data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Given two integers $a$ and $b$, if they are not coprime and neither is a multiple of the other, they are called a \"league\" pair. Let $A$ be an $n$-element subset of the set $M = \\{1, 2, \\cdots, 2014\\}$ such that every pair of numbers in $A$ is a league pair. Determine the ... | math | {
"style": "rule",
"ground_truth": "504"
} | {
"split": "train",
"index": 31249,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define $a \\Delta b = a^2 -b $. What is the value of $ (2^{4 \\Delta13})\\Delta(3^{3\\Delta5})$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-17"
} | {
"split": "train",
"index": 31250,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the board is written the number 98. Every minute the number is erased and replaced with the product of its digits increased by 15. What number will be on the board in an hour? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 31251,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the set of three-digit numbers composed of \\\\(0\\\\), \\\\(1\\\\), \\\\(2\\\\), and \\\\(3\\\\) without repeating digits be \\\\(A\\\\). If a number is randomly selected from \\\\(A\\\\), the probability that the number is exactly even is. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "\\dfrac{5}{9}"
} | {
"split": "train",
"index": 31252,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify first, then evaluate: $(\\frac{{x-3}}{{{x^2}-1}}-\\frac{2}{{x+1}})\\div \\frac{x}{{{x^2}-2x+1}}$, where $x=(\\frac{1}{2})^{-1}+\\left(\\pi -1\\right)^{0}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2}{3}"
} | {
"split": "train",
"index": 31253,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain bookstore currently has $7700$ yuan in funds, planning to use all of it to purchase a total of $20$ sets of three types of books, A, B, and C. Among them, type A books cost $500$ yuan per set, type B books cost $400$ yuan per set, and type C books cost $250$ yuan per ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 31254,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S'$ be the set of all real values of $x$ with $0 < x < \\frac{\\pi}{2}$ such that $\\sin x$, $\\cos x$, and $\\cot x$ form the side lengths (in some order) of a right triangle. Compute the sum of $\\cot^2 x$ over all $x$ in $S'$. Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 31255,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of integers $c$ such that the equation \\[\\left||20|x|-x^2|-c\\right|=21\\]has $12$ distinct real solutions. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "57"
} | {
"split": "train",
"index": 31256,
"solution": "Graph $y=|20|x|-x^2|$ (If you are having trouble, look at the description in the next two lines and/or the diagram in Solution 1). Notice that we want this to be equal to $c-21$ and $c+21$.\n We see that from left to right, the graph first dips from very positive t... |
deepscaler | [
{
"role": "user",
"content": "It is known that $\\tan\\alpha$ and $\\tan\\beta$ are the two roots of the equation $x^2+6x+7=0$, and $\\alpha, \\beta \\in \\left(-\\frac{\\pi}{2}, \\frac{\\pi}{2}\\right)$. What is the value of $\\alpha + \\beta$? Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "- \\frac{3\\pi}{4}"
} | {
"split": "train",
"index": 31257,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A room is 25 feet long and 15 feet wide. Find the ratio of the length of the room to its perimeter and the ratio of the width of the room to its perimeter. Express both your answers in the form $a:b$. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "3:16"
} | {
"split": "train",
"index": 31258,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the areas of the three squares in the figure, what is the area of the interior triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 31259,
"solution": "1. **Identify the sides of the squares**: Given the areas of the squares are $25$, $144$, and $169$, we find the side lengths by taking the square root of each area:\n - $\\sqrt{25} = 5$\n - $\\sqrt{144} = 12$\n - $\\sqrt{169} = 13$\n\n2. **Determine the conf... |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ that satisfy $|\\overrightarrow{a}|=3$, $|\\overrightarrow{b}|=2\\sqrt{3}$, and $\\overrightarrow{a} \\perp (\\overrightarrow{a}+\\overrightarrow{b})$, find the projection of $\\overrightarrow{b}$ onto $\\overrightar... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"split": "train",
"index": 31260,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers between 1 and 300 are multiples of both 2 and 5 but not of either 3 or 8? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 31261,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $G, A_{1}, A_{2}, A_{3}, A_{4}, B_{1}, B_{2}, B_{3}, B_{4}, B_{5}$ be ten points on a circle such that $G A_{1} A_{2} A_{3} A_{4}$ is a regular pentagon and $G B_{1} B_{2} B_{3} B_{4} B_{5}$ is a regular hexagon, and $B_{1}$ lies on minor arc $G A_{1}$. Let $B_{5} B_{3}$ in... | math | {
"style": "rule",
"ground_truth": "12^{\\circ}"
} | {
"split": "train",
"index": 31262,
