data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(2,0)$, $(2,2)$, and $(0,2)$. What is the probability that $x^2 + y^2 < y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{32}"
} | {
"split": "train",
"index": 5200,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the function $f(x+1)$ is an odd function, and the function $f(x-1)$ is an even function, and $f(0) = 2$, determine the value of $f(4)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 5201,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of points $(a, b)$ with $0 \\leq a, b \\leq 1$ such that the equation $x^{4}+a x^{3}-b x^{2}+a x+1=0$ has at least one real root. Determine the area of the graph of $S$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 5202,
"solution": "After dividing the equation by $x^{2}$, we can rearrange it as $(x+\\frac{1}{x})^{2}+a(x+\\frac{1}{x})-b-2=0$. Let $y=x+\\frac{1}{x}$. We can check that the range of $x+\\frac{1}{x}$ as $x$ varies over the nonzero reals is $(-\\infty,-2] \\cup[2, \\infty)$. Thus, th... |
deepscaler | [
{
"role": "user",
"content": "Three circles of radius 1 are externally tangent to each other and internally tangent to a larger circle. What is the radius of the large circle? Express your answer as a common fraction in simplest radical form.\n\n[asy]\ndraw(Circle((0,-0.58),2.15),linewidth(0.7));\ndraw(Circ... | math | {
"style": "rule",
"ground_truth": "\\frac{3+2\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 5203,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A company needs to transport two types of products, $A$ and $B$, with the following volumes and masses per unit as shown in the table:\n\n| | Volume $(m^{3}/$unit) | Mass (tons$/$unit) |\n|----------|-----------------------|--------------------|\n| $A$ type | $0.8$ ... | math | {
"style": "rule",
"ground_truth": "2100"
} | {
"split": "train",
"index": 5204,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Borya and Vova play the following game on an initially white $8 \\times 8$ board. Borya goes first and, on each of his turns, colors any four white cells black. After each of his turns, Vova colors an entire row or column white. Borya aims to color as many cells black as possib... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 5205,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain city, vehicle license plates are numbered consecutively from \"10000\" to \"99999\". How many license plates out of these 90,000 have the digit 9 appearing at least once and where the sum of the digits is a multiple of 9? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "4168"
} | {
"split": "train",
"index": 5206,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a$, $b$, $c$, $d$, $e$, and $f$ are integers such that $8x^3 + 64 = (ax^2 + bx + c)(dx^2 + ex + f)$ for all $x$, then what is $a^2 + b^2 + c^2 + d^2 + e^2 + f^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "356"
} | {
"split": "train",
"index": 5207,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 32 ones written on the board. Each minute, Carlsson erases any two numbers, writes their sum on the board, and then eats an amount of candy equal to the product of the two erased numbers. What is the maximum number of candies he could eat in 32 minutes? Let's think st... | math | {
"style": "rule",
"ground_truth": "496"
} | {
"split": "train",
"index": 5208,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A self-employed person plans to distribute two types of goods, A and B. According to a survey, when the investment amount is $x(x\\geqslant 0)$ ten thousand yuan, the profits obtained from distributing goods A and B are $f(x)$ ten thousand yuan and $g(x)$ ten thousand yuan, res... | math | {
"style": "rule",
"ground_truth": "12.6"
} | {
"split": "train",
"index": 5209,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A chest of gold coins is divided among 10 pirates where the kth pirate takes k/10 of the remaining coins. Determine the smallest number of coins initially in the chest such that each pirate gets a positive whole number of coins and find the number of coins the 10th pirate recei... | math | {
"style": "rule",
"ground_truth": "362880"
} | {
"split": "train",
"index": 5210,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S$ be the set of points $(x, y)$ in the Cartesian plane that satisfy\n\\[\\Big|\\big| |x|-2\\big|-1\\Big|+\\Big|\\big| |y|-2\\big|-1\\Big|=1.\\]What is the total length of all the lines that make up $S$? Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "64\\sqrt{2}"
} | {
"split": "train",
"index": 5211,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two squares of a $7\\times 7$ checkerboard are painted yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane board. How many inequivalent color schemes are possible? Let's think step by... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 5212,
