data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Suppose $x$ and $y$ are real numbers such that $-1<x<y<1$. Let $G$ be the sum of the geometric series whose first term is $x$ and whose ratio is $y$, and let $G^{\\prime}$ be the sum of the geometric series whose first term is $y$ and ratio is $x$. If $G=G^{\\prime}$, find $x+y... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 4600,
"solution": "We note that $G=x /(1-y)$ and $G^{\\prime}=y /(1-x)$. Setting them equal gives $x /(1-y)=$ $y /(1-x) \\Rightarrow x^{2}-x=y^{2}-x \\Rightarrow(x+y-1)(x-y)=0$, so we get that $x+y-1=0 \\Rightarrow x+y=1$."
} |
deepscaler | [
{
"role": "user",
"content": "Dima took the fractional-linear function \\(\\frac{a x + 2b}{c x + 2d}\\), where \\(a, b, c, d\\) are positive numbers, and summed it with the remaining 23 functions obtained from it by permuting the numbers \\(a, b, c, d\\). Find the root of the sum of all these functions, ind... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 4601,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One leg of a right triangle is 12 inches, and the measure of the angle opposite that leg is $30^\\circ$. What is the number of inches in the hypotenuse of the triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 4602,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find any solution to the rebus\n\n$$\n\\overline{A B C A}=182 \\cdot \\overline{C D}\n$$\n\nwhere \\( A, B, C, D \\) are four distinct non-zero digits (the notation \\(\\overline{X Y \\ldots Z}\\) denotes the decimal representation of a number).\n\nAs an answer, write the four-... | math | {
"style": "rule",
"ground_truth": "2916"
} | {
"split": "train",
"index": 4603,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of each of the four congruent L-shaped regions of this 100-inch by 100-inch square is 3/16 of the total area. How many inches long is the side of the center square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 4604,
"solution": "1. **Identify the total area of the large square**: The large square has side lengths of 100 inches, so its total area is:\n \\[\n 100 \\times 100 = 10,000 \\text{ square inches}\n \\]\n\n2. **Calculate the total area occupied by the L-shaped regions**: Each L... |
deepscaler | [
{
"role": "user",
"content": "How many subsets of two elements can be removed from the set $\\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\\}$ so that the mean (average) of the remaining numbers is 6? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 4605,
"solution": "\n1. **Calculate the total sum of the original set**: \n The set given is $\\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\\}$. The sum of all elements in this set can be calculated using the formula for the sum of an arithmetic series:\n \\[\n \\text{Sum} = \\frac{n}{2}... |
deepscaler | [
{
"role": "user",
"content": "Circle $C_0$ has radius $1$, and the point $A_0$ is a point on the circle. Circle $C_1$ has radius $r<1$ and is internally tangent to $C_0$ at point $A_0$. Point $A_1$ lies on circle $C_1$ so that $A_1$ is located $90^{\\circ}$ counterclockwise from $A_0$ on $C_1$. Circle $C_2$... | math | {
"style": "rule",
"ground_truth": "110"
} | {
"split": "train",
"index": 4606,
"solution": "Let the center of circle $C_i$ be $O_i$. Note that $O_0BO_1$ is a right triangle, with right angle at $B$. Also, $O_1B=\\frac{11}{60}O_0B$, or $O_0B = \\frac{60}{61}O_0O_1$. It is clear that $O_0O_1=1-r=\\frac{49}{60}$, so $O_0B=\\frac{60}{61}\\times\\frac{49}{60}=\... |
deepscaler | [
{
"role": "user",
"content": "For certain real values of $a, b, c,$ and $d_{},$ the equation $x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $13+i$ and the sum of the other two roots is $3+4i,$ where $i^2 = -1.$ Find $b.$ Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 4607,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the sequence $\\{a\\_n\\}(n\\geqslant 1,n\\in\\mathbb{N})$ that satisfies $a\\_1=2$, $a\\_2=6$, and $a\\_{n+2}-2a\\_{n+1}+a\\_n=2$. If $[x]$ represents the largest integer not exceeding $x$, then the value of $\\left[\\frac{2017}{a\\_1}+\\frac{2017}{a\\_2}+...+\\frac{2... | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"split": "train",
