data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "$6^6 + 6^6 + 6^6 + 6^6 + 6^6 + 6^6 = $ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6^7"
} | {
"split": "train",
"index": 4300,
"solution": "1. **Identify the Expression**: The problem gives the expression $6^6 + 6^6 + 6^6 + 6^6 + 6^6 + 6^6$.\n\n2. **Simplify the Expression**: We observe that the expression consists of six terms, each equal to $6^6$. This can be rewritten using the distributive property ... |
deepscaler | [
{
"role": "user",
"content": "The sides of triangle $PQR$ are tangent to a circle with center $C$ as shown. Given that $\\angle PQR = 65^\\circ$ and $\\angle QRC = 30^\\circ$, find $\\angle QPR$, in degrees.\n\n[asy]\nunitsize(1.0 cm);\n\npair Q, P, R, C;\n\nQ = (2.43,3.46);\nP = (0,0);\nR = (4.43,0);\nC = ... | math | {
"style": "rule",
"ground_truth": "55^\\circ"
} | {
"split": "train",
"index": 4301,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Billy is hiking in Colorado. He walks eastward four miles, then turns $60$ degrees northward and walks six miles. How far is he from his starting point? Express your answer in simplest radical form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{19}"
} | {
"split": "train",
"index": 4302,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(\\alpha)=\\dfrac{\\sin(\\alpha-3\\pi)\\cdot \\cos(2\\pi-\\alpha)\\cdot \\sin(-\\alpha+\\frac{3}{2}\\pi)}{\\cos(-\\pi-\\alpha)\\cdot \\sin(-\\pi-\\alpha)}$,\n(1) Simplify $f(\\alpha)$;\n(2) If $\\sin(\\alpha-\\frac{3}{2}\\pi)=\\frac{1}{5}$, find the value of $f(\\alpha)... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{5}"
} | {
"split": "train",
"index": 4303,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify first, then evaluate: $\\left(\\dfrac{a+2}{a^{2}-2a}+\\dfrac{8}{4-a^{2}}\\right)\\div \\dfrac{a^{2}-4}{a}$, where $a$ satisfies the equation $a^{2}+4a+1=0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{3}"
} | {
"split": "train",
"index": 4304,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For real numbers $a,$ $b,$ and $c,$ and a real scalar $\\lambda,$ consider the matrix\n\\[\\begin{pmatrix} a + \\lambda & b & c \\\\ b & c + \\lambda & a \\\\ c & a & b + \\lambda \\end{pmatrix}.\\] Determine all possible values of \\[\\frac{a}{b + c} + \\frac{b}{a + c} + \\fra... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 4305,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the expression $c \\cdot a^b - d$, the values of $a$, $b$, $c$, and $d$ are 0, 1, 2, and 3, although not necessarily in that order. What is the maximum possible value of the result? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 4306,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a convex quadrilateral inscribed around a circle, the products of opposite sides are equal. The angle between a side and one of the diagonals is $20^{\\circ}$. Find the angle between this side and the other diagonal. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 4307,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the integers \\( a, b, c \\) satisfy \\( 0 \\leq a \\leq 10 \\), \\( 0 \\leq b \\leq 10 \\), \\( 0 \\leq c \\leq 10 \\), and \\( 10 \\leq a + b + c \\leq 20 \\), then how many ordered triples \\((a, b, c)\\) meet the conditions? Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "286"
} | {
"split": "train",
"index": 4308,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the smallest positive integer $ n$ such that there exists positive integers $ a_1,a_2,\\cdots,a_n$, that smaller than or equal to $ 15$ and are not necessarily distinct, such that the last four digits of the sum,\n\n\\[ a_1!\\plus{}a_2!\\plus{}\\cdots\\plus{}a_n!\\]\n... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 4309,
"solution": "\nWe are tasked with finding the smallest positive integer \\( n \\) such that there exist positive integers \\( a_1, a_2, \\ldots, a_n \\) where each \\( a_i \\) is less than or equal to 15, and the last four digits of the sum \\( a_1! + a_2! + \\cdots + a_n! \\) i... |
deepscaler | [
{
"role": "user",
"content": "Determine the longest side of the polygon formed by the system:\n$$\n\\begin{cases}\nx + y \\leq 4 \\\\\nx + 2y \\geq 4 \\\\\nx \\geq 0 \\\\\ny \\geq 0\n\\end{cases}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"split": "train",
