, calculate the sum of $x_{1}$ and $x_{2}$. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{9}"
} | {
"split": "train",
"index": 20334,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a triangle $\\triangle ABC$ whose side lengths form an arithmetic sequence with a common difference of $2$, and the sine of its largest angle is $\\frac{\\sqrt{3}}{2}$, find the perimeter of this triangle. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 20335,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The mean, median, and mode of the $7$ data values $60, 100, x, 40, 50, 200, 90$ are all equal to $x$. 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": "90"
} | {
"split": "train",
"index": 20336,
"solution": "1. **Given Information**: The mean, median, and mode of the data set $60, 100, x, 40, 50, 200, 90$ are all equal to $x$.\n\n2. **Calculating the Mean**:\n The mean $\\mu$ of a data set is calculated by the formula:\n \\[\n \\mu = \\frac{\\text{sum of all data... |
deepscaler | [
{
"role": "user",
"content": "The hypotenuse $c$ and one arm $a$ of a right triangle are consecutive integers. The square of the second arm is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "c+a"
} | {
"split": "train",
"index": 20337,
"solution": "1. Let's denote the sides of the right triangle as $a$, $b$, and $c$, where $c$ is the hypotenuse and $a$ and $b$ are the legs. According to the Pythagorean theorem, we have:\n \\[\n a^2 + b^2 = c^2\n \\]\n\n2. Given that $c$ and $a$ are consecutive integers,... |
deepscaler | [
{
"role": "user",
"content": "A point $(x,y)$ in the plane is called a lattice point if both $x$ and $y$ are integers. The area of the largest square that contains exactly three lattice points in its interior is closest to Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5.0"
} | {
"split": "train",
"index": 20338,
"solution": "To solve this problem, we need to find the largest square that contains exactly three lattice points in its interior. We will consider the properties of lattice points and the geometry of squares.\n\n1. **Understanding Lattice Points and Squares**:\n A lattice po... |
deepscaler | [
{
"role": "user",
"content": "Last year, Isabella took 7 math tests and received 7 different scores, each an integer between 91 and 100, inclusive. After each test she noticed that the average of her test scores was an integer. Her score on the seventh test was 95. What was her score on the sixth test? $\\t... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 20339,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that plane vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ are non-zero vectors, $|\\overrightarrow{a}|=2$, and $\\overrightarrow{a} \\bot (\\overrightarrow{a}+2\\overrightarrow{b})$, calculate the projection of vector $\\overrightarrow{b}$ in the direction of vec... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 20340,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the quadratic equation $ax^{2}+bx+c=0$ with $a > 0$ and $b, c \\in \\mathbb{R}$, and the roots of the equation lying in the interval $(0, 2)$, determine the minimum value of the real number $a$ given that $25a+10b+4c \\geqslant 4$ for $c \\geqslant 1$. Let's think step by... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{25}"
} | {
"split": "train",
"index": 20341,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that point \\( P \\) lies on the hyperbola \\( \\frac{x^2}{16} - \\frac{y^2}{9} = 1 \\), and the distance from \\( P \\) to the right directrix of this hyperbola is the arithmetic mean of the distances from \\( P \\) to the two foci of the hyperbola, find the x-coordinate... | math | {
"style": "rule",
"ground_truth": "-\\frac{64}{5}"
} | {
"split": "train",
"index": 20342,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The three medians of a triangle has lengths $3, 4, 5$ . What is the length of the shortest side of this triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10}{3}"
} | {
"split": "train",
"index": 20343,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ n(\\ge2) $ be a positive integer. Find the minimum $ m $, so that there exists $x_{ij}(1\\le i ,j\\le n)$ satisfying:\n(1)For every $1\\le i ,j\\le n, x_{ij}=max\\{x_{i1},x_{i2},...,x_{ij}\\} $ or $ x_{ij}=max\\{x_{1j},x_{2j},...,x_{ij}\\}.$\n(2)For every $1\\le i \\le n$... | math | {
"style": "rule",
"ground_truth": "1 + \\left\\lceil \\frac{n}{2} \\right\\rceil"
} | {
"split": "train",
"index": 20344,
"solution": "\nLet \\( n (\\geq 2) \\) be a positive integer. We aim to find the minimum \\( m \\) such that there exists \\( x_{ij} \\) (for \\( 1 \\leq i, j \\leq n \\)) satisfying the following conditions:\n1. For every \\( 1 \\leq i, j \\leq n \\), \\( x_{ij} = \\max \\{ x_... |
deepscaler | [
{
"role": "user",
"content": "The circle is divided into 30 equal parts by 30 points on the circle. Randomly selecting 3 different points, what is the probability that these 3 points form an equilateral triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/406"
} | {
"split": "train",
"index": 20345,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $x = 2-t$ and $y = 4t + 7$. If $x = -3$, what is $y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 20346,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 8 white balls and 2 red balls in a bag. Each time a ball is randomly taken out, a white ball is put back in. What is the probability that all red balls are taken out exactly by the 4th draw? Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "353/5000"
} | {
"split": "train",
"index": 20347,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\tau(n)$ denote the number of positive integer divisors of $n$. Find the sum of the six least positive integers $n$ that are solutions to $\\tau (n) + \\tau (n+1) = 7$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "540"
} | {
"split": "train",
"index": 20348,
"solution": "In order to obtain a sum of $7$, we must have:\neither a number with $5$ divisors (a fourth power of a prime) and a number with $2$ divisors (a prime), or\na number with $4$ divisors (a semiprime or a cube of a prime) and a number with $3$ divisors (a square of a p... |
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