data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system $xOy$, the equation of line $C_1$ is $y=-\\sqrt{3}x$, and the parametric equations of curve $C_2$ are given by $\\begin{cases}x=-\\sqrt{3}+\\cos\\varphi\\\\y=-2+\\sin\\varphi\\end{cases}$. Establish a polar coordinate system with the coordin... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 21549,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $XYZ$ with right angle at $Z$, $\\angle XYZ < 45^\\circ$ and $XY = 6$. A point $Q$ on $\\overline{XY}$ is chosen such that $\\angle YQZ = 3\\angle XQZ$ and $QZ = 2$. Determine the ratio $\\frac{XQ}{YQ}$ in simplest form. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "\\frac{7 + 3\\sqrt{5}}{2}"
} | {
"split": "train",
"index": 21550,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If a positive integer is a multiple of 3, and we sum the cubes of its digits to perform the first operation; then, we take the new number and sum the cubes of its digits to perform the second operation; if we repeat the above operations several times, you will find that eventua... | math | {
"style": "rule",
"ground_truth": "153"
} | {
"split": "train",
"index": 21551,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a television program, five children (Tian Tian, Shi Tou, Kimi, Cindy, Angela) need to switch fathers (each child can choose any one of the other four fathers except their own). Calculate the total number of different combinations of choices for the five children. Let's think... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 21552,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the positive integer $N$, such that numbers $N$ and $N^2$ end in the same sequence of four digits $abcd$ where $a$ is not zero, under the modulus $8000$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "625"
} | {
"split": "train",
"index": 21553,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P(x)$ be a quadratic polynomial with complex coefficients whose $x^2$ coefficient is $1.$ Suppose the equation $P(P(x))=0$ has four distinct solutions, $x=3,4,a,b.$ Find the sum of all possible values of $(a+b)^2.$ Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "85"
} | {
"split": "train",
"index": 21554,
"solution": "Either $P(3) = P(4)$ or not. We first see that if $P(3) = P(4)$ it's easy to obtain by Vieta's that $(a+b)^2 = 49$. Now, take $P(3) \\neq P(4)$ and WLOG $P(3) = P(a), P(4) = P(b)$. Now, consider the parabola formed by the graph of $P$. It has vertex $\\frac{3+a}{2}... |
deepscaler | [
{
"role": "user",
"content": "Mike had a bag of candies, and all candies were whole pieces that cannot be divided. Initially, Mike ate $\\frac{1}{4}$ of the candies. Then, he shared $\\frac{1}{3}$ of the remaining candies with his sister, Linda. Next, both Mike and his father ate 12 candies each from the re... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 21555,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a vertical wooden pillar, a rope is tied to its top, with 4 feet of the rope hanging down to the ground. Additionally, when pulling the rope, it runs out when 8 feet away from the base of the pillar. Determine the length of the rope. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 21556,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the line y=kx+b is tangent to the graph of the function f(x)=1/2 x^2 + ln x, find the minimum value of k-b. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{2}"
} | {
"split": "train",
"index": 21557,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three circles are drawn around vertices \\( A, B, \\) and \\( C \\) of a regular hexagon \\( ABCDEF \\) with side length 2 units, such that the circles touch each other externally. What is the radius of the smallest circle? Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "2 - \\sqrt{3}"
} | {
"split": "train",
"index": 21558,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fair coin is tossed 4 times. What is the probability of getting at least two consecutive heads? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{8}"
} | {
"split": "train",
"index": 21559,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given non-zero vectors \\\\(a\\\\) and \\\\(b\\\\) satisfying \\\\(|b|=2|a|\\\\) and \\\\(a \\perp (\\sqrt{3}a+b)\\\\), find the angle between \\\\(a\\\\) and \\\\(b\\\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{5\\pi}{6}"
} | {
"split": "train",
"index": 21560,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive perfect squares less than \\(10^8\\) are multiples of 36? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1666"
} | {
"split": "train",