"solution": "There are 4 cases: 1. The center square is occupied, in which there are $12$ cases. 2. The center square isn't occupied and the two squares that are opposite to each other with respect to the center square, in which there are $12$ cases. 3. The center square isn't o... |
deepscaler | [
{
"role": "user",
"content": "In some 16 cells of an $8 \\times 8$ board, rooks are placed. What is the minimum number of pairs of rooks that can attack each other in this configuration? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 5213,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\frac{\\ln{x}}{x + a}$ where $a \\in \\mathbb{R}$: \n(1) If the tangent to the curve $y = f(x)$ at the point $(1, f(1))$ is perpendicular to the line $x + y + 1 = 0$, find the value of $a$. \n(2) Discuss the number of real roots of the equation $f(... | math | {
"style": "rule",
"ground_truth": "a = 0"
} | {
"split": "train",
"index": 5214,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ramanujan and Hardy played a game where they both picked a complex number. If the product of their numbers was $32-8i$, and Hardy picked $5+3i$, what number did Ramanujan pick? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4-4i"
} | {
"split": "train",
"index": 5215,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In three-dimensional space, find the number of lattice points that have a distance of 4 from the origin. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 5216,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a$, $b$, $c$, and $d$ are real numbers satisfying:\n\\begin{align*}\na+b+c &= 3, \\\\\na+b+d &= -2, \\\\\na+c+d &= 8, \\text{ and} \\\\\nb+c+d &= -1,\n\\end{align*}\nwhat is $ab + cd$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{190}{9}"
} | {
"split": "train",
"index": 5217,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the equation \\(2x^3 - 7x^2 + 7x + p = 0\\) has three distinct roots, and these roots form a geometric progression. Find \\(p\\) and solve this equation. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 5218,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiaopang and Xiaoya both have their birthdays in May, and both fall on a Wednesday. Xiaopang's birthday is later, and the sum of their birth dates is 38. What is Xiaopang's birthday in May? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 5219,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On this 5 by 5 grid of points, what fraction of the larger square's area is inside the shaded square? Express your answer as a common fraction.\n\n[asy]\nfill((2,2)--(3,3)--(2,4)--(1,3)--cycle,gray(0.7));\ndot((0,0));\ndot((0,1));\ndot((0,2));\ndot((0,3));\ndot((0,4));\ndot((1,... | math | {
"style": "rule",
"ground_truth": "\\frac 18"
} | {
"split": "train",
"index": 5220,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each pair of real numbers $a \\ne b$, define the operation $\\star$ as \\[\n(a \\star b) = \\frac{a + b}{a - b}.\n\\]What is the value of $((1 \\star 2) \\star 4)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{7}"
} | {
"split": "train",
"index": 5221,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a unique two-digit positive integer $t$ for which the last two digits of $13 \\cdot t$ are $26$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "62"
} | {
"split": "train",
"index": 5222,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Shelby drives her scooter at a speed of $30$ miles per hour if it is not raining, and $20$ miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of $16$ miles in $40$ minutes. How many minutes did she drive in the... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 5223,
"solution": "1. **Define Variables:**\n Let $x$ be the time (in minutes) Shelby drove in the rain. Therefore, the time she drove when it was not raining is $40 - x$ minutes.\n\n2. **Convert Speeds to Miles per Minute:**\n - Shelby's speed in non-rainy conditions is $30$ mile... |
deepscaler | [
{
"role": "user",
"content": "In the diagram below, $\\overline{AB}\\parallel \\overline{CD}$ and $\\angle AXE$ is $108^\\circ$ less than 3 times $\\angle CYX$. Find $\\angle BXY$.\n\n[asy]\n\nunitsize(1inch);\n\npair A,B,C,D,X,Y,EE,F;\n\nA = (0,0);\n\nB=(1,0);\n\nC = (0,0.8);\n\nD=(1,0.8);\n\nEE = (0.35,-... | math | {
"style": "rule",
"ground_truth": "54^\\circ"
} | {
"split": "train",