"solution": "Note that $G B_{3}$ is a diameter of the circle. As a result, $A_{2}, A_{3}$ are symmetric with respect to $G B_{3}$, as are $B_{1}, B_{5}$. Therefore, $B_{1} A_{2}$ and $B_{5} A_{3}$ intersect along line $G B_{3}$, so in fact, $B_{1}, A_{2}, G_{1}, G_{2}$ are coll... |
deepscaler | [
{
"role": "user",
"content": "In a bag, there are three balls of different colors: red, yellow, and blue, each color having one ball. Each time a ball is drawn from the bag, its color is recorded and then the ball is put back. The drawing stops when all three colors of balls have been drawn, what is the pro... | math | {
"style": "rule",
"ground_truth": "\\frac{14}{81}"
} | {
"split": "train",
"index": 31263,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers less than $500$ can be written as the sum of two positive perfect cubes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 31264,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One of the angles of a trapezoid is $60^{\\circ}$. Find the ratio of its bases if it is known that a circle can be inscribed in this trapezoid and also circumscribed around this trapezoid. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1:3"
} | {
"split": "train",
"index": 31265,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify first, then evaluate: $(1+\\frac{4}{a-1})÷\\frac{a^2+6a+9}{a^2-a}$, where $a=2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"split": "train",
"index": 31266,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all ordered pairs of positive integers$ (x, y)$ such that:$$x^3+y^3=x^2+42xy+y^2.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(7, 1), (1, 7), (22,22)"
} | {
"split": "train",
"index": 31267,
"solution": "To find all ordered pairs of positive integers \\((x, y)\\) such that \\(x^3 + y^3 = x^2 + 42xy + y^2\\), we start by rewriting the given equation as follows:\n\n\\[\nx^3 + y^3 - x^2 - y^2 - 42xy = 0\n\\]\n\nWe rearrange and factor the left-hand side:\n\n\\[\n(x^3 ... |
deepscaler | [
{
"role": "user",
"content": "In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l$ are $\\left\\{\\begin{array}{l}x=2t+1,\\\\ y=2t\\end{array}\\right.$ (where $t$ is a parameter). Taking the coordinate origin $O$ as the pole and the non-negative half-axis of the $x$-axi... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{14}}{2}"
} | {
"split": "train",
"index": 31268,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\{a_k\\}_{k=1}^{11}$ of real numbers defined by $a_1=0.5$, $a_2=(0.51)^{a_1}$, $a_3=(0.501)^{a_2}$, $a_4=(0.511)^{a_3}$, and in general,\n\n$a_k=\\begin{cases}\n(0.\\underbrace{501\\cdots 01}_{k+1\\text{ digits}})^{a_{k-1}} & \\text{if } k \\text{ is odd,} ... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 31269,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of the base of a hemisphere is $144\\pi$. A cylinder of the same radius as the hemisphere and height equal to the radius of the hemisphere is attached to its base. What is the total surface area of the combined solid (hemisphere + cylinder)? Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "576\\pi"
} | {
"split": "train",
"index": 31270,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A man and a dog are walking. The dog waits for the man to start walking along a path and then runs to the end of the path and back to the man a total of four times, always moving at a constant speed. The last time the dog runs back to the man, it covers the remaining distance o... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 31271,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A single-elimination ping-pong tournament has $2^{2013}$ players, seeded in order of ability. If the player with seed $x$ plays the player with seed $y$, then it is possible for $x$ to win if and only if $x \\leq y+3$. For how many players $P$ it is possible for $P$ to win? (In... | math | {
"style": "rule",
"ground_truth": "6038"
} | {
"split": "train",
"index": 31272,
"solution": "We calculate the highest seed $n$ that can win. Below, we say that a player $x$ vicariously defeats a player $y$ if $x$ defeats $y$ directly or indirectly through some chain (i.e. $x$ defeats $x_{1}$, who defeated $x_{2}, \\ldots$, who defeated $x_{n}$, who defeate... |
deepscaler | [
{
"role": "user",
"content": "In a 10 by 10 table \\(A\\), some numbers are written. Let \\(S_1\\) be the sum of all numbers in the first row, \\(S_2\\) in the second row, and so on. Similarly, let \\(t_1\\) be the sum of all numbers in the first column, \\(-t_2\\) in the second column, and so on. A new tab... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 31273,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of two numbers is 50 and their difference is 6. What is their product? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "616"
} | {
"split": "train",
"index": 31274,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a new function $\\$N$ such that $\\$N = 0.75N + 2$. Calculate $\\$(\\$(\\$30))$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17.28125"
} | {
"split": "train",