"index": 21561,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the remainder of $19^{1999}$ divided by 25? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 21562,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest possible sum of two perfect squares such that their difference is 175 and both squares are greater or equal to 36. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "625"
} | {
"split": "train",
"index": 21563,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two semicircles, each with radius \\(\\sqrt{2}\\), are tangent to each other. If \\( AB \\parallel CD \\), determine the length of segment \\( AD \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{2}"
} | {
"split": "train",
"index": 21564,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, the circle has radius \\(\\sqrt{5}\\). Rectangle \\(ABCD\\) has \\(C\\) and \\(D\\) on the circle, \\(A\\) and \\(B\\) outside the circle, and \\(AB\\) tangent to the circle. What is the area of \\(ABCD\\) if \\(AB = 4AD\\)? Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "16/5"
} | {
"split": "train",
"index": 21565,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that one of the roots of the function $f(x)=ax+b$ is $2$, find the roots of the function $g(x)=bx^{2}-ax$. 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": 21566,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square has area $\\frac14$. What is the side length of the square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac12"
} | {
"split": "train",
"index": 21567,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\gcd(83^9+1,83^9+83^2+1)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 21568,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A marathon of 42 km started at 11:30 AM and the winner finished at 1:45 PM on the same day. What was the average speed of the winner, in km/h? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18.6"
} | {
"split": "train",
"index": 21569,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the values of $a$ and $b$ such that $a + b^2$ can be calculated, where $x = a \\pm b i$ are the solutions to the equation $5x^2 + 7 = 2x - 10$. Express your answer as a fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{89}{25}"
} | {
"split": "train",
"index": 21570,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $\\overrightarrow{a}+2\\overrightarrow{b}$, where $\\overrightarrow{a}=(2,0)$ and $|\\overrightarrow{b}|=1$, and then calculate the magnitude of this vector. 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": 21571,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\sin a= \\frac{ \\sqrt{5}}{5}$, $a\\in\\left( \\frac{\\pi}{2},\\pi\\right)$, find:\n\n$(1)$ The value of $\\sin 2a$;\n\n$(2)$ The value of $\\tan \\left( \\frac{\\pi}{3}+a\\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5 \\sqrt{3}-8"
} | {
"split": "train",
"index": 21572,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Joann ate a total of 100 lollipops in five days. Each day after the first day she ate six more than she had eaten on the previous day. How many lollipops did she eat on the third day? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 21573,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence \\( a_{1}, a_{2}, \\cdots, a_{n}, \\cdots \\) that satisfies \\( a_{1}=a_{2}=1 \\) and \\( a_{3}=2 \\), and for any natural number \\( n \\), \\( a_{n} a_{n+1} a_{n+2} \\neq 1 \\), and \\( a_{n} a_{n+1} a_{n+2} a_{n+3}=a_{n}+a_{n+1}+a_{n+2}+a_{n+3} \\), find ... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 21574,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $r_{1}, \\ldots, r_{n}$ be the distinct real zeroes of the equation $x^{8}-14 x^{4}-8 x^{3}-x^{2}+1=0$. Evaluate $r_{1}^{2}+\\cdots+r_{n}^{2}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 21575,
"solution": "Observe that $x^{8}-14 x^{4}-8 x^{3}-x^{2}+1 =\\left(x^{8}+2 x^{4}+1\\right)-\\left(16 x^{4}+8 x^{3}+x^{2}\\right) =\\left(x^{4}+4 x^{2}+x+1\\right)\\left(x^{4}-4 x^{2}-x+1\\right)$. The polynomial $x^{4}+4 x^{2}+x+1=x^{4}+\\frac{15}{4} x^{2}+\\left(\\frac{x}{2}+1\... |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has vertices $A(0,8)$, $B(2,0)$, $C(8,0)$. A vertical line intersects $AC$ at $R$ and $\\overline{BC}$ at $S$, forming triangle $RSC$. If the area of $\\triangle RSC$ is 12.5, determine the positive difference of the $x$ and $y$ coordinates of point $R$. Let's ... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 21576,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Some vertices (the vertices of the unit squares) of a \\(6 \\times 6\\) grid are colored red. We need to ensure that for any sub-grid \\(k \\times k\\) where \\(1 \\leq k \\leq 6\\), at least one red point exists on its boundary. Find the minimum number of red points needed to ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 21577,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A particular novel contains $40,\\!000$ words. If the author completed the novel in $80$ hours, on average how many words per hour did she write? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "500"