"index": 5224,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a row of 10 chairs, one of which is broken and cannot be used, Mary and James randomly select their seats. What is the probability that they do not sit next to each other? 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": 5225,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given points E and D are on sides AB and BC of triangle ABC, where AE:EB=1:3 and CD:DB=1:2, find the value of EF/FC + AF/FD. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 5226,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $a+b+c=0$ and $a^2+b^2+c^2=1$, find the values of $ab+bc+ca$ and $a^4+b^4+c^4$. 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": 5227,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the equilateral triangle \\(ABC\\), point \\(T\\) is its centroid, point \\(R\\) is the reflection of \\(T\\) across the line \\(AB\\), and point \\(N\\) is the reflection of \\(T\\) across the line \\(BC\\).\n\nDetermine the ratio of the areas of triangles \\(ABC\\) and \\(... | math | {
"style": "rule",
"ground_truth": "3:1"
} | {
"split": "train",
"index": 5228,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the square roots of a number are $2a+3$ and $a-18$, then this number is ____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "169"
} | {
"split": "train",
"index": 5229,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a $5 \\times 5$ grid of squares. Vladimir colors some of these squares red, such that the centers of any four red squares do not form an axis-parallel rectangle (i.e. a rectangle whose sides are parallel to those of the squares). What is the maximum number of squares h... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 5230,
"solution": "We claim that the answer is 12. We first show that if 13 squares are colored red, then some four form an axis-parallel rectangle. Note that we can swap both columns and rows without affecting whether four squares form a rectangle, so we may assume without loss of ge... |
deepscaler | [
{
"role": "user",
"content": "Let $a$, $b$, and $c$ be positive real numbers. Find the minimum value of\n\\[\n\\frac{(a^2 + 4a + 4)(b^2 + 4b + 4)(c^2 + 4c + 4)}{abc}.\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 5231,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a right circular cone ($S-ABC$), $SA =2$, the midpoints of $SC$ and $BC$ are $M$ and $N$ respectively, and $MN \\perp AM$. Determine the surface area of the sphere that circumscribes the right circular cone ($S-ABC$). Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "12\\pi"
} | {
"split": "train",
"index": 5232,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle of radius 3 is centered at point $A$. An equilateral triangle with side length 6 has one vertex tangent to the edge of the circle at point $A$. Calculate the difference between the area of the region that lies inside the circle but outside the triangle and the area of ... | math | {
"style": "rule",
"ground_truth": "9(\\sqrt{3} - \\pi)"
} | {
"split": "train",
"index": 5233,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f\\left( x \\right)=\\sqrt{3}\\sin\\left( \\omega x-\\frac{\\pi }{6} \\right)(\\omega > 0)$, the distance between two adjacent highest points on the graph is $\\pi$.\n\n(1) Find the value of $\\omega$ and the equation of the axis of symmetry for the function... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{5}-1}{8}"
} | {
"split": "train",
"index": 5234,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be the number of ways $10$ dollars can be changed into dimes and quarters, with at least one of each coin being used. Then $n$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 5235,
"solution": "To solve this problem, we need to find the number of ways to express $10$ dollars using dimes ($0.10$ dollars) and quarters ($0.25$ dollars), with the condition that at least one of each type of coin is used.\n\n1. **Convert the total amount into cents for simplicit... |
deepscaler | [
{
"role": "user",
"content": "Peter borrows $2000$ dollars from John, who charges an interest of $6\\%$ per month (which compounds monthly). What is the least integer number of months after which Peter will owe more than three times as much as he borrowed? Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 5236,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the figure, $\\overline{DC}$ is parallel to $\\overline{AB}$. We have $\\angle DCA = 40^\\circ$ and $\\angle ABC = 73^\\circ$. Find $\\angle ACB$.\n\n[asy]\npair A,B,C,D;\nA = (0,0);\nB= (1,0);\nD = (-0.1,0.6);\nC = (0.8,0.6);\ndraw((-0.2,0)--(1.2,0));\ndraw((-0.2,0.6)--(1.... | math | {
"style": "rule",
"ground_truth": "67^\\circ"
} | {
"split": "train",