"index": 31275,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "a) In how many ways can a rectangle $8 \\times 2$ be divided into $1 \\times 2$ rectangles? \n\nb) Imagine and describe a shape that can be divided into $1 \\times 2$ rectangles in exactly 555 ways. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 31276,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the sum $$\\frac{3^1}{9^1 - 1} + \\frac{3^2}{9^2 - 1} + \\frac{3^4}{9^4 - 1} + \\frac{3^8}{9^8 - 1} + \\cdots.$$ 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": 31277,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a directed graph with $2013$ vertices, there is exactly one edge between any two vertices and for every vertex there exists an edge outwards this vertex. We know that whatever the arrangement of the edges, from every vertex we can reach $k$ vertices using at most two edg... | math | {
"style": "rule",
"ground_truth": "2012"
} | {
"split": "train",
"index": 31278,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, \\( AB \\) is the diameter of circle \\( O \\) with a length of 6 cm. One vertex \\( E \\) of square \\( BCDE \\) is on the circumference of the circle, and \\( \\angle ABE = 45^\\circ \\). Find the difference in area between the non-shaded region of circle \\( ... | math | {
"style": "rule",
"ground_truth": "10.26"
} | {
"split": "train",
"index": 31279,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that a rectangle with length $3x$ inches and width $x + 5$ inches has the property that its area and perimeter have equal values, what is $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 31280,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the congruence $11n \\equiv 7 \\pmod{43}$, as a residue modulo 43. (Give an answer between 0 and 42.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 31281,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose you have an equilateral triangle divided into 9 smaller equilateral triangles with the bottom side horizontal. Starting from the top corner labeled \\( A \\), you must walk to the bottom right corner labeled \\( B \\), and are only allowed to take steps along the edges ... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 31282,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of $\\cos 96^\\circ \\cos 24^\\circ - \\sin 96^\\circ \\sin 66^\\circ$. 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": 31283,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $a \\gt 0$, $b \\gt 0$, if ${a}^{2}+{b}^{2}-\\sqrt{3}ab=1$, determine the maximum value of $\\sqrt{3}{a}^{2}-ab$. 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": 31284,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $x$, if $8^{4x-6}=\\left(\\frac{1}{2}\\right)^{x+5}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 31285,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In three-dimensional space, find the number of lattice points that have a distance of 3 from the origin.\n\nNote: A point is a lattice point if all its coordinates are integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 31286,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the numbers \\( x, y, z \\in \\left[0, \\frac{\\pi}{2}\\right] \\), find the maximum value of the expression\n\n\\[ A = \\sin(x-y) + \\sin(y-z) + \\sin(z-x). \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} - 1"
} | {
"split": "train",
"index": 31287,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a WeChat group, there are five people playing the red envelope game: A, B, C, D, and E. There are 4 red envelopes, each person can grab at most one, and all red envelopes must be grabbed. Among the 4 red envelopes, there are two 2-yuan envelopes, one 3-yuan envelope, and one... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 31288,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, $sin2C=\\sqrt{3}sinC$. \n$(1)$ Find the value of $\\angle C$; \n$(2)$ If $b=6$ and the perimeter of $\\triangle ABC$ is $6\\sqrt{3}+6$, find the area of $\\triangle ABC$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "6\\sqrt{3}"
} | {
"split": "train",
"index": 31289,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $$\\frac {1}{3}$$≤a≤1, if the function f(x)=ax<sup>2</sup>-2x+1 has its maximum value M(a) and minimum value N(a) in the interval [1,3], let g(a)=M(a)-N(a).\n(1) Find the expression for g(a);\n(2) Describe the intervals where g(a) is increasing and decreasing (no proof re... | math | {
"style": "rule",
"ground_truth": "\\frac {1}{2}"
} | {
"split": "train",
"index": 31290,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that a water tower stands 60 meters high and contains 150,000 liters of water, and a model of the tower holds 0.15 liters, determine the height of Liam's model tower. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.6"
} | {
"split": "train",
"index": 31291,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that \\( f(x) \\) and \\( g(x) \\) are functions which satisfy \\( f(g(x)) = x^3 \\) and \\( g(f(x)) = x^4 \\) for all \\( x \\ge 1 \\). If \\( g(81) = 81 \\), compute \\( [g(3)]^4 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 31292,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of \\( z \\) in the carpet installation cost chart? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1261.40"
} | {
"split": "train",
"index": 31293,
"solution": "Using the cost per square metre, \\( z = 1261.40 \\)."