} | {
"split": "train",
"index": 21578,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A mail carrier delivers mail to the nineteen houses on the east side of Elm Street. The carrier notices that no two adjacent houses ever get mail on the same day, but that there are never more than two houses in a row that get no mail on the same day. How many different pattern... | math | {
"style": "rule",
"ground_truth": "351"
} | {
"split": "train",
"index": 21579,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\dbinom{50}{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1225"
} | {
"split": "train",
"index": 21580,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $(a_1, a_2, a_3,\\ldots,a_{13})$ be a permutation of $(1,2,3,\\ldots,13)$ for which\n$$a_1 > a_2 > a_3 > a_4 > a_5 > a_6 > a_7 \\mathrm{\\ and \\ } a_7 < a_8 < a_9 < a_{10} < a_{11} < a_{12} < a_{13}.$$\nFind the number of such permutations. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "924"
} | {
"split": "train",
"index": 21581,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cassie leaves Escanaba at 8:30 AM heading for Marquette on her bike. She bikes at a uniform rate of 12 miles per hour. Brian leaves Marquette at 9:00 AM heading for Escanaba on his bike. He bikes at a uniform rate of 16 miles per hour. They both bike on the same 62-mile route b... | math | {
"style": "rule",
"ground_truth": "11:00"
} | {
"split": "train",
"index": 21582,
"solution": "1. **Define Variables:**\n Let $x$ be the number of hours after Cassie leaves Escanaba when both of them meet. Cassie leaves at 8:30 AM, so if they meet $x$ hours later, the time will be $8:30$ AM + $x$ hours.\n\n2. **Calculate Distance Traveled by Cassie:**\n ... |
deepscaler | [
{
"role": "user",
"content": "Given that $(1-3x)^6 = a + a_1x + a_2x^2 + a_3x^3 + a_4x^4 + a_5x^5 + a_6x^6$, find the total sum of elements in all subsets containing 2 elements of the set $\\{a_1, a_2, a_3, a_4, a_5, a_6\\}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "315"
} | {
"split": "train",
"index": 21583,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p>2$ be a prime number. $\\mathbb{F}_{p}[x]$ is defined as the set of all polynomials in $x$ with coefficients in $\\mathbb{F}_{p}$ (the integers modulo $p$ with usual addition and subtraction), so that two polynomials are equal if and only if the coefficients of $x^{k}$ a... | math | {
"style": "rule",
"ground_truth": "4 p(p-1)"
} | {
"split": "train",
"index": 21584,
"solution": "Answer: $4 p(p-1)$ Solution 1. First, notice that $(\\operatorname{deg} f)(\\operatorname{deg} g)=p^{2}$ and both polynomials are clearly nonconstant. Therefore there are three possibilities for the ordered pair $(\\operatorname{deg} f, \\operatorname{deg} g)$, whi... |
deepscaler | [
{
"role": "user",
"content": "A school uses a systematic sampling method to conduct a vision test on 50 out of the 800 students in the first year. The 800 students are numbered from 1 to 800 and are evenly divided into 50 groups in ascending order of their numbers, with group numbers from 1 to 50. It is kno... | math | {
"style": "rule",
"ground_truth": "94"
} | {
"split": "train",
"index": 21585,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each square in a $3 \\times 10$ grid is colored black or white. Let $N$ be the number of ways this can be done in such a way that no five squares in an 'X' configuration (as shown by the black squares below) are all white or all black. Determine $\\sqrt{N}$. Let's think step by... | math | {
"style": "rule",
"ground_truth": "25636"
} | {
"split": "train",
"index": 21586,
"solution": "Note that we may label half of the cells in our board the number 0 and the other half 1, in such a way that squares labeled 0 are adjacent only to squares labeled 1 and vice versa. In other words, we make this labeling in a 'checkerboard' pattern. Since cells in an... |
deepscaler | [
{
"role": "user",
"content": "Two medians of a triangle with unequal sides are $3$ inches and $6$ inches. Its area is $3 \\sqrt{15}$ square inches. The length of the third median in inches, is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{6}"
} | {
"split": "train",
"index": 21587,
"solution": "1. **Understanding the Problem**: We are given a triangle with two medians of lengths 3 inches and 6 inches, and the area of the triangle is $3 \\sqrt{15}$ square inches. We need to find the length of the third median.\n\n2. **Labeling the Triangle**: Let's denote ... |
deepscaler | [
{
"role": "user",