"index": 5237,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A given odd function $f(x)$, defined on $\\mathbb{R}$, is symmetric about the line $x=1$, and $f(-1) = 1$. Find the value of $f(1) + f(2) + f(3) + \\ldots + f(2009)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 5238,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice and Bob play a game around a circle divided into 15 equally spaced points, numbered 1 through 15. Alice moves 7 points clockwise per turn, and Bob moves 4 points counterclockwise per turn. Determine how many turns will be required for Alice and Bob to land on the same poi... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 5239,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $81^{3/4}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 5240,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Which digit is represented by $\\Theta$ if $252/\\Theta=\\underline{3\\Theta}+\\Theta$, where $\\underline{3\\Theta}$ represents a two-digit number with $3$ in the tens digit and $\\Theta$ in the ones digit? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 5241,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the average number of pairs of consecutive integers in a randomly selected subset of $5$ distinct integers chosen from the set $\\{ 1, 2, 3, \\dots, 30\\}$? (For example the set $\\{1, 17, 18, 19, 30\\}$ has $2$ pairs of consecutive integers.) Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 5242,
"solution": "We define an outcome as $\\left( a_1 ,\\cdots, a_5 \\right)$ with $1 \\leq a_1 < a_2 < a_3 < a_4 < a_5 \\leq 30$.\nWe denote by $\\Omega$ the sample space. Hence, $| \\Omega | = \\binom{30}{5}$.\n\n#### Case 1: There is only 1 pair of consecutive integers.\n- **Case... |
deepscaler | [
{
"role": "user",
"content": "Let $A$ and $B$ be two subsets of the set $\\{1,2, \\cdots, 20\\}$ such that $A \\cap B = \\varnothing$, and if $n \\in A$, then $2n + 2 \\in B$. Let $M(A)$ be the sum of the elements in $A$. Find the maximum value of $M(A)$. Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 5243,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the smallest positive integer $n \\geq 3$ for which $$A \\equiv 2^{10 n}\\left(\\bmod 2^{170}\\right)$$ where $A$ denotes the result when the numbers $2^{10}, 2^{20}, \\ldots, 2^{10 n}$ are written in decimal notation and concatenated (for example, if $n=2$ we have $A... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 5244,
"solution": "Note that $$2^{10 n}=1024^{n}=1.024^{n} \\times 10^{3 n}$$ So $2^{10 n}$ has roughly $3 n+1$ digits for relatively small $n$'s. (Actually we have that for $0<x<1$, $$(1+x)^{2}=1+2 x+x^{2}<1+3 x$$ Therefore, $1.024^{2}<1.03^{2}<1.09,1.09^{2}<1.27,1.27^{2}<1.81<2$, an... |
deepscaler | [
{
"role": "user",
"content": "Four A's, four B's, four C's, and four D's are placed in the sixteen spaces of a 4x4 grid so that each row and column contains one of each letter. If A is placed in the upper left corner, determine the number of possible arrangements. Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 5245,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let there be three individuals, labeled A, B, and C, to be allocated across seven laboratories in preparation for experiments. Each laboratory can accommodate no more than two people. Calculate the total number of distinct allocation schemes. Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "336"
} | {
"split": "train",
"index": 5246,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\tan \\alpha= \\sqrt {2}$, then $2\\sin ^{2}\\alpha-\\sin \\alpha\\cos \\alpha+\\cos ^{2}\\alpha=$ \\_\\_\\_\\_\\_\\_ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {5- \\sqrt {2}}{3}"
} | {
"split": "train",
"index": 5247,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The four-member math team at Pecanridge Middle School is chosen from the math club, which has three girls and five boys. How many different teams made up of two girls and two boys could be chosen? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 5248,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of $\\frac{2468_{10}}{111_{3}} - 3471_{9} + 1234_{7}$. Express your answer in base 10. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1919"
} | {
"split": "train",
"index": 5249,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $F$ is taken on the extension of side $AD$ of rectangle $ABCD$. $BF$ intersects diagonal $AC$ at $E$ and side $DC$ at $G$. If $EF = 40$ and $GF = 15$, then $BE$ equals:\n[Insert diagram similar to above, with F relocated, set different values for EF and GF] Let's think st... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 5250,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the arithmetic square root of $m$ is $3$, and the square roots of $n$ are $a+4$ and $2a-16$. \n$(1)$ Find the values of $m$ and $n$. \n$(2)$ Find $\\sqrt[3]{{7m-n}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 5251,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a sequence $x_1,x_2,\\ldots,x_n$ of real numbers, we define its $\\textit{price}$ as \\[\\max_{1\\le i\\le n}|x_1+\\cdots +x_i|.\\] Given $n$ real numbers, Dave and George want to arrange them into a sequence with a low price. Diligent Dave checks all possible ways and find... | math | {