"index": 4310,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The line $ax+2by=1$ intersects the circle $x^{2}+y^{2}=1$ at points $A$ and $B$ (where $a$ and $b$ are real numbers), and $\\triangle AOB$ is a right-angled triangle ($O$ is the origin). The maximum distance between point $P(a,b)$ and point $Q(0,0)$ is ______. Let's think step ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 4311,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The minimum positive period of the function $f(x) = \\sin \\omega x + \\sqrt{3}\\cos \\omega x + 1$ ($\\omega > 0$) is $\\pi$. When $x \\in [m, n]$, $f(x)$ has at least 5 zeros. The minimum value of $n-m$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "2\\pi"
} | {
"split": "train",
"index": 4312,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the digits of the result of the expression $\\underbrace{99 \\cdots 99}_{2021 \\text{ digits}} \\times \\underbrace{99 \\cdots 99}_{2020 \\text{ digits}}$ is $\\qquad$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18189"
} | {
"split": "train",
"index": 4313,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of a certain quadratic $y = ax^2 + bx + c$ is a parabola with vertex $(-4,0)$ which passes through the point $(1,-75)$. 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": 4314,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the asymptotic line of the hyperbola $\\frac{x^2}{a}+y^2=1$ has a slope of $\\frac{5π}{6}$, determine 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": 4315,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let ellipse $C$:$\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1\\left(a \\gt b \\gt 0\\right)$ have foci $F_{1}(-c,0)$ and $F_{2}(c,0)$. Point $P$ is the intersection point of $C$ and the circle $x^{2}+y^{2}=c^{2}$. The bisector of $\\angle PF_{1}F_{2}$ intersects $PF_{2}$... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}-1"
} | {
"split": "train",
"index": 4316,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( z \\) be a complex number that satisfies\n\\[ |z - 2| + |z - 7i| = 10. \\]\nFind the minimum value of \\( |z| \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1.4"
} | {
"split": "train",
"index": 4317,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $29x + y = 2035$. Find the number of such distinct triangles whose area is a positive integer. Let's think ste... | math | {
"style": "rule",
"ground_truth": "1225"
} | {
"split": "train",
"index": 4318,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The line segment connecting the focus F of the parabola $y^2=4x$ and the point M(0,1) intersects the parabola at point A. Let O be the origin, then the area of △OAM is _____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {3}{2} - \\sqrt {2}"
} | {
"split": "train",
"index": 4319,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We consider an $n \\times n$ table, with $n\\ge1$. Aya wishes to color $k$ cells of this table so that that there is a unique way to place $n$ tokens on colored squares without two tokens are not in the same row or column. What is the maximum value of $k$ for which Aya's wish i... | math | {
"style": "rule",
"ground_truth": "\\frac{n(n+1)}{2}"
} | {
"split": "train",
"index": 4320,
"solution": "\nGiven an \\( n \\times n \\) table, where \\( n \\geq 1 \\), the task is to determine the maximum number of cells \\( k \\) that can be colored such that there is a unique way to place \\( n \\) tokens on the colored cells. Importantly, no two tokens should be in ... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=1-2\\sin (x+ \\frac {π}{8})[\\sin (x+ \\frac {π}{8})-\\cos (x+ \\frac {π}{8})]$, $x\\in R$.\n(I) Find the smallest positive period of the function $f(x)$;\n(II) Find the maximum and minimum values of the function $f(x+ \\frac {π}{8})$ on the interval $[... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 4321,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Of the numbers $\\frac{7}{10}, \\frac{4}{5}$ and $\\frac{3}{4}$, which number is the arithmetic mean of the other two? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac34"
} | {
"split": "train",
"index": 4322,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If two distinct numbers are selected at random from the first seven prime numbers, what is the probability that their sum is an even number? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{7}"
} | {
"split": "train",
"index": 4323,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the circulation of the vector field\n\n$$\n\\vec{a}=\\frac{y}{3} \\vec{i} 3-3 x \\vec{j}+x \\vec{k}\n$$\n\nalong the closed contour $\\Gamma$\n\n$$\n\\left\\{\\begin{array}{l}\nx=2 \\cos t \\\\\ny=2 \\sin t \\\\\nz=1-2 \\cos t - 2 \\sin t\n\\end{array} \\quad t \\in [0,2\\... | math | {
"style": "rule",
"ground_truth": "-\\frac{52 \\pi}{3}"
} | {
"split": "train",