"index": 4608,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(\\sin \\alpha + \\cos \\alpha) = \\sin \\alpha \\cdot \\cos \\alpha$, determine the domain of $f(x)$ and the value of $f\\left(\\sin \\frac{\\pi}{6}\\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{8}"
} | {
"split": "train",
"index": 4609,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two numbers $90$ and $m$ share exactly three positive divisors. What is the greatest of these three common divisors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 4610,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When flipping a fair coin, what is the probability that the first two flips are both heads? 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{1}{4}"
} | {
"split": "train",
"index": 4611,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Choose $4-4$: Parameter Equation Lecture. In the plane rectangular coordinate system $xOy$, with $O$ as the pole and the non-negative half-axis of $x$ as the polar axis, establish a polar coordinate system. The polar coordinates of point $P$ are $(2\\sqrt{3}, \\dfrac{\\pi}{6})$... | math | {
"style": "rule",
"ground_truth": "\\dfrac{11\\sqrt{5}}{10} - 1"
} | {
"split": "train",
"index": 4612,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the positive factors of 48? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "124"
} | {
"split": "train",
"index": 4613,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $V$ be a rectangular prism with integer side lengths. The largest face has area 240 and the smallest face has area 48. A third face has area $x$, where $x$ is not equal to 48 or 240. What is the sum of all possible values of $x$? Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "260"
} | {
"split": "train",
"index": 4614,
"solution": "Let the length, width, and height of the prism be $s_{1}, s_{2}, s_{3}$. Without loss of generality, assume that $s_{1} \\leq s_{2} \\leq s_{3}$. Then, we have that $s_{1} s_{2}=48$ and $s_{2} s_{3}=240$. Noting that $s_{1} \\leq s_{2}$, we must have $\\left(s_{1}, ... |
deepscaler | [
{
"role": "user",
"content": "If each of two intersecting lines intersects a hyperbola and neither line is tangent to the hyperbola, then the possible number of points of intersection with the hyperbola is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2, 3, or 4"
} | {
"split": "train",
"index": 4615,
"solution": "To solve this problem, we need to analyze the possible number of intersection points between two lines and a hyperbola, given that neither line is tangent to the hyperbola.\n\n1. **Understanding the Hyperbola**: Consider the standard hyperbola given by the equation ... |
deepscaler | [
{
"role": "user",
"content": "Given the binomial $(ax+b)^{n}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+…+a_{n}x^{n}(a，b∈R，n∈N^{*})$.\n\n(1) When $a=b=1$, $n=6$, find:\n ① The value of $a_{1}+a_{2}+a_{3}+…+a_{n}$.\n ② The value of $a_{1}+2a_{2}+3a_{3}+…+na_{n}$.\n\n(2) When $a=1$, $b=- \\sqrt {3}$, $n=8$, find th... | math | {
"style": "rule",
"ground_truth": "256"
} | {
"split": "train",
"index": 4616,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p = 101$ and let $S$ be the set of $p$ -tuples $(a_1, a_2, \\dots, a_p) \\in \\mathbb{Z}^p$ of integers. Let $N$ denote the number of functions $f: S \\to \\{0, 1, \\dots, p-1\\}$ such that\n\n\t\n- $f(a + b) + f(a - b) \\equiv 2\\big(f(a) + f(b)\\big) \\pmod{p}... | math | {
"style": "rule",
"ground_truth": "5152"
} | {
"split": "train",
"index": 4617,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The mathematician John is having trouble remembering his girlfriend Alicia's 7-digit phone number. He remembers that the first four digits consist of one 1, one 2, and two 3s. He also remembers that the fifth digit is either a 4 or 5. While he has no memory of the sixth digit, ... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 4618,