} |
deepscaler | [
{
"role": "user",
"content": "The notation $[x]$ stands for the greatest integer that is less than or equal to $x$. Calculate $[-1.2]$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 31294,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are four points that are $5$ units from the line $y=13$ and $13$ units from the point $(7,13)$. What is the sum of the $x$- and $y$-coordinates of all four of these points? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 31295,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the least positive integer $N$ such that the set of $1000$ consecutive integers beginning with $1000\\cdot N$ contains no square of an integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "282"
} | {
"split": "train",
"index": 31296,
"solution": "Let $x$ be the number being squared. Based on the reasoning above, we know that $N$ must be at least $250$, so $x$ has to be at least $500$. Let $k$ be $x-500$. We can write $x^2$ as $(500+k)^2$, or $250000+1000k+k^2$. We can disregard $250000$ and $1000k$, since t... |
deepscaler | [
{
"role": "user",
"content": "A total of $731$ objects are put into $n$ nonempty bags where $n$ is a positive integer. These bags can be distributed into $17$ red boxes and also into $43$ blue boxes so that each red and each blue box contain $43$ and $17$ objects, respectively. Find the minimu... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 31297,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One dimension of a cube is tripled, another is decreased by `a/2`, and the third dimension remains unchanged. The volume gap between the new solid and the original cube is equal to `2a^2`. Calculate the volume of the original cube. Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 31298,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The planet Xavier follows an elliptical orbit with its sun at one focus. At its nearest point (perigee), it is 2 astronomical units (AU) from the sun, while at its furthest point (apogee) it is 12 AU away. When Xavier is midway along its orbit, as shown, how far is it from th... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 31299,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For what values of $x$ is \\[\\frac{x-10x^2+25x^3}{8-x^3}\\]nonnegative? Answer as an interval. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "[0,2)"
} | {
"split": "train",
"index": 31300,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four red beads, two white beads, and one green bead are placed in a line in random order. What is the probability that no two neighboring beads are the same color?\nA) $\\frac{1}{15}$\nB) $\\frac{2}{15}$\nC) $\\frac{1}{7}$\nD) $\\frac{1}{30}$\nE) $\\frac{1}{21}$ Let's think ste... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{15}"
} | {
"split": "train",
"index": 31301,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The infinite sequence $S=\\{s_1,s_2,s_3,\\ldots\\}$ is defined by $s_1=7$ and $s_n=7^{s_{n-1}}$ for each integer $n>1$. What is the remainder when $s_{100}$ is divided by $5$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 31302,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each prime $p$, a polynomial $P(x)$ with rational coefficients is called $p$-good if and only if there exist three integers $a, b$, and $c$ such that $0 \\leq a<b<c<\\frac{p}{3}$ and $p$ divides all the numerators of $P(a)$, $P(b)$, and $P(c)$, when written in simplest form... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 31303,
"solution": "By Vieta, the sum of the roots is $-10(\\bmod p)$. However, since the three roots are less than $p/3$, it follows that the roots are $\\left(p-a^{\\prime}\\right)/3,\\left(p-b^{\\prime}\\right)/3,\\left(p-c^{\\prime}\\right)/3$, where there are finitely many choice... |
deepscaler | [
{
"role": "user",
"content": "A small fish is holding 17 cards, labeled 1 through 17, which he shuffles into a random order. Then, he notices that although the cards are not currently sorted in ascending order, he can sort them into ascending order by removing one card and putting it back in a different pos... | math | {
"style": "rule",
"ground_truth": "256"
} | {
"split": "train",
"index": 31304,