"index": 4324,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle with center O is tangent to the coordinate axes and to the hypotenuse of a $45^\\circ$-$45^\\circ$-$90^\\circ$ triangle ABC, where AB = 2. Determine the exact radius of the circle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 + \\sqrt{2}"
} | {
"split": "train",
"index": 4325,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a, b$, and $c$ are complex numbers satisfying $$\\begin{aligned} a^{2}+a b+b^{2} & =1+i \\\\ b^{2}+b c+c^{2} & =-2 \\\\ c^{2}+c a+a^{2} & =1 \\end{aligned}$$ compute $(a b+b c+c a)^{2}$. (Here, $\\left.i=\\sqrt{-1}.\\right)$ Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "\\frac{-11-4 i}{3}"
} | {
"split": "train",
"index": 4326,
"solution": "More generally, suppose $a^{2}+a b+b^{2}=z, b^{2}+b c+c^{2}=x$, $c^{2}+c a+a^{2}=y$ for some complex numbers $a, b, c, x, y, z$. We show that $$f(a, b, c, x, y, z)=\\left(\\frac{1}{2}(a b+b c+c a) \\sin 120^{\\circ}\\right)^{2}-\\left(\\frac{1}{4}\\right)^{2}\\left[... |
deepscaler | [
{
"role": "user",
"content": "Consider the $12$-sided polygon $ABCDEFGHIJKL$, as shown. Each of its sides has length $4$, and each two consecutive sides form a right angle. Suppose that $\\overline{AG}$ and $\\overline{CH}$ meet at $M$. What is the area of quadrilateral $ABCM$? Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "88/5"
} | {
"split": "train",
"index": 4327,
"solution": "\n1. **Identify the Key Points and Setup:**\n - We are given a 12-sided polygon with each side of length 4 and each angle being a right angle.\n - We need to find the area of quadrilateral $ABCM$ where lines $\\overline{AG}$ and $\\overline{CH}$ intersect at poi... |
deepscaler | [
{
"role": "user",
"content": "Each point in the hexagonal lattice shown is one unit from its nearest neighbor. How many equilateral triangles have all three vertices in the lattice? [asy]size(75);\ndot(origin);\ndot(dir(0));\ndot(dir(60));\ndot(dir(120));\ndot(dir(180));\ndot(dir(240));\ndot(dir(300));\n[/a... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 4328,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any positive integer $x$, define $\\operatorname{Accident}(x)$ to be the set of ordered pairs $(s, t)$ with $s \\in \\{0,2,4,5,7,9,11\\}$ and $t \\in\\{1,3,6,8,10\\}$ such that $x+s-t$ is divisible by 12. For any nonnegative integer $i$, let $a_{i}$ denote the number of $x ... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 4329,
"solution": "Modulo twelve, the first set turns out to be $\\{-1 \\cdot 7,0 \\cdot 7, \\ldots, 5 \\cdot 7\\}$ and the second set turns out to be be $\\{6 \\cdot 7, \\ldots, 10 \\cdot 7\\}$. We can eliminate the factor of 7 and shift to reduce the problem to $s \\in\\{0,1, \\ldot... |
deepscaler | [
{
"role": "user",
"content": "Given the parametric equation of curve $C\\_1$ as $\\begin{cases} x=a\\cos \\theta \\\\ y=b\\sin \\theta \\end{cases}$ $(a > b > 0, \\theta$ is the parameter$)$, and the point $M(1, \\frac{ \\sqrt{3}}{2})$ on curve $C\\_1$ corresponds to the parameter $\\theta= \\frac{ \\pi}{3}... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{4}"
} | {
"split": "train",
"index": 4330,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_0=-2,b_0=1$, and for $n\\geq 0$, let\n\\begin{align*}a_{n+1}&=a_n+b_n+\\sqrt{a_n^2+b_n^2},\\\\b_{n+1}&=a_n+b_n-\\sqrt{a_n^2+b_n^2}.\\end{align*}Find $\\frac{1}{a_{2012}} + \\frac{1}{b_{2012}}.$ 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": 4331,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $6 \\div 0.\\overline{6}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 4332,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the triangle shown, $n$ is a positive integer, and $\\angle A > \\angle B > \\angle C$. How many possible values of $n$ are there? [asy]\ndraw((0,0)--(1,0)--(.4,.5)--cycle);\nlabel(\"$A$\",(.4,.5),N); label(\"$B$\",(1,0),SE); label(\"$C$\",(0,0),SW);\nlabel(\"$2n + 12$\",(.... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 4333,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Right triangles \\(ABC\\) and \\(ABD\\) share a common hypotenuse \\(AB = 5\\). Points \\(C\\) and \\(D\\) are located on opposite sides of the line passing through points \\(A\\) and \\(B\\), with \\(BC = BD = 3\\). Point \\(E\\) lies on \\(AC\\), and \\(EC = 1\\). Point \\(F\... | math | {
"style": "rule",
"ground_truth": "9.12"
} | {
"split": "train",