"solution": "There are $\\frac{4!}{2!}=12$ possibilities for the first four digits. There are two possibilities for the fifth digit. There are 10 possibilities for the sixth digit, and this uniquely determines the seventh digit. So he has to dial $12 \\cdot 2 \\cdot 10=240$ numb... |
deepscaler | [
{
"role": "user",
"content": "If the inequality system $\\left\\{\\begin{array}{l}{x-m＞0}\\\\{x-2＜0}\\end{array}\\right.$ has only one positive integer solution, then write down a value of $m$ that satisfies the condition: ______. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"split": "train",
"index": 4619,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $3x-5=10x+9$, what is $4(x+7)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 4620,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $2^{11} \\times 6^{5}=4^{x} \\times 3^{y}$ for some positive integers $x$ and $y$, what is the value of $x+y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 4621,
"solution": "Manipulating the left side, $2^{11} \\times 6^{5}=2^{11} \\times(2 \\times 3)^{5}=2^{11} \\times 2^{5} \\times 3^{5}=2^{16} \\times 3^{5}$. Since $4^{x} \\times 3^{y}=2^{16} \\times 3^{5}$ and $x$ and $y$ are positive integers, then $y=5$ (because $4^{x}$ has no fac... |
deepscaler | [
{
"role": "user",
"content": "The sequence $b_1, b_2, b_3, \\dots$ satisfies $b_1 = 25,$ $b_{12} = 125,$ and for all $n \\ge 3,$ $b_n$ is the arithmetic mean of the first $n - 1$ terms. Find $b_2.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "225"
} | {
"split": "train",
"index": 4622,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an integer sequence \\(\\{a_i\\}\\) defined as follows:\n\\[ a_i = \\begin{cases} \ni, & \\text{if } 1 \\leq i \\leq 5; \\\\\na_1 a_2 \\cdots a_{i-1} - 1, & \\text{if } i > 5.\n\\end{cases} \\]\n\nFind the value of \\(\\sum_{i=1}^{2019} a_i^2 - a_1 a_2 \\cdots a_{2019}\\)... | math | {
"style": "rule",
"ground_truth": "1949"
} | {
"split": "train",
"index": 4623,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five positive integers (not necessarily all different) are written on five cards. Boris calculates the sum of the numbers on every pair of cards. He obtains only three different totals: 57, 70, and 83. What is the largest integer on any card? Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 4624,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "People have long been exploring the numerical solution of high-degree equations. Newton gave a numerical solution method for high-degree algebraic equations in his book \"Fluxions.\" This method for finding the roots of equations has been widely used in the scientific community... | math | {
"style": "rule",
"ground_truth": "-\\frac{11}{8}"
} | {
"split": "train",
"index": 4625,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of squares of all distinct complex numbers $x$ satisfying the equation $0=4 x^{10}-7 x^{9}+5 x^{8}-8 x^{7}+12 x^{6}-12 x^{5}+12 x^{4}-8 x^{3}+5 x^{2}-7 x+4$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{7}{16}"
} | {
"split": "train",
"index": 4626,
"solution": "For convenience denote the polynomial by $P(x)$. Notice $4+8=7+5=12$ and that the consecutive terms $12 x^{6}-12 x^{5}+12 x^{4}$ are the leading terms of $12 \\Phi_{14}(x)$, which is suggestive. Indeed, consider $\\omega$ a primitive 14 -th root of unity; since $\\o... |
deepscaler | [
{
"role": "user",
"content": "At 7:00, five sheep, designated as A, B, C, D, and E, have distances to Wolf Castle forming an arithmetic sequence with a common difference of 20 meters. At 8:00, these same five sheep have distances to Wolf Castle forming another arithmetic sequence, but with a common differen... | math | {
"style": "rule",
"ground_truth": "140"
} | {
"split": "train",
"index": 4627,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There exist $r$ unique nonnegative integers $n_1 > n_2 > \\cdots > n_r$ and $r$ unique integers $a_k$ ($1\\le k\\le r$) with each $a_k$ either $1$ or $- 1$ such that \\[a_13^{n_1} + a_23^{n_2} + \\cdots + a_r3^{n_r} = 2008.\\] Find $n_1 + n_2 + \\cdots + n_r$. Let's think step ... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 4628,