"solution": "Instead of looking at moves which put the cards in order, we start with the cards in order and consider possible starting positions by backtracking one move: each of 17 cards can be moved to 16 new places. But moving card $k$ between card $k+1$ and card $k+2$ is eq... |
deepscaler | [
{
"role": "user",
"content": "Two cubical dice each have removable numbers $1$ through $6$. The twelve numbers on the two dice are removed, put into a bag, then drawn one at a time and randomly reattached to the faces of the cubes, one number to each face. The dice are then rolled and the numbers on the two... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 31305,
"solution": "To solve this problem, we need to consider the probability of rolling a sum of 7 with two dice, where the numbers on the dice have been randomly reassigned.\n\n#### Step 1: Understand the possible sums\nThe possible combinations of two numbers summing to 7 are:\n- ... |
deepscaler | [
{
"role": "user",
"content": "Class 5(2) has 28 female students, which is 6 more than the male students. The ratio of female to male students is ____, and the percentage of male students in the whole class is ____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{25}"
} | {
"split": "train",
"index": 31306,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagram shows a karting track circuit. The start and finish are at point $A$, and the kart driver can return to point $A$ and continue on the circuit as many times as desired.\n\nThe time taken to travel from $A$ to $B$ or from $B$ to $A$ is one minute. The time taken to tr... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 31307,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When $15$ is appended to a list of integers, the mean is increased by $2$. When $1$ is appended to the enlarged list, the mean of the enlarged list is decreased by $1$. How many integers were in the original list? Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 31308,
"solution": "Let $n$ be the number of integers in the original list and $m$ be the mean of these integers. The sum of the integers in the original list is then $mn$.\n\n1. **Appending 15 to the list:**\n When 15 is added to the list, the new list has $n+1$ integers, and the n... |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $k$ such that $1^2+2^2+3^2+\\ldots+k^2$ is a multiple of $360$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "360"
} | {
"split": "train",
"index": 31309,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all rational roots of\n\\[4x^4 - 3x^3 - 13x^2 + 5x + 2 = 0\\]Enter all the rational roots, separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2,-\\frac{1}{4}"
} | {
"split": "train",
"index": 31310,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rectangle $ABCD$ is inscribed in triangle $EFG$ such that side $AD$ of the rectangle is on side $EG$ of the triangle, as shown. The triangle's altitude from $F$ to side $EG$ is 7 inches, and $EG = 10 \\text{ inches}$. The length of segment $AB$ is equal to half the length of se... | math | {
"style": "rule",
"ground_truth": "\\frac{1225}{72}"
} | {
"split": "train",
"index": 31311,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three tenths plus four thousandths. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.304"
} | {
"split": "train",
"index": 31312,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the 729 smallest positive integers written in base 9 use 7 or 8 (or both) as a digit? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "386"
} | {
"split": "train",
"index": 31313,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine how many integers $n$ between 1 and 15 (inclusive) result in a fraction $\\frac{n}{30}$ that has a repeating decimal. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 31314,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If a positive integer multiple of 864 is picked randomly, with each multiple having the same probability of being picked, what is the probability that it is divisible by 1944? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"split": "train",
"index": 31315,
"solution": "The probability that a multiple of $864=2^{5} 3^{3}$ is divisible by $1944=2^{3} 3^{5}$ is the same as the probability that a multiple of $2^{2}$ is divisible by $3^{2}$, which since 4 and 9 are relatively prime is $\\frac{1}{9}$."