"content": "If $4u-5v=23$ and $2u+4v=-8$, find the value of $u+v$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 21588,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a triangle with $m(\\widehat{ABC}) = 90^{\\circ}$ . The circle with diameter $AB$ intersects the side $[AC]$ at $D$ . The tangent to the circle at $D$ meets $BC$ at $E$ . If $|EC| =2$ , then what is $|AC|^2 - |AE|^2$ ? Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 21589,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the triangle $\\triangle ABC$ is an isosceles right triangle with $\\angle ABC = 90^\\circ$, and sides $\\overline{AB}$ and $\\overline{BC}$ are each tangent to a circle at points $B$ and $C$ respectively, with the circle having center $O$ and located inside the triangle,... | math | {
"style": "rule",
"ground_truth": "1 - \\frac{\\pi}{2}"
} | {
"split": "train",
"index": 21590,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of the orthogonal projection of a circle with a radius of 1 on the plane $\\alpha$ is 1. Find the length of the orthogonal projection of this circle on a line perpendicular to the plane $\\alpha$. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{\\pi^2 - 1}}{\\pi}"
} | {
"split": "train",
"index": 21591,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When every vector on the line $y = \\frac{5}{2} x + 4$ is projected onto a certain vector $\\mathbf{w},$ the result is always the vector $\\mathbf{p}.$ Find the vector $\\mathbf{p}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -40/29 \\\\ 16/29 \\end{pmatrix}"
} | {
"split": "train",
"index": 21592,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Jennifer plans to build a fence around her garden in the shape of a rectangle, with $24$ fence posts, and evenly distributing the remaining along the edges, with $6$ yards between each post, and with the longer side of the garden, including corners, having three time... | math | {
"style": "rule",
"ground_truth": "855"
} | {
"split": "train",
"index": 21593,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right rectangular prism, with edge lengths $\\log_{5}x, \\log_{6}x,$ and $\\log_{8}x,$ must satisfy the condition that the sum of the squares of its face diagonals is numerically equal to 8 times the volume. What is $x?$\nA) $24$\nB) $36$\nC) $120$\nD) $\\sqrt{240}$\nE) $240$... | math | {
"style": "rule",
"ground_truth": "\\sqrt{240}"
} | {
"split": "train",
"index": 21594,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the sum $2^{-2} + 2^{-3} + 2^{-4} + 2^{-5} + 2^{-6} + 2^{-7} \\pmod{17}$.\n\nExpress your answer as an integer from $0$ to $16$, inclusive. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 21595,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle $PQRS$, $PQ=7$ and $QR =4$. Points $X$ and $Y$ are on $\\overline{RS}$ so that $RX = 2$ and $SY=3$. Lines $PX$ and $QY$ intersect at $Z$. Find the area of $\\triangle PQZ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19.6"
} | {
"split": "train",
"index": 21596,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Raashan, Sylvia, and Ted play the following game. Each starts with $1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $1$ to that player. What... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 21597,
"solution": "1. **Initial Setup and State Description:**\n Each player starts with $1. The possible states of money distribution after each round are $(1-1-1)$ and $(2-1-0)$ in some permutation. The state $(3-0-0)$ is not possible because:\n - A player cannot give money to ... |
deepscaler | [
{
"role": "user",
"content": "A sample of 121 integers is given, each between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let $D$ be the difference between the mode and the arithmetic mean of the sample. What is the largest possible value of $\\lfloor ... | math | {
"style": "rule",
"ground_truth": "947"
} | {
"split": "train",
"index": 21598,
"solution": "With the same reasoning as Solution 1, in order to get largest possible value of D, we can construct that our set of numbers as $\\underbrace{1,1,1...1,}_\\text{n times}\\underbrace{2,2,2...2,}_\\text{n times}\\underbrace{3,3,3...3,}_\\text{n times}........\\underb... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sin x-2\\cos x= \\sqrt {5}$, find the value of $\\tan x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\dfrac{1}{2}"
} | {
"split": "train",
"index": 21599,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $y=2\\sin \\left(3x+ \\dfrac{\\pi}{4}\\right)$, determine the shift required to obtain its graph from the graph of the function $y=2\\sin 3x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\pi}{12}"
} | {
"split": "train",