"solution": "In base $3$, we find that $\\overline{2008}_{10} = \\overline{2202101}_{3}$. In other words,\n$2008 = 2 \\cdot 3^{6} + 2 \\cdot 3^{5} + 2 \\cdot 3^3 + 1 \\cdot 3^2 + 1 \\cdot 3^0$\n In order to rewrite as a sum of perfect powers of $3$, we can use the fact that $2 \... |
deepscaler | [
{
"role": "user",
"content": "If $\\log_2 x^2 + \\log_{1/2} x = 5,$ compute $x.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 4629,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the diameter $d=\\sqrt[3]{\\dfrac{16}{9}V}$, find the volume $V$ of the sphere with a radius of $\\dfrac{1}{3}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 4630,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x, y$ and $z$ be consecutive integers such that\n\\[\\frac 1x+\\frac 1y+\\frac 1z >\\frac{1}{45}.\\]\nFind the maximum value of $x + y + z$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "402"
} | {
"split": "train",
"index": 4631,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Assume the quartic $x^{4}-a x^{3}+b x^{2}-a x+d=0$ has four real roots $\\frac{1}{2} \\leq x_{1}, x_{2}, x_{3}, x_{4} \\leq 2$. Find the maximum possible value of $\\frac{\\left(x_{1}+x_{2}\\right)\\left(x_{1}+x_{3}\\right) x_{4}}{\\left(x_{4}+x_{2}\\right)\\left(x_{4}+x_{3}\\r... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{4}"
} | {
"split": "train",
"index": 4632,
"solution": "We can rewrite the expression as $$\\frac{x_{4}^{2}}{x_{1}^{2}} \\cdot \\frac{\\left(x_{1}+x_{1}\\right)\\left(x_{1}+x_{2}\\right)\\left(x_{1}+x_{3}\\right)\\left(x_{1}+x_{4}\\right)}{\\left(x_{4}+x_{1}\\right)\\left(x_{4}+x_{2}\\right)\\left(x_{4}+x_{3}\\right)\\le... |
deepscaler | [
{
"role": "user",
"content": "Calculate the volume of the tetrahedron with vertices at points \\( A_{1}, A_{2}, A_{3}, A_{4} \\) and its height dropped from the vertex \\( A_{4} \\) onto the face \\( A_{1} A_{2} A_{3} \\).\n\nGiven points:\n\\( A_{1}(1, -1, 1) \\)\n\n\\( A_{2}(-2, 0, 3) \\)\n\n\\( A_{3}(2, ... | math | {
"style": "rule",
"ground_truth": "\\frac{33}{\\sqrt{101}}"
} | {
"split": "train",
"index": 4633,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\cos \\left( \\arctan \\frac{3}{4} \\right)$ and $\\tan \\left( \\arctan \\frac{3}{4} \\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 4634,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ units be the increase in circumference of a circle resulting from an increase in $\\pi$ units in the diameter. Then $P$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\pi^2"
} | {
"split": "train",
"index": 4635,
"solution": "1. **Understanding the Problem:**\n - We are given that the diameter of a circle increases by $\\pi$ units.\n - We need to find the corresponding increase in the circumference of the circle.\n\n2. **Using the Formula for Circumference:**\n - The circumference ... |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be a point inside triangle $ABC$ such that\n\\[\\overrightarrow{PA} + 2 \\overrightarrow{PB} + 3 \\overrightarrow{PC} = \\mathbf{0}.\\]Find the ratio of the area of triangle $ABC$ to the area of triangle $APC.$ Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 4636,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two circles have the same center O. Point X is the midpoint of segment OP. What is the ratio of the area of the circle with radius OX to the area of the circle with radius OP? Express your answer as a common fraction.\n\n[asy]\nimport graph;\ndraw(Circle((0,0),20));\ndraw(Circl... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 4637,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Every card in a deck has a picture of one shape - circle, square, or triangle, which is painted in one of the three colors - red, blue, or green. Furthermore, each color is applied in one of three shades - light, medium, or dark. The deck has 27 cards, with every shape-color-sh... | math | {
"style": "rule",
"ground_truth": "117"
} | {
"split": "train",