} |
deepscaler | [
{
"role": "user",
"content": "The lengths of two line segments are $a$ units and $b$ units respectively. Then the correct relation between them is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{a+b}{2} \\geq \\sqrt{ab}"
} | {
"split": "train",
"index": 31316,
"solution": "To determine the correct relationship between $a$ and $b$, we can use the Arithmetic Mean-Geometric Mean Inequality (AM-GM Inequality). The AM-GM Inequality states that for any non-negative real numbers $x_1, x_2, \\ldots, x_n$, the following inequality holds:\n\\[... |
deepscaler | [
{
"role": "user",
"content": "The real number \\( a \\) is such that \\( 2a - \\frac{1}{a} = 3 \\). What is \\( 16a^{4} + \\frac{1}{a^{4}} \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "161"
} | {
"split": "train",
"index": 31317,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a new diagram showing the miles traveled by bikers Alberto, Bjorn, and Carlos over a period of 6 hours. The straight lines represent their paths on a coordinate plot where the y-axis represents miles and x-axis represents hours. Alberto's line passes through the points (0,0)... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 31318,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$ABCD$ is a rectangular sheet of paper that has been folded so that corner $B$ is matched with point $B'$ on edge $AD.$ The crease is $EF,$ where $E$ is on $AB$ and $F$ is on $CD.$ The dimensions $AE=8, BE=17,$ and $CF=3$ are given. The perimeter of rectangle $ABCD$ is $m/n,$ w... | math | {
"style": "rule",
"ground_truth": "293"
} | {
"split": "train",
"index": 31319,
"solution": "Use the prepared diagram for this solution.\n Call the intersection of DF and B'C' G. AB'E is an 8-15-17 right triangle, and so are B'DG and C'FG. Since C'F is 3, then using the properties of similar triangles GF is 51/8. DF is 22, so DG is 125/8. Finally, DB can t... |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\lfloor 3.2\\rfloor$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 31320,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle $ABCD,$ $P$ is a point on side $\\overline{BC}$ such that $BP = 18$ and $CP = 6.$ If $\\tan \\angle APD = 2,$ find $AB.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 31321,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $\\log_{10}{4}+2\\log_{10}{5}+3\\log_{10}{2}+6\\log_{10}{5}+\\log_{10}{8}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 31322,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\sin^2 6^\\circ + \\sin^2 12^\\circ + \\sin^2 18^\\circ + \\dots + \\sin^2 174^\\circ.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{31}{2}"
} | {
"split": "train",
"index": 31323,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Mike walks to his college, averaging 70 steps per minute with each step being 80 cm long, and it takes him 20 minutes to get there, determine how long it takes Tom to reach the college, given that he averages 120 steps per minute, but his steps are only 50 cm long. L... | math | {
"style": "rule",
"ground_truth": "18.67"
} | {
"split": "train",
"index": 31324,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The first term of a given sequence is 1, and each successive term is the sum of all the previous terms of the sequence plus the square of the first term. What is the value of the first term which exceeds 10000? Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "16384"
} | {
"split": "train",
"index": 31325,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the last digit of $1^{1}+2^{2}+3^{3}+\\cdots+100^{100}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 31326,
"solution": "Let $L(d, n)$ be the last digit of a number ending in $d$ to the $n$th power. For $n \\geq 1$, we know that $L(0, n)=0, L(1, n)=1, L(5, n)=5, L(6, n)=6$. All numbers ending in odd digits in this series are raised to odd powers; for odd $n, L(3, n)=3$ or 7, $L(7, n)... |
deepscaler | [
{
"role": "user",
"content": "Sherry and Val are playing a game. Sherry has a deck containing 2011 red cards and 2012 black cards, shuffled randomly. Sherry flips these cards over one at a time, and before she flips each card over, Val guesses whether it is red or black. If Val guesses correctly, she wins 1... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4023}"
} | {
"split": "train",
"index": 31327,