"index": 4334,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $f(x)$ satisfies\n\\[f(2^x) + xf(2^{-x}) = 1\\]for all real numbers $x.$ Find $f(2).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 4335,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a $9 \\times 9$ chessboard, determine the total number of rectangles and the number of those rectangles that are squares. Then, find the simplified fraction of the ratio of the number of squares to the total number of rectangles and compute $m + n$ where the fraction is expr... | math | {
"style": "rule",
"ground_truth": "154"
} | {
"split": "train",
"index": 4336,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In product inspection, the method of sampling inspection is often used. Now, 4 products are randomly selected from 100 products (among which there are 3 defective products) for inspection. The number of ways to exactly select 2 defective products is ____. (Answer with a number)... | math | {
"style": "rule",
"ground_truth": "13968"
} | {
"split": "train",
"index": 4337,
"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 $y=f(x)$, evaluate $f(-\\frac{{5π}}... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 4338,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At a nursery, 2006 babies sit in a circle. Suddenly each baby pokes the baby immediately to either its left or its right, with equal probability. What is the expected number of unpoked babies? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1003}{2}"
} | {
"split": "train",
"index": 4339,
"solution": "The probability that any given baby goes unpoked is $1 / 4$. So the answer is $2006 / 4=1003 / 2$."
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{a\\_n\\}$ that satisfies $a\\_1=1$ and $a\\_n= \\frac{2S\\_n^2}{2S\\_n-1}$ for $n\\geqslant 2$, where $S\\_n$ is the sum of the first $n$ terms of the sequence, find the value of $S\\_{2016}$. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4031}"
} | {
"split": "train",
"index": 4340,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the share of the Japanese yen in the currency structure of the National Wealth Fund (NWF) as of 01.12.2022 using one of the following methods:\n\nFirst method:\n\na) Find the total amount of NWF funds placed in Japanese yen as of 01.12.2022:\n\n\\[ J P Y_{22} = 1388.0... | math | {
"style": "rule",
"ground_truth": "-12.6"
} | {
"split": "train",
"index": 4341,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(6,0)$, $(6,2)$, and $(0,2)$. What is the probability that $x + 1 < y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{24}"
} | {
"split": "train",
"index": 4342,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Zan has created this iterative rule for generating sequences of whole numbers:\n\n1) If a number is 25 or less, double the number.\n\n2) If a number is greater than 25, subtract 12 from it.\n\nLet $F$ be the first number in a sequence generated by the rule above. $F$ is a \"swe... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 4343,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are exactly three integers $x$ satisfying the inequality\n\\[x^2 + bx + 2 \\le 0.\\]How many integer values of $b$ are possible? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 4344,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(n)$ is the sum of the digits of $n^2+1$ (where $n \\in \\mathbb{N}^*$). For example, since $14^2+1=197$, and $1+9+7=17$, thus $f(14)=17$. Let $f_1(n)=f(n)$, $f_2(n)=f(f_1(n))$, ..., $f_{k+1}(n)=f(f_k(n))$, where $k \\in \\mathbb{N}^*$, then $f_{2005}(8)=$ ． Let's think s... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 4345,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f\\left( x \\right)=\\frac{{{e}^{x}}-a}{{{e}^{x}}+1}\\left( a\\in R \\right)$ defined on $R$ as an odd function.\n\n(1) Find the range of the function $y=f\\left( x \\right)$;\n\n(2) When ${{x}_{1}},{{x}_{2}}\\in \\left[ \\ln \\frac{1}{2},\\ln 2 \\right]$, t... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 4346,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$ABC$ is an equilateral triangle with sides equal to 2cm. $\\overline{BC}$ is extended its own length to $D$, and $E$ is the midpoint of $\\overline{AB}$. Suppose $\\overline{ED}$ meets $\\overline{AC}$ at $F$. Find the area of the quadrilateral $BEFC$ in square centimeters.\n\... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 4347,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest number, written using only ones and zeros, that would be divisible by 225. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11111111100"
} | {
"split": "train",
"index": 4348,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A X B Y$ be a cyclic quadrilateral, and let line $A B$ and line $X Y$ intersect at $C$. Suppose $A X \\cdot A Y=6, B X \\cdot B Y=5$, and $C X \\cdot C Y=4$. Compute $A B^{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{242}{15}"