"style": "rule",
"ground_truth": "c=2"
} | {
"split": "train",
"index": 5252,
"solution": "\nLet's consider the problem of arranging a sequence of \\( n \\) real numbers to minimize the \\textit{price} defined as:\n\\[\n\\max_{1 \\leq i \\leq n} \\left| x_1 + x_2 + \\cdots + x_i \\right|.\n\\]\n\nDave's approach determines the optimal sequence with the mi... |
deepscaler | [
{
"role": "user",
"content": "The postal department stipulates that for letters weighing up to $100$ grams (including $100$ grams), each $20$ grams requires a postage stamp of $0.8$ yuan. If the weight is less than $20$ grams, it is rounded up to $20$ grams. For weights exceeding $100$ grams, the initial po... | math | {
"style": "rule",
"ground_truth": "5.6"
} | {
"split": "train",
"index": 5253,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a positive integer $n$ , let \n\n\\[S_n=\\int_0^1 \\frac{1-(-x)^n}{1+x}dx,\\ \\ T_n=\\sum_{k=1}^n \\frac{(-1)^{k-1}}{k(k+1)}\\]\n\nAnswer the following questions:\n\n(1) Show the following inequality.\n\n\\[\\left|S_n-\\int_0^1 \\frac{1}{1+x}dx\\right|\\leq \\frac{1}{n+1}\... | math | {
"style": "rule",
"ground_truth": "2 \\ln 2 - 1"
} | {
"split": "train",
"index": 5254,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m$ and $n$ be positive integers satisfying the conditions\n$\\quad\\bullet\\ \\gcd(m+n,210)=1,$\n$\\quad\\bullet\\ m^m$ is a multiple of $n^n,$ and\n$\\quad\\bullet\\ m$ is not a multiple of $n.$\nFind the least possible value of $m+n.$\n Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "407"
} | {
"split": "train",
"index": 5255,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $31.25 : x : 13.75 = 100 : 9.6 : y$, find the value of $x + y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 5256,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a > 0$, $b > 0$, and $\\frac{1}{a}$, $\\frac{1}{2}$, $\\frac{1}{b}$ form an arithmetic sequence, find the minimum value of $a+9b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 5257,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Gari is seated in a jeep, and at the moment, has one 10-peso coin, two 5-peso coins, and six 1-peso coins in his pocket. If he picks four coins at random from his pocket, what is the probability that these will be enough to pay for his jeepney fare of 8 pesos? Let's think step ... | math | {
"style": "rule",
"ground_truth": "37/42"
} | {
"split": "train",
"index": 5258,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an after-school program for juniors and seniors, there is a debate team with an equal number of students from each class on the team. Among the $28$ students in the program, $25\\%$ of the juniors and $10\\%$ of the seniors are on the debate team. How many juniors are in the... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 5259,
"solution": "1. **Define Variables:**\n Let $x$ be the number of juniors on the debate team and $y$ be the number of seniors on the debate team. Given that the number of juniors and seniors on the debate team is equal, we have $x = y$.\n\n2. **Percentage Relationships:**\n -... |
deepscaler | [
{
"role": "user",
"content": "How many positive factors of 36 are also multiples of 4? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 5260,
"solution": "To solve this problem, we first need to find the prime factorization of 36. Then, we will determine which factors of 36 are also multiples of 4.\n\n1. **Prime Factorization of 36**:\n \\[\n 36 = 2^2 \\cdot 3^2\n \\]\n This factorization shows that 36 is comp... |
deepscaler | [
{
"role": "user",
"content": "The sum of seven consecutive even numbers is 686. What is the smallest of these seven numbers? Additionally, calculate the median and mean of this sequence. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "98"
} | {
"split": "train",
"index": 5261,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be a triangle with $C A=C B=5$ and $A B=8$. A circle $\\omega$ is drawn such that the interior of triangle $A B C$ is completely contained in the interior of $\\omega$. Find the smallest possible area of $\\omega$. Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "16 \\pi"
} | {
"split": "train",
"index": 5262,
"solution": "We need to contain the interior of $\\overline{A B}$, so the diameter is at least 8. This bound is sharp because the circle with diameter $\\overline{A B}$ contains all of $A B C$. Hence the minimal area is $16 \\pi$."