"solution": "We will prove by induction on $r+b$ that the expected profit for guessing if there are $r$ red cards, $b$ black cards, and where $g$ guesses must be red, is equal to $(b-r)+\\frac{2(r-b)}{(r+b)} g$. It is not difficult to check that this holds in the cases $(r, b, ... |
deepscaler | [
{
"role": "user",
"content": "Find the largest real number $\\lambda$ with the following property: for any positive real numbers $p,q,r,s$ there exists a complex number $z=a+bi$($a,b\\in \\mathbb{R})$ such that $$ |b|\\ge \\lambda |a| \\quad \\text{and} \\quad (pz^3+2qz^2+2rz+s) \\cdot (qz^3+2pz^2+2sz+r) =... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 31328,
"solution": "\nTo find the largest real number \\(\\lambda\\) such that for any positive real numbers \\(p, q, r, s\\), there exists a complex number \\(z = a + bi\\) (\\(a, b \\in \\mathbb{R}\\)) satisfying\n\\[\n|b| \\ge \\lambda |a|\n\\]\nand\n\\[\n(pz^3 + 2qz^2 + 2rz + s) \... |
deepscaler | [
{
"role": "user",
"content": "Let \\(\\theta\\) be an angle in the second quadrant, and if \\(\\tan (\\theta+ \\frac {\\pi}{3})= \\frac {1}{2}\\), calculate the value of \\(\\sin \\theta+ \\sqrt {3}\\cos \\theta\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac {2 \\sqrt {5}}{5}"
} | {
"split": "train",
"index": 31329,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{a},$ $\\mathbf{b},$ $\\mathbf{c},$ $\\mathbf{d}$ be four distinct unit vectors in space such that\n\\[\\mathbf{a} \\cdot \\mathbf{b} = \\mathbf{a} \\cdot \\mathbf{c} = \\mathbf{b} \\cdot \\mathbf{c} =\\mathbf{b} \\cdot \\mathbf{d} = \\mathbf{c} \\cdot \\mathbf{d}... | math | {
"style": "rule",
"ground_truth": "-\\frac{53}{55}"
} | {
"split": "train",
"index": 31330,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a$ , $b$ are integers and $s=a^3+b^3-60ab(a+b)\\geq 2012$ , find the least possible value of $s$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2015"
} | {
"split": "train",
"index": 31331,
"solution": "\nGiven the inequality \\( s = a^3 + b^3 - 60ab(a + b) \\geq 2012 \\), we are tasked with finding the least possible value of \\( s \\).\n\nTo solve this, we start by rewriting \\( s \\) in terms of a simplified expression:\n\\[\ns = a^3 + b^3 - 60ab(a + b).\n\\]\n... |
deepscaler | [
{
"role": "user",
"content": "A Senate committee has 8 Republicans and 6 Democrats. In how many ways can we form a subcommittee with 3 Republicans and 2 Democrats? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "840"
} | {
"split": "train",
"index": 31332,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The coefficient of $x^5$ in the expansion of $(1+x)^2(1-x)^5$ is \\_\\_\\_\\_\\_\\_ (Answer with a number). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 31333,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a} =(\\cos x,\\sin x)$, $\\overrightarrow{b} =(3,-\\sqrt{3} )$, with $x\\in[0,\\pi]$.\n$(1)$ If $\\overrightarrow{a} \\parallel \\overrightarrow{b}$, find the value of $x$;\n$(2)$ Let $f(x)=\\overrightarrow{a} \\cdot \\overrightarrow{b}$, find th... | math | {
"style": "rule",
"ground_truth": "-2 \\sqrt {3}"
} | {
"split": "train",
"index": 31334,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal{S}$ be the set of all perfect squares whose rightmost three digits in base $10$ are $256$. Let $\\mathcal{T}$ be the set of all numbers of the form $\\frac{x-256}{1000}$, where $x$ is in $\\mathcal{S}$. In other words, $\\mathcal{T}$ is the set of numbers that re... | math | {
"style": "rule",
"ground_truth": "170"
} | {
"split": "train",
"index": 31335,
"solution": "The condition implies $x^2\\equiv 256 \\pmod{1000}$. Rearranging and factoring, \\[(x-16)(x+16)\\equiv 0\\pmod {1000}.\\] This can be expressed with the system of congruences \\[\\begin{cases} (x-16)(x+16)\\equiv 0\\pmod{125} \\\\ (x-16)(x+16)\\equiv 0\\pmod{8} \\e... |
deepscaler | [
{
"role": "user",
"content": "An ideal gas is used as the working substance of a heat engine operating cyclically. The cycle consists of three stages: isochoric pressure reduction from $3 P_{0}$ to $P_{0}$, isobaric density increase from $\\rho_{0}$ to $3 \\rho_{0}$, and a return to the initial state, repre... | math | {
"style": "rule",
"ground_truth": "1/9"
} | {
"split": "train",
"index": 31336,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express $7.\\overline{123}$ as a common fraction in lowest terms. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{593}{111}"