"index": 21600,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parabola $y^{2}=2px\\left(p \\gt 0\\right)$ with focus $F$ and the intersection point $E$ of its directrix with the $x$-axis, let line $l$ pass through $E$.<br/>$(1)$ If line $l$ is tangent to the parabola at point $M$, then $\\angle EMF=$____.<br/>$(2)$ Let $p=6$. If... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 21601,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\alpha$ is an angle in the third quadrant, $f(\\alpha) = \\frac {\\sin(\\pi-\\alpha)\\cdot \\cos(2\\pi-\\alpha)\\cdot \\tan(-\\alpha-\\pi)}{\\tan(-\\alpha )\\cdot \\sin(-\\pi -\\alpha)}$.\n1. Simplify $f(\\alpha)$;\n2. If $\\cos\\left(\\alpha- \\frac {3}{2}\\pi\\ri... | math | {
"style": "rule",
"ground_truth": "\\frac {1}{2}"
} | {
"split": "train",
"index": 21602,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the vertex of angle α is at the origin of the coordinate system, its initial side coincides with the non-negative half-axis of the x-axis, and its terminal side passes through the point (-√3,2), find the value of tan(α - π/6). Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "-3\\sqrt{3}"
} | {
"split": "train",
"index": 21603,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a convex hexagon $A B C D E F$ with all six side lengths equal, and internal angles $\\angle A$, $\\angle B$, and $\\angle C$ are $134^{\\circ}$, $106^{\\circ}$, and $134^{\\circ}$ respectively. Find the measure of the internal angle $\\angle E$. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "134"
} | {
"split": "train",
"index": 21604,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\phi$ is the Golden Ratio, we know that $\\frac1\\phi = \\phi - 1$ . Define a new positive real number, called $\\phi_d$ , where $\\frac1{\\phi_d} = \\phi_d - d$ (so $\\phi = \\phi_1$ ). Given that $\\phi_{2009} = \\frac{a + \\sqrt{b}}{c}$ , $a, b, c$ positive ... | math | {
"style": "rule",
"ground_truth": "4038096"
} | {
"split": "train",
"index": 21605,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given points $A(-2,0)$ and $B(0,2)$, let point $C$ be a moving point on the circle $x^{2}-2x+y^{2}=0$. Determine the minimum area of $\\triangle ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3-\\sqrt{2}"
} | {
"split": "train",
"index": 21606,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the extensions of the medians \\(A K\\), \\(B L\\), and \\(C M\\) of triangle \\(A B C\\), points \\(P\\), \\(Q\\), and \\(R\\) are taken such that \\(K P = \\frac{1}{2} A K\\), \\(L Q = \\frac{1}{2} B L\\), and \\(M R = \\frac{1}{2} C M\\). Find the area of triangle \\(P Q ... | math | {
"style": "rule",
"ground_truth": "25/16"
} | {
"split": "train",
"index": 21607,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sale prices at the Ajax Outlet Store are $50\\%$ below original prices. On Saturdays an additional discount of $20\\%$ off the sale price is given. What is the Saturday price of a coat whose original price is $\\$ 180$? Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "$72"
} | {
"split": "train",
"index": 21608,
"solution": "1. **Calculate the first discount of 50% on the original price**: The original price of the coat is $\\$180$. A 50% discount reduces the price by half:\n \\[\n 180 \\times 50\\% = 180 \\times \\frac{1}{2} = 90\n \\]\n So, the price after the first discount ... |
deepscaler | [
{
"role": "user",
"content": "Given that a school has 5 top students and 3 teachers, where each teacher mentors no more than 2 students, calculate the number of different mentorship arrangements possible. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 21609,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $k$ be a positive real number. In the $X-Y$ coordinate plane, let $S$ be the set of all points of the form $(x,x^2+k)$ where $x\\in\\mathbb{R}$ . Let $C$ be the set of all circles whose center lies in $S$ , and which are tangent to $X$ -axis. Find the minimum ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 21610,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $x$ is a multiple of $46200$, determine the greatest common divisor of $f(x) = (3x + 5)(5x + 3)(11x + 6)(x + 11)$ and $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "990"
} | {
"split": "train",
"index": 21611,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many positive integers $k$ do the lines with equations $9x+4y=600$ and $kx-4y=24$ intersect at a point whose coordinates are positive integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 21612,
"solution": "Suppose that $k$ is a fixed, but unknown, positive integer. Suppose also that the lines with equations $9x+4y=600$ and $kx-4y=24$ intersect at the point with positive integer coordinates $(x, y)$. Since $9x+4y=600$ and $kx-4y=24$, adding these equations, we get $9x... |
deepscaler | [
{
"role": "user",
"content": "If \\( 3-5+7=6-x \\), what is the value of \\( x \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 21613,
"solution": "Simplifying the left side of the equation, we obtain \\( 5=6-x \\). Therefore, \\( x=6-5=1 \\)."