} | {
"split": "train",
"index": 4349,
"solution": "Observe that $$\\begin{aligned} & \\triangle A C X \\sim \\triangle Y C B \\Longrightarrow \\frac{A C}{A X}=\\frac{C Y}{B Y} \\\\ & \\triangle A C Y \\sim \\triangle X C B \\Longrightarrow \\frac{A C}{A Y}=\\frac{C X}{B X} \\end{aligned}$$ Mulitplying these two equa... |
deepscaler | [
{
"role": "user",
"content": "Given $f(x)$ be a differentiable function, satisfying $\\lim_{x \\to 0} \\frac{f(1)-f(1-x)}{2x} = -1$, find the slope of the tangent line to the curve $y=f(x)$ at the point $(1, f(1))$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 4350,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be an isosceles trapezoid with $AD=BC$ and $AB<CD.$ Suppose that the distances from $A$ to the lines $BC,CD,$ and $BD$ are $15,18,$ and $10,$ respectively. Let $K$ be the area of $ABCD.$ Find $\\sqrt2 \\cdot K.$ Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "270"
} | {
"split": "train",
"index": 4351,
"solution": "1. **Identify the Trapezoid and Points**: Let $ABCD$ be an isosceles trapezoid with $AD = BC$ and $AB < CD$. Define $P$, $Q$, and $R$ as the feet of the perpendiculars from $A$ to $BC$, $CD$, and $BD$, respectively.\n\n2. **Use of Simson Line**: Since $ABCD$ is an i... |
deepscaler | [
{
"role": "user",
"content": "Given the symbol $R_k$ represents an integer whose base-ten representation is a sequence of $k$ ones, find the number of zeros in the quotient $Q=R_{24}/R_4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 4352,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Matt now needs to place five identical, dotless dominoes (shaded 1 by 2 rectangles) on a 6 by 5 grid. He wishes to create a path from the upper left-hand corner $C$ to the lower right-hand corner $D$. In the path, consecutive dominoes must touch at their sides and not just at t... | math | {
"style": "rule",
"ground_truth": "126"
} | {
"split": "train",
"index": 4353,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\frac{1}{3}$ of $x$ is equal to 4, what is $\\frac{1}{6}$ of $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 4354,
"solution": "Since $\\frac{1}{3}$ of $x$ is equal to 4, then $x$ is equal to $3 \\times 4$ or 12. Thus, $\\frac{1}{6}$ of $x$ is equal to $12 \\div 6=2$. Alternatively, since $\\frac{1}{6}$ is one-half of $\\frac{1}{3}$, then $\\frac{1}{6}$ of $x$ is equal to one-half of $\\frac... |
deepscaler | [
{
"role": "user",
"content": "Given the sets $A=\\{x|x^2 - mx + m^2 - 19 = 0\\}$, $B=\\{x|x^2 - 5x + 6 = 0\\}$, and $C=\\{2, -4\\}$. If $A \\cap B \\neq \\emptyset$ and $A \\cap C = \\emptyset$, find the value of the real number $m$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 4355,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the distance from point \\( M_{0} \\) to the plane passing through three points \\( M_{1}, M_{2}, M_{3} \\).\n\n\\( M_{1}(0, -1, -1) \\)\n\n\\( M_{2}(-2, 3, 5) \\)\n\n\\( M_{3}(1, -5, -9) \\)\n\n\\( M_{0}(-4, -13, 6) \\) Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{45}"
} | {
"split": "train",
"index": 4356,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that point P is on the curve $y=\\frac{1}{2}{{e}^{x}}$ and point Q is on the curve $y=x$, determine the minimum value of the distance $|PQ|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}(1 - \\ln 2)"
} | {
"split": "train",
"index": 4357,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\sin (x-\\frac{5π}{12})=\\frac{1}{3}$, find $\\cos (\\frac{2021π}{6}-2x)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{9}"
} | {
"split": "train",
"index": 4358,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle of radius $r$ has chords $\\overline{AB}$ of length $10$ and $\\overline{CD}$ of length 7. When $\\overline{AB}$ and $\\overline{CD}$ are extended through $B$ and $C$, respectively, they intersect at $P$, which is outside of the circle. If $\\angle{APD}=60^\\circ$ and ... | math | {
"style": "rule",
"ground_truth": "73"
} | {
"split": "train",
"index": 4359,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among the numbers $1^{2}, 2^{2}, 3^{2}, \\cdots, 95^{2}$, how many of these 95 numbers have an odd tens digit? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 4360,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A particle moves through the first quadrant as follows. During the first minute it moves from the origin to $(1,0)$. Thereafter, it continues to follow the directions indicated in the figure, going back and forth between the positive x and y axes, moving one unit of distance pa... | math | {