"index": 4638,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the sequence \\(1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \\cdots, 200, 200, \\cdots, 200\\), each number \\(n\\) appears \\(n\\) times consecutively, where \\(n \\in \\mathbf{N}\\) and \\(1 \\leq n \\leq 200\\). What is the median of this sequence? Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "142"
} | {
"split": "train",
"index": 4639,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Juan rolls a fair regular decagonal die marked with numbers from 1 to 10. Then Amal rolls a fair eight-sided die marked with numbers from 1 to 8. What is the probability that the product of the two rolls is a multiple of 4? Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "\\frac{19}{40}"
} | {
"split": "train",
"index": 4640,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer $a$ is input into a machine. If $a$ is odd, the output is $a+3$. If $a$ is even, the output is $a+5$. This process can be repeated using each successive output as the next input. If the input is $a=15$ and the machine is used 51 times, what is the final outpu... | math | {
"style": "rule",
"ground_truth": "218"
} | {
"split": "train",
"index": 4641,
"solution": "If $a$ is odd, the output is $a+3$, which is even because it is the sum of two odd integers. If $a$ is even, the output is $a+5$, which is odd, because it is the sum of an even integer and an odd integer. Starting with $a=15$ and using the machine 2 times, we obtain... |
deepscaler | [
{
"role": "user",
"content": "A train takes 60 seconds to pass through a 1260-meter-long bridge and 90 seconds to pass through a 2010-meter-long tunnel. What is the speed of the train in meters per second, and what is the length of the train? Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"split": "train",
"index": 4642,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S_{7}$ denote all the permutations of $1,2, \\ldots, 7$. For any \\pi \\in S_{7}$, let $f(\\pi)$ be the smallest positive integer $i$ such that \\pi(1), \\pi(2), \\ldots, \\pi(i)$ is a permutation of $1,2, \\ldots, i$. Compute \\sum_{\\pi \\in S_{7}} f(\\pi)$. Let's think ... | math | {
"style": "rule",
"ground_truth": "29093"
} | {
"split": "train",
"index": 4643,
"solution": "Extend the definition of $f$ to apply for any permutation of $1,2, \\ldots, n$, for any positive integer $n$. For positive integer $n$, let $g(n)$ denote the number of permutations \\pi$ of $1,2, \\ldots, n$ such that $f(\\pi)=n$. We have $g(1)=1$. For fixed $n, k$ ... |
deepscaler | [
{
"role": "user",
"content": "Determine how many ordered pairs of positive integers $(x, y)$, where $x < y$, have a harmonic mean of $5^{20}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 4644,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular octahedron $A B C D E F$ is given such that $A D, B E$, and $C F$ are perpendicular. Let $G, H$, and $I$ lie on edges $A B, B C$, and $C A$ respectively such that \\frac{A G}{G B}=\\frac{B H}{H C}=\\frac{C I}{I A}=\\rho. For some choice of $\\rho>1, G H, H I$, and $I ... | math | {
"style": "rule",
"ground_truth": "(1+\\sqrt{5}) / 2"
} | {
"split": "train",
"index": 4645,
"solution": "Let $J$ lie on edge $C E$ such that \\frac{E J}{J C}=\\rho. Then we must have that $H I J$ is another face of the icosahedron, so in particular, $H I=H J$. But since $B C$ and $C E$ are perpendicular, $H J=H C \\sqrt{2}$. By the Law of Cosines, $H I^{2}=H C^{2}+C I^... |
deepscaler | [
{
"role": "user",
"content": "A parallelogram is generated by the vectors $\\begin{pmatrix} 3 \\\\ 1 \\\\ 1 \\end{pmatrix}$ and $\\begin{pmatrix} 1 \\\\ -1 \\\\ -1 \\end{pmatrix}$. Find the cosine of the angle $\\theta$ between the diagonals of the parallelogram. Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 4646,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To popularize knowledge of fire safety, a certain school organized a competition on related knowledge. The competition is divided into two rounds, and each participant must participate in both rounds. If a participant wins in both rounds, they are considered to have won the com... | math | {
"style": "rule",
"ground_truth": "\\frac{223}{300}"
} | {
"split": "train",