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $(2^8 + 4^5)(2^3 - (-2)^3)^7$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1280 \\cdot 16^7"
} | {
"split": "train",
"index": 5263,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the greatest common divisor of $8!$ and $9!$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40320"
} | {
"split": "train",
"index": 5264,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Several energy-saving devices with a total weight of 120 kg were delivered to the factory. It is known that the total weight of the three lightest devices is 31 kg, and the total weight of the three heaviest devices is 41 kg. How many energy-saving devices were delivered to the... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 5265,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For every integer $n \\ge 1$ , the function $f_n : \\left\\{ 0, 1, \\cdots, n \\right\\} \\to \\mathbb R$ is defined recursively by $f_n(0) = 0$ , $f_n(1) = 1$ and \\[ (n-k) f_n(k-1) + kf_n(k+1) = nf_n(k) \\] for each $1 \\le k < n$ . Let $S_N = f_{N+1}(1) + f_{N+2}(2)... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 5266,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the equation $x^3 - 12x^2 + 27x - 18 = 0$ with roots $a$, $b$, $c$, find the value of $\\frac{1}{a^3} + \\frac{1}{b^3} + \\frac{1}{c^3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{24}"
} | {
"split": "train",
"index": 5267,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function\n\\[ f(x) = x^2 - (k^2 - 5ak + 3)x + 7 \\quad (a, k \\in \\mathbb{R}) \\]\nfor any \\( k \\in [0, 2] \\), if \\( x_1, x_2 \\) satisfy\n\\[ x_1 \\in [k, k+a], \\quad x_2 \\in [k+2a, k+4a], \\]\nthen \\( f(x_1) \\geq f(x_2) \\). Find the maximum value of the po... | math | {
"style": "rule",
"ground_truth": "\\frac{2 \\sqrt{6} - 4}{5}"
} | {
"split": "train",
"index": 5268,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T = (1+i)^{19} - (1-i)^{19}$, where $i=\\sqrt{-1}$. Determine $|T|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "512\\sqrt{2}"
} | {
"split": "train",
"index": 5269,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Factor $w^4-16$ as far as possible, where the factors are monic polynomials with real coefficients. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(w-2)(w+2)(w^2+4)"
} | {
"split": "train",
"index": 5270,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the function $g(x)$ represented by the line segments in the graph below. The graph consists of four line segments as follows: connecting points (-3, -4) to (-1, 0), (-1, 0) to (0, -1), (0, -1) to (2, 3), and (2, 3) to (3, 2). Find the sum of the $x$-coordinates where $... | math | {
"style": "rule",
"ground_truth": "-1.5"
} | {
"split": "train",
"index": 5271,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six positive integers are written on the faces of a cube. Each vertex is labeled with the product of the three numbers on the faces adjacent to that vertex. If the sum of the numbers on the vertices is equal to $1287$, what is the sum of the numbers written on the faces? Let's ... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 5272,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The points on this graph show the end-of-year sales amounts for each year. During what year after 1994 did sales increase the most number of dollars? Express your answer as a 4-digit year.\n\n[asy]unitsize(101/10);\nreal textsize = 10, largetext = 12;\ndraw((10,0)--(0,0)--(0,1... | math | {
"style": "rule",
"ground_truth": "1998"
} | {
"split": "train",
"index": 5273,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a $40\\%$ chance of rain on Saturday and a $30\\%$ chance of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is $\\frac{a}{b}$, wh... | math | {
"style": "rule",
"ground_truth": "107"
} | {
"split": "train",
"index": 5274,
"solution": "Let $x$ be the probability that it rains on Sunday given that it doesn't rain on Saturday. We then have $\\dfrac{3}{5}x+\\dfrac{2}{5}2x = \\dfrac{3}{10} \\implies \\dfrac{7}{5}x=\\dfrac{3}{10}$ $\\implies x=\\dfrac{3}{14}$. Therefore, the probability that it doesn't... |
deepscaler | [
{
"role": "user",