"style": "rule",
"ground_truth": "(44,35)"
} | {
"split": "train",
"index": 4361,
"solution": "1. **Understanding the Movement Pattern**: The particle starts at the origin and moves in a pattern that encloses squares of increasing size. Each square is $n \\times n$ where $n$ starts from 1 and increases by 1 for each new square. The movement pattern is such th... |
deepscaler | [
{
"role": "user",
"content": "Given that \\( p \\) is a prime number, the decimal part of \\( \\sqrt{p} \\) is \\( x \\). The decimal part of \\( \\frac{1}{x} \\) is \\( \\frac{\\sqrt{p} - 31}{75} \\). Find all prime numbers \\( p \\) that satisfy these conditions. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "2011"
} | {
"split": "train",
"index": 4362,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two circles of radius 3 are centered at $(3,0)$ and at $(0,3)$. What is the area of the intersection of the interiors of the two circles? Express your answer in fully expanded form in terms of $\\pi$. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "\\frac{9\\pi - 18}{2}"
} | {
"split": "train",
"index": 4363,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circle $C$ has radius 6 cm. How many square centimeters are in the area of the largest possible inscribed triangle having one side as a diameter of circle $C$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 4364,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine all six-digit numbers \\( p \\) that satisfy the following properties:\n(1) \\( p, 2p, 3p, 4p, 5p, 6p \\) are all six-digit numbers;\n(2) Each of the six-digit numbers' digits is a permutation of \\( p \\)'s six digits. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "142857"
} | {
"split": "train",
"index": 4365,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive multiples of nine are two-digit numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 4366,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sam and Lee run at equal and constant rates. They also cycle and skateboard at equal and constant rates. Sam covers $120$ kilometers after running for $4$ hours, cycling for $5$ hours, and skateboarding for $3$ hours while Lee covers $138$ kilometers after running for $5$ hours... | math | {
"style": "rule",
"ground_truth": "436"
} | {
"split": "train",
"index": 4367,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many digits are in the product $4^5 \\cdot 5^{10}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 4368,
"solution": "To find the number of digits in the product $4^5 \\cdot 5^{10}$, we first calculate each term separately and then use logarithmic properties to determine the number of digits in the product.\n\n1. **Calculate $4^5$:**\n \\[\n 4^5 = (2^2)^5 = 2^{10} = 1024\n \\... |
deepscaler | [
{
"role": "user",
"content": "The sides of rectangle $ABCD$ have lengths $12$ and $14$. An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD$. Find the maximum possible area of such a triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36\\sqrt{3}"
} | {
"split": "train",
"index": 4369,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If I have a $5\\times5$ chess board, in how many ways can I place five distinct pawns on the board such that each column and each row of the board contains no more than one pawn? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14400"
} | {
"split": "train",
"index": 4370,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ with right angle at $C$, $\\angle BAC < 45^\\circ$ and $AB = 4$. Point $P$ on $\\overline{AB}$ is chosen such that $\\angle APC = 2\\angle ACP$ and $CP = 1$. The ratio $\\frac{AP}{BP}$ can be represented in the form $p + q\\sqrt{r}$, where $p$, $q$, $r$ are posit... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 4371,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest possible sum of two consecutive integers whose product is less than 400? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 4372,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define the operation \"□\" as: $a□b=a^2+2ab-b^2$. Let the function $f(x)=x□2$, and the equation related to $x$ is $f(x)=\\lg|x+2|$ ($x\\neq -2$) has exactly four distinct real roots $x_1$, $x_2$, $x_3$, $x_4$. Find the value of $x_1+x_2+x_3+x_4$. Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"split": "train",
"index": 4373,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bug is on one exterior vertex of solid $S$, a $3 \\times 3 \\times 3$ cube that has its center $1 \\times 1 \\times 1$ cube removed, and wishes to travel to the opposite exterior vertex. Let $O$ denote the outer surface of $S$ (formed by the surface of the $3 \\times 3 \\time... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{29}}{3 \\sqrt{5}} \\text{ OR } \\frac{\\sqrt{145}}{15}"