} | {
"split": "train",
"index": 31337,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alex picks his favorite point $(x, y)$ in the first quadrant on the unit circle $x^{2}+y^{2}=1$, such that a ray from the origin through $(x, y)$ is $\\theta$ radians counterclockwise from the positive $x$-axis. He then computes $\\cos ^{-1}\\left(\\frac{4 x+3 y}{5}\\right)$ an... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 31338,
"solution": "$x=\\cos (\\theta), y=\\sin (\\theta)$. By the trig identity you never thought you'd need, $\\frac{4 x+3 y}{5}=\\cos (\\theta-\\phi)$, where $\\phi$ has sine $3 / 5$ and cosine $4 / 5$. Now $\\theta-\\phi=\\theta$ is impossible, since $\\phi \\neq 0$, so we must ha... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $AB > AC$, $\\angle BAC = 45^\\circ$. Point $E$ is the intersection of the external angle bisector of $\\angle BAC$ with the circumcircle of $\\triangle ABC$. Point $F$ is on $AB$ such that $EF \\perp AB$. Given $AF = 1$ and $BF = 5$, find the area of $\\tr... | math | {
"style": "rule",
"ground_truth": "6 \\sqrt{2}"
} | {
"split": "train",
"index": 31339,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many terms are in the arithmetic sequence $13$, $16$, $19$, $\\dotsc$, $70$, $73$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 31340,
"solution": "1. **Identify the first term and common difference**: The given sequence is $13, 16, 19, \\dots, 70, 73$. The first term ($a$) is $13$ and the common difference ($d$) can be calculated as $16 - 13 = 3$.\n\n2. **Formulate the general term**: The general term of an a... |
deepscaler | [
{
"role": "user",
"content": "Vasya wrote a note on a piece of paper, folded it in quarters, and wrote \"MAME\" on top. He then unfolded the note, added something more, folded it again randomly along the crease lines (not necessarily as before), and left it on the table with a random side facing up. Find th... | math | {
"style": "rule",
"ground_truth": "1/8"
} | {
"split": "train",
"index": 31341,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In quadrilateral ABCD, m∠B = m∠C = 120°, AB = 4, BC = 6, and CD = 7. Diagonal BD = 8. Calculate the area of ABCD. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16.5\\sqrt{3}"
} | {
"split": "train",
"index": 31342,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function \\(\\frac{(4^t - 2t)t}{16^t}\\), find the maximum value for real values of \\(t\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 31343,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=x^{3}+ax^{2}+bx+c$, it reaches a maximum value of $7$ when $x=-1$, and it reaches a minimum value when $x=3$. Find the values of $a$, $b$, $c$, and this minimum value. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-25"
} | {
"split": "train",
"index": 31344,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In three-dimensional space, the volume of the geometric body formed by points whose distance to line segment $A B$ is no greater than three units is $216 \\pi$. What is the length of the line segment $A B$? Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 31345,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A and B bought the same number of sheets of stationery. A put 1 sheet of stationery into each envelope and had 40 sheets of stationery left after using all the envelopes. B put 3 sheets of stationery into each envelope and had 40 envelopes left after using all the sheets of sta... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 31346,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A palindrome is a positive integer whose digits are the same when read forwards or backwards. For example, 2002 is a palindrome. What is the smallest number which can be added to 2002 to produce a larger palindrome? Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "110"
} | {
"split": "train",
"index": 31347,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $\\angle C= \\frac{\\pi}{2}$, $\\angle B= \\frac{\\pi}{6}$, and $AC=2$. $M$ is the midpoint of $AB$. $\\triangle ACM$ is folded along $CM$ such that the distance between $A$ and $B$ is $2\\sqrt{2}$. The surface area of the circumscribed sphere of the tetrah... | math | {
"style": "rule",
"ground_truth": "16\\pi"
} | {
"split": "train",
"index": 31348,
"solution": ""
} |
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