"content": "In trapezoid $PQRS$, the lengths of the bases $PQ$ and $RS$ are 10 and 20, respectively. The height of the trapezoid from $PQ$ to $RS$ is 6 units. The legs of the trapezoid are extended beyond $P$ and $Q$ to meet at point $T$. What is the ratio of the area of triangle $TPQ$ to ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 5275,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x$ is positive and $x^2 = 729$, 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": "27"
} | {
"split": "train",
"index": 5276,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A chord of a circle is perpendicular to a radius at the midpoint of the radius. The ratio of the area of the larger of the two regions into which the chord divides the circle to the smaller can be expressed in the form $\\frac{a\\pi+b\\sqrt{c}}{d\\pi-e\\sqrt{f}},$ where $a, b, ... | math | {
"style": "rule",
"ground_truth": "592"
} | {
"split": "train",
"index": 5277,
"solution": "Let $r$ be the length of the radius of the circle. A right triangle is formed by half of the chord, half of the radius (since the chord bisects it), and the radius. Thus, it is a $30^\\circ$ - $60^\\circ$ - $90^\\circ$ triangle, and the area of two such triangles is... |
deepscaler | [
{
"role": "user",
"content": "A positive integer is said to be bi-digital if it uses two different digits, with each digit used exactly twice. For example, 1331 is bi-digital, whereas 1113, 1111, 1333, and 303 are not. Determine the exact value of the integer \\( b \\), the number of bi-digital positive int... | math | {
"style": "rule",
"ground_truth": "243"
} | {
"split": "train",
"index": 5278,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Team A and Team B each have $n$ members. It is known that each member of Team A shakes hands with each member of Team B exactly once (members within the same team do not shake hands). From these $n^2$ handshakes, two are randomly selected. Let event A be the event that exactly ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 5279,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a regular tetrahedron $P-ABC$, where points $P$, $A$, $B$, and $C$ are all on the surface of a sphere with radius $\\sqrt{3}$. If $PA$, $PB$, and $PC$ are mutually perpendicular, calculate the distance from the center of the sphere to the plane $ABC$. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 5280,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 2011 positive numbers with both their sum and the sum of their reciprocals equal to 2012. Let $x$ be one of these numbers. Find the maximum value of $x + \\frac{1}{x}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8045}{2012}"
} | {
"split": "train",
"index": 5281,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an isosceles trapezoid, one base is \\(40 \\text{ cm}\\) and the other is \\(24 \\text{ cm}\\). The diagonals of this trapezoid are mutually perpendicular. Find its area. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1024"
} | {
"split": "train",
"index": 5282,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Numbering the pages of an encyclopedia required 6869 digits. How many pages does it contain? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1994"
} | {
"split": "train",
"index": 5283,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the vector $$\\overrightarrow {a_{k}} = (\\cos \\frac {k\\pi}{6}, \\sin \\frac {k\\pi}{6} + \\cos \\frac {k\\pi}{6})$$ for k=0, 1, 2, …, 12, find the value of $$\\sum\\limits_{k=0}^{11} (\\overrightarrow {a_{k}} \\cdot \\overrightarrow {a_{k+1}})$$. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "9\\sqrt{3}"
} | {
"split": "train",
"index": 5284,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A box contains six cards. Three of the cards are black on both sides, one card is black on one side and red on the other, and two of the cards are red on both sides. You pick a card uniformly at random from the box and look at a random side. Given that the side you see is red... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"split": "train",
"index": 5285,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( f(x) = x^2 + px + q \\). It is known that the inequality \\( |f(x)| > \\frac{1}{2} \\) has no solutions on the interval \\([3, 5]\\). Find \\(\\underbrace{f(f(\\ldots f}_{2017}\\left(\\frac{7+\\sqrt{15}}{2}\\right)) \\ldots)\\). Round the answer to hundredths if necessa... | math | {