} | {
"split": "train",
"index": 4374,
"solution": "By $\\left(^{*}\\right)$, the shortest route in $O$ has length $2 \\sqrt{1.5^{2}+3^{2}}=3 \\sqrt{5}$. By $\\left({ }^{* *}\\right)$, the shortest route overall (in $S$ ) has length $2 \\sqrt{1.5^{2}+1^{2}+2^{2}}=\\sqrt{3^{2}+2^{2}+4^{2}}=\\sqrt{29}$. Therefore the d... |
deepscaler | [
{
"role": "user",
"content": "In the AU tribe's language, there are two letters - \"a\" and \"u\". Certain sequences of these letters form words, where each word contains no fewer than one and no more than 13 letters. It is known that if any two words are written consecutively, the resulting sequence will n... | math | {
"style": "rule",
"ground_truth": "16256"
} | {
"split": "train",
"index": 4375,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Snow White has a row of 101 plaster dwarfs in her garden, arranged by weight from heaviest to lightest, with the weight difference between each pair of adjacent dwarfs being the same. Once, Snow White weighed the dwarfs and discovered that the first, heaviest dwarf weighs exact... | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"split": "train",
"index": 4376,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that four integers \\( a, b, c, d \\) are all even numbers, and \\( 0 < a < b < c < d \\), with \\( d - a = 90 \\). If \\( a, b, c \\) form an arithmetic sequence and \\( b, c, d \\) form a geometric sequence, then find the value of \\( a + b + c + d \\). Let's think step... | math | {
"style": "rule",
"ground_truth": "194"
} | {
"split": "train",
"index": 4377,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = 2\\sin x \\cos x + 2\\sqrt{3}\\cos^2 x$.\n(1) Find the smallest positive period of the function $f(x)$;\n(2) When $x \\in \\left[-\\frac{\\pi}{3}, \\frac{\\pi}{3}\\right]$, find the maximum and minimum values of the function $f(x)$. Let's think step b... | math | {
"style": "rule",
"ground_truth": "2 + \\sqrt{3}"
} | {
"split": "train",
"index": 4378,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( x[n] \\) denote \\( x \\) raised to the power of \\( x \\), repeated \\( n \\) times. What is the minimum value of \\( n \\) such that \\( 9[9] < 3[n] \\)? \n\n(For example, \\( 3[2] = 3^3 = 27 \\); \\( 2[3] = 2^{2^2} = 16 \\).) Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 4379,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The target below is made up of concentric circles with diameters $4$ , $8$ , $12$ , $16$ , and $20$ . The area of the dark region is $n\\pi$ . Find $n$ .\n\n[asy]\nsize(150);\ndefaultpen(linewidth(0.8));\nint i;\nfor(i=5;i>=1;i=i-1)\n{\nif (floor(i/2)==i/2)\n{\nfilldraw(... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 4380,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p,$ $q,$ $r$ be positive real numbers. Find the smallest possible value of\n\\[4p^3 + 6q^3 + 24r^3 + \\frac{8}{3pqr}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 4381,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the polynomial $f(x)=3x^{9}+3x^{6}+5x^{4}+x^{3}+7x^{2}+3x+1$, calculate the value of $v_{5}$ when $x=3$ using Horner's method. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "761"
} | {
"split": "train",
"index": 4382,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), the side lengths \\(AC = 14\\) and \\(AB = 6\\) are given. A circle with center \\(O\\), constructed on side \\(AC\\) as its diameter, intersects side \\(BC\\) at point \\(K\\). It is given that \\(\\angle BAK = \\angle ACB\\). Find the area of triangle \... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 4383,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many ordered pairs of positive integers $(x, y)$, with $x < y$, is the harmonic mean of $x$ and $y$ equal to $9^{15}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 4384,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a complex number $z$ with imaginary part $164$ and a positive integer $n$ such that\n\\[\\frac {z}{z + n} = 4i.\\]\n Find $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "697"
} | {
"split": "train",
"index": 4385,
"solution": "\\[\\frac {z}{z+n}=4i\\]\n\\[1-\\frac {n}{z+n}=4i\\]\n\\[1-4i=\\frac {n}{z+n}\\]\n\\[\\frac {1}{1-4i}=\\frac {z+n}{n}\\]\n\\[\\frac {1+4i}{17}=\\frac {z}{n}+1\\]\n Since their imaginary part has to be equal,\n\\[\\frac {4i}{17}=\\frac {164i}{n}\\]\n\\[n=\\frac {(164... |
deepscaler | [
{
"role": "user",