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ is inscribed in a circle. The measure of the non-overlapping minor arcs $AB$, $BC$ and $CA$ are, respectively, $x+75^{\\circ} , 2x+25^{\\circ},3x-22^{\\circ}$. Then one interior angle of the triangle is: Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "61^{\\circ}"
} | {
"split": "train",
"index": 21614,
"solution": "1. **Identify the relationship between the arcs and the angles**: Since triangle $ABC$ is inscribed in a circle, the measure of each interior angle of the triangle is half the measure of its corresponding arc. This is due to the Inscribed Angle Theorem, which state... |
deepscaler | [
{
"role": "user",
"content": "How many multiples of 5 are between 100 and 400? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 21615,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any positive integers $a$ and $b$ with $b>1$, let $s_{b}(a)$ be the sum of the digits of $a$ when it is written in base $b$. Suppose $n$ is a positive integer such that $$\\sum_{i=1}^{\\left\\lfloor\\log _{23} n\\right\\rfloor} s_{20}\\left(\\left\\lfloor\\frac{n}{23^{i}}\\... | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 21616,
"solution": "First we will prove that $$s_{a}(n)=n-(a-1)\\left(\\sum_{i=1}^{\\infty}\\left\\lfloor\\frac{n}{a^{i}}\\right\\rfloor\\right)$$ If $n=\\left(n_{k} n_{k-1} \\cdots n_{1} n_{0}\\right)_{a}$, then the digit $n_{i}$ contributes $n_{i}$ to the left side of the sum, while... |
deepscaler | [
{
"role": "user",
"content": "Given that $|\\overrightarrow{a}|=1$, $|\\overrightarrow{b}|=\\sqrt{2}$, and $\\overrightarrow{a} \\bot (\\overrightarrow{a}-\\overrightarrow{b})$, find the projection of vector $\\overrightarrow{a}$ in the direction of vector $\\overrightarrow{b}$. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 21617,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the numbers $1, 2, 3, 4, 5$, 3 numbers are randomly drawn (with replacement) to form a three-digit number. What is the probability that the sum of its digits equals 9? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{19}{125}"
} | {
"split": "train",
"index": 21618,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each day 289 students are divided into 17 groups of 17. No two students are ever in the same group more than once. What is the largest number of days that this can be done? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 21619,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points $A, B$, and $C$ lie in that order on line $\\ell$, such that $A B=3$ and $B C=2$. Point $H$ is such that $C H$ is perpendicular to $\\ell$. Determine the length $C H$ such that $\\angle A H B$ is as large as possible. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "\\sqrt{10}"
} | {
"split": "train",
"index": 21620,
"solution": "Let $\\omega$ denote the circumcircle of triangle $A B H$. Since $A B$ is fixed, the smaller the radius of $\\omega$, the bigger the angle $A H B$. If $\\omega$ crosses the line $C H$ in more than one point, then there exists a smaller circle that goes through $A$ ... |
deepscaler | [
{
"role": "user",
"content": "Compute the number of positive real numbers $x$ that satisfy $\\left(3 \\cdot 2^{\\left\\lfloor\\log _{2} x\\right\\rfloor}-x\\right)^{16}=2022 x^{13}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 21621,
"solution": "Let $f(x)=3 \\cdot 2^{\\left\\lfloor\\log _{2} x\\right\\rfloor}-x$. Note that for each integer $i$, if $x \\in\\left[2^{i}, 2^{i+1}\\right)$, then $f(x)=3 \\cdot 2^{i}-x$. This is a line segment from $\\left(2^{i}, 2^{i+1}\\right)$ to $\\left(2^{i+1}, 2^{i}\\right... |
deepscaler | [
{
"role": "user",
"content": "In the rhombus \\(ABCD\\), the angle \\(\\angle ABC = 60^{\\circ}\\). A circle is tangent to the line \\(AD\\) at point \\(A\\), and the center of the circle lies inside the rhombus. Tangents to the circle, drawn from point \\(C\\), are perpendicular. Find the ratio of the peri... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3} + \\sqrt{7}}{\\pi}"
} | {
"split": "train",