"style": "rule",
"ground_truth": "1.56"
} | {
"split": "train",
"index": 5286,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "While finding the sine of a certain angle, an absent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. The two least positive real values of $x$ for which the sine of $x$ degrees is the same as the s... | math | {
"style": "rule",
"ground_truth": "900"
} | {
"split": "train",
"index": 5287,
"solution": "Note that $x$ degrees is equal to $\\frac{\\pi x}{180}$ radians. Also, for $\\alpha \\in \\left[0 , \\frac{\\pi}{2} \\right]$, the two least positive angles $\\theta > \\alpha$ such that $\\sin{\\theta} = \\sin{\\alpha}$ are $\\theta = \\pi-\\alpha$, and $\\theta = ... |
deepscaler | [
{
"role": "user",
"content": "Distribute 5 students into 3 groups: Group A, Group B, and Group C, with Group A having at least two people, and Groups B and C having at least one person each, and calculate the number of different distribution schemes. Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 5288,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Regarding the value of \\\\(\\pi\\\\), the history of mathematics has seen many creative methods for its estimation, such as the famous Buffon's Needle experiment and the Charles' experiment. Inspired by these, we can also estimate the value of \\\\(\\pi\\\\) through designing ... | math | {
"style": "rule",
"ground_truth": "\\dfrac {78}{25}"
} | {
"split": "train",
"index": 5289,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the parabola $y=x^2-5x+2$ is symmetric about the point $(3,2)$ with $y=ax^2+bx+c$, find the value of $3a+3c+b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"split": "train",
"index": 5290,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the approximate change in the function \\( y = 3x^{2} + 2 \\) at \\( x = 2 \\) with \\( \\Delta x = 0.001 \\). Determine the absolute and relative errors of the calculation. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.012"
} | {
"split": "train",
"index": 5291,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the difference between the sum of the numbers $3$, $-4$, and $-5$ and the sum of their absolute values. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-18"
} | {
"split": "train",
"index": 5292,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $n$ is a positive integer, and given that $\\mathop{\\text{lcm}}[24,n]=72$ and $\\mathop{\\text{lcm}}[n,27]=108$, what is $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 5293,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, the length of the side opposite angle $A$ is equal to 2, the vector $\\overrightarrow {m} = (2, 2\\cos^2 \\frac {B+C}{2}-1)$, and the vector $\\overrightarrow {n} = (\\sin \\frac {A}{2}, -1)$. \n(1) Find the size of angle $A$ when the dot product ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 5294,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all the solutions to $2^{|x|} + 3|x| = 18.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 5295,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the coefficient of $x^2$ in the expansion of $(1+ax)(1+x)^5$ is $5$, determine the value of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 5296,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A hyperbola in the coordinate plane passing through the points $(2,5)$ , $(7,3)$ , $(1,1)$ , and $(10,10)$ has an asymptote of slope $\\frac{20}{17}$ . The slope of its other asymptote can be expressed in the form $-\\frac{m}{n}$ , where $m$ and $n$ are relatively pr... | math | {
"style": "rule",
"ground_truth": "1720"
} | {
"split": "train",
"index": 5297,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the quadratic function $f(x)=x^{2}+mx+n$.\n(1) If $f(x)$ is an even function with a minimum value of $1$, find the analytic expression of $f(x)$;\n(2) Under the condition of (1), for the function $g(x)= \\frac {6x}{f(x)}$, solve the inequality $g(2^{x}) > 2^{x}$ with resp... | math | {
"style": "rule",
"ground_truth": "\\frac {1}{2}"
} | {
"split": "train",
"index": 5298,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\cos 270^\\circ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 5299,
"solution": ""
} |
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