"content": "The length of the median to the hypotenuse of an isosceles, right triangle is $10$ units. What is the length of a leg of the triangle, in units? Express your answer in simplest radical form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10\\sqrt{2}"
} | {
"split": "train",
"index": 4386,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a parking lot, there are seven parking spaces numbered from 1 to 7. Now, two different trucks and two different buses are to be parked at the same time, with each parking space accommodating at most one vehicle. If vehicles of the same type are not parked in adjacent spaces,... | math | {
"style": "rule",
"ground_truth": "840"
} | {
"split": "train",
"index": 4387,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nThe square of a natural number \\( a \\) gives a remainder of 8 when divided by a natural number \\( n \\). The cube of the number \\( a \\) gives a remainder of 25 when divided by \\( n \\). Find \\( n \\). Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "113"
} | {
"split": "train",
"index": 4388,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular tetrahedron has an edge length of $2$. Calculate the surface area of the sphere circumscribed around this tetrahedron. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6\\pi"
} | {
"split": "train",
"index": 4389,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle has a radius of 3 units. A line segment of length 3 units is tangent to the circle at its midpoint. Determine the area of the region consisting of all such line segments.\nA) $1.5\\pi$\nB) $2.25\\pi$\nC) $3\\pi$\nD) $4.5\\pi$ Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "2.25\\pi"
} | {
"split": "train",
"index": 4390,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(a) A natural number \\( n \\) is less than 120. What is the maximum remainder that the number 209 can leave when divided by \\( n \\)?\n\n(b) A natural number \\( n \\) is less than 90. What is the maximum remainder that the number 209 can leave when divided by \\( n \\)? Let'... | math | {
"style": "rule",
"ground_truth": "69"
} | {
"split": "train",
"index": 4391,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mary typed a six-digit number, but the two $1$ s she typed didn't show. What appeared was $2002$ . How many different six-digit numbers could she have typed? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 4392,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Fiona is people-watching again. She spies a group of ten high schoolers and starts playing a game by herself, in which she looks at a pair of people from the group of ten and tries to guess whether they like or dislike each other. How many pairs of friends can she observe befor... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 4393,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three students $A, B$ and $C$ are traveling from a location on the National Highway No. $5$ on direction to Hanoi for participating the HOMC $2018$ . At beginning, $A$ takes $B$ on the motocycle, and at the same time $C$ rides the bicycle. After one hour and a half,... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 4394,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $r$ and $s$ be positive integers such that\\[\\frac{5}{11} < \\frac{r}{s} < \\frac{4}{9}\\]and $s$ is as small as possible. What is $s - r$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 4395,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. Given that the volume of this set is $\\displaystyle {{m+n\\pi}\\over\np}$, where $m$, $n$, and $p$ are positive integers, and $n$ and $p$ are re... | math | {
"style": "rule",
"ground_truth": "505"
} | {
"split": "train",
"index": 4396,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $0 < α < \\dfrac {π}{2}$, and $\\cos (2π-α)-\\sin (π-α)=- \\dfrac { \\sqrt {5}}{5}$,\n(1) Find the value of $\\sin α+\\cos α$;\n(2) Find the value of $\\sin (2α- \\dfrac {π}{4})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac {7 \\sqrt {2}}{10}"
} | {
"split": "train",
"index": 4397,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all real roots \\( x \\) of the equation \\( \\left(2^{x}-4\\right)^{3}+\\left(4^{x}-2\\right)^{3}=\\left(4^{x}+2^{x}-6\\right)^{3} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3.5"
} | {
"split": "train",
"index": 4398,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the base ten equivalent of $54321_6$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7465"
} | {
"split": "train",
"index": 4399,
"solution": ""
} |
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