"index": 21622,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the four vertices of the tetrahedron $P-ABC$ are all on the surface of a sphere with radius $3$, and $PA$, $PB$, $PC$ are mutually perpendicular, find the maximum value of the lateral surface area of the tetrahedron $P-ABC$. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 21623,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among the natural numbers not exceeding 10,000, calculate the number of odd numbers with distinct digits. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2605"
} | {
"split": "train",
"index": 21624,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$ABCD$ is a convex quadrilateral such that $AB=2$, $BC=3$, $CD=7$, and $AD=6$. It also has an incircle. Given that $\\angle ABC$ is right, determine the radius of this incircle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1+\\sqrt{13}}{3}"
} | {
"split": "train",
"index": 21625,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the polynomial $f(x) = 2x^7 + x^6 + x^4 + x^2 + 1$, calculate the value of $V_2$ using Horner's method when $x=2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 21626,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiao Ming observed a faucet that was continuously dripping water due to damage. In order to investigate the waste caused by the water leakage, Xiao Ming placed a graduated cylinder under the faucet to collect water and recorded the total amount of water in the cylinder every mi... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 21627,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $XYZ,$ $\\angle Y = 45^\\circ,$ $\\angle Z = 90^\\circ,$ and $XZ = 6.$ Find $YZ.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{2}"
} | {
"split": "train",
"index": 21628,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate: $(723)(723)-(722)(724)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 21629,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For some particular value of $N$, when $(a+b+c+d+1)^N$ is expanded and like terms are combined, the resulting expression contains exactly $1001$ terms that include all four variables $a, b,c,$ and $d$, each to some positive power. What is $N$? Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 21630,
"solution": "1. **Identify the form of the terms**: We are interested in terms that include all four variables $a, b, c, d$ each raised to some positive power in the expansion of $(a+b+c+d+1)^N$. This means we are looking for terms of the form $a^x b^y c^z d^w 1^t$ where $x, y,... |
deepscaler | [
{
"role": "user",
"content": "What is the value of $x$ if $|x-1| = |x-2|$? 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{3}{2}"
} | {
"split": "train",
"index": 21631,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x) = -7x^4 + 3x^3 + x - 5$, and $g(x)$ is a polynomial such that the degree of $f(x) + g(x)$ is 1, then what is the degree of $g(x)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 21632,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the polar coordinate system, circle $C$ is centered at point $C\\left(2, -\\frac{\\pi}{6}\\right)$ with a radius of $2$.\n\n$(1)$ Find the polar equation of circle $C$;\n\n$(2)$ Find the length of the chord cut from circle $C$ by the line $l$: $\\theta = -\\frac{5\\pi}{12} (... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 21633,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Bob the bomb-defuser has stumbled upon an active bomb. He opens it up, and finds the red and green wires conveniently located for him to cut. Being a seasoned member of the bomb-squad, Bob quickly determines that it is the green wire that he should cut, and puts his wirecutters... | math | {
"style": "rule",
"ground_truth": "\\frac{23}{30}"
} | {
"split": "train",
"index": 21634,
"solution": "Suppose Bob makes $n$ independent decisions, with probabilities of switching $p_{1}, p_{2}, \\ldots, p_{n}$. Then in the expansion of the product $$P(x)=\\left(p_{1}+\\left(1-p_{1}\\right) x\\right)\\left(p_{2}+\\left(1-p_{2}\\right) x\\right) \\cdots\\left(p_{n}+\... |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive integer $n$ such that $\\frac{5^{n+1}+2^{n+1}}{5^{n}+2^{n}}>4.99$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 21635,
"solution": "Writing $5^{n+1}=5 \\cdot 5^{n}$ and $2^{n+1}=2 \\cdot 2^{n}$ and cross-multiplying yields $0.01 \\cdot 5^{n}>2.99 \\cdot 2^{n}$, and re-arranging yields $(2.5)^{n}>299$. A straightforward calculation shows that the smallest $n$ for which this is true is $n=7$."
} |
deepscaler | [
{
"role": "user",
"content": "Two circles of radius 3 are centered at $(0,3)$ and at $(3,0)$. What is the area of the intersection of the interiors of these two circles? Express your answer in terms of $\\pi$ in its simplest form. 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": 21636,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $f$ is defined on the set of integers and satisfies \\[f(n)= \\begin{cases} n-3 & \\mbox{if }n\\ge 1000 \\\\ f(f(n+5)) & \\mbox{if }n<1000. \\end{cases}\\]Find $f(84)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "997"
} | {
"split": "train",
"index": 21637,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Anna, Stephanie, and James all start running around a track at 8:00. Anna completes a lap every 4 minutes, Stephanie finishes a lap every 7 minutes, and James finishes a lap every 6 minutes. What is the earliest time when all three meet back at the beginning? Let's think step b... | math | {
"style": "rule",
"ground_truth": "9:24"
} | {
"split": "train",
"index": 21638,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a tetrahedron \\(A B C D\\) with the following edge lengths: \\(A B = 2\\), \\(C D = 2\\sqrt{5}\\), \\(A C = B D = 3\\), and \\(A D = B C = \\sqrt{5}\\), find the volume of the tetrahedron. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4/3"
} | {
"split": "train",
"index": 21639,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the diagonals of a rhombus are always perpendicular bisectors of each other, what is the area of a rhombus with side length $\\sqrt{89}$ units and diagonals that differ by 6 units? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 21640,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Joyce made $12$ of her first $30$ shots in the first three games of this basketball game, so her seasonal shooting average was $40\\%$. In her next game, she took $10$ shots and raised her seasonal shooting average to $50\\%$. How many of these $10$ shots did she make? Let's ... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 21641,
"solution": "1. **Calculate Initial Performance**: Joyce made 12 out of her first 30 shots. This gives her an initial shooting average:\n \\[\n \\frac{12}{30} = 0.4 \\text{ or } 40\\%\n \\]\n\n2. **Determine Total Shots After Four Games**: After the next game, where she t... |
deepscaler | [
{
"role": "user",
"content": "A circle centered at $A$ with a radius of $1$ and a circle centered at $B$ with a radius of $4$ are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. What is the radius of the third circle? [asy]\ndraw((-3,0)--... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"split": "train",
"index": 21642,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes fiv... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 21643,
"solution": "To solve this problem, we need to find the number of steps in the staircase such that Dash takes 19 fewer jumps than Cozy. Let $n$ be the total number of steps in the staircase. Cozy jumps 2 steps at a time, and Dash jumps 5 steps at a time. We can express the numb... |
deepscaler | [
{
"role": "user",
"content": "The sum of two prime numbers is $85$. What is the product of these two prime numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "166"
} | {
"split": "train",
"index": 21644,
"solution": "1. **Identify the nature of the prime numbers**: The sum of two prime numbers is given as $85$, which is an odd number. Since the sum of two odd numbers is even and the sum of two even numbers is even, one of these primes must be even and the other must be odd (bec... |
deepscaler | [
{
"role": "user",
"content": "A merchant's cumulative sales from January to May reached 38.6 million yuan. It is predicted that the sales in June will be 5 million yuan, the sales in July will increase by x% compared to June, and the sales in August will increase by x% compared to July. The total sales in S... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 21645,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define the operation \"\" such that $ab = a^2 + 2ab - b^2$. Let the function $f(x) = x2$, and the equation $f(x) = \\lg|x + 2|$ (where $x \\neq -2$) 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... | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"split": "train",
"index": 21646,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of natural numbers \\( k \\) that do not exceed 291000 and such that \\( k^{2} - 1 \\) is divisible by 291. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4000"
} | {
"split": "train",
"index": 21647,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The top of one tree is $16$ feet higher than the top of another tree. The heights of the two trees are in the ratio $3:4$. In feet, how tall is the taller tree? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 21648,
"solution": "1. **Define Variables:**\n Let the height of the taller tree be $h$ feet. According to the problem, the top of one tree is $16$ feet higher than the top of another tree. Therefore, the height of the shorter tree is $h - 16$ feet.\n\n2. **Set Up the Ratio:**\n T... |
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