"index": 4647,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest natural number whose digits sum up to 47. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "299999"
} | {
"split": "train",
"index": 4648,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $156 + 44 + 26 + 74$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 4649,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The recruits were standing in a row, one behind the other, facing the same direction. Among them were three brothers: Peter, Nicholas, and Denis. There were 50 people ahead of Peter, 100 ahead of Nicholas, and 170 ahead of Denis. Upon the command \"About face!\", everyone turne... | math | {
"style": "rule",
"ground_truth": "211"
} | {
"split": "train",
"index": 4650,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The digits of the positive integer $N$ consist only of 1s and 0s, and $225$ divides $N$. What is the minimum value of $N$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "111,111,100"
} | {
"split": "train",
"index": 4651,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest three-digit positive integer which can be written in the form \\( p q^{2} r \\), where \\( p, q \\), and \\( r \\) are distinct primes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "126"
} | {
"split": "train",
"index": 4652,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Supposed that $x$ and $y$ are nonzero real numbers such that $\\frac{3x+y}{x-3y}=-2$. What is the value of $\\frac{x+3y}{3x-y}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 4653,
"solution": "1. Start with the given equation:\n \\[\n \\frac{3x+y}{x-3y} = -2\n \\]\n\n2. Cross-multiply to eliminate the fraction:\n \\[\n 3x + y = -2(x - 3y)\n \\]\n\n3. Distribute the -2 on the right-hand side:\n \\[\n 3x + y = -2x + 6y\n \\]\n\n4. Rearrang... |
deepscaler | [
{
"role": "user",
"content": "Arc $AC$ is a quarter-circle with center $B$. The shaded region $ABC$ is \"rolled\" along a straight board $PQ$ until it reaches its original orientation for the first time with point $B$ landing at point $B^{\\prime}$. If $BC = \\frac{2}{\\pi}$ cm, what is the length of the pa... | math | {
"style": "rule",
"ground_truth": "3\\text{ cm}"
} | {
"split": "train",
"index": 4654,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an All-Area track meet, $216$ sprinters enter a $100-$meter dash competition. The track has $6$ lanes, so only $6$ sprinters can compete at a time. At the end of each race, the five non-winners are eliminated, and the winner will compete again in a later race. How many races... | math | {
"style": "rule",
"ground_truth": "43"
} | {
"split": "train",
"index": 4655,
"solution": "\nTo determine the number of races required to find the champion sprinter, we need to consider how the competition progresses. Each race has 6 sprinters, and only the winner of each race advances to the next round, while the other 5 are eliminated.\n\n#### Detailed ... |
deepscaler | [
{
"role": "user",
"content": "A parallelogram is defined by the vectors $\\begin{pmatrix} 3 \\\\ 0 \\\\ 4 \\end{pmatrix}$ and $\\begin{pmatrix} 1 \\\\ 2 \\\\ 3 \\end{pmatrix}$. Determine the cosine of the angle $\\theta$ between the diagonals of this parallelogram. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "\\frac{-11}{3\\sqrt{69}}"
} | {
"split": "train",
"index": 4656,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a square ABCD with side length 4 units. Points P and R are the midpoints of sides AB and CD, respectively. Points Q is located at the midpoint of side BC, and point S is located at the midpoint of side AD. Calculate the fraction of the square's total area that is shade... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 4657,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Aces are playing the Kings in a playoff series, where the first team to win 5 games wins the series. Each game's outcome leads to the Aces winning with a probability of $\\dfrac{7}{10}$, and there are no draws. Calculate the probability that the Aces win the series. Let's t... | math | {
"style": "rule",
"ground_truth": "90\\%"
} | {
"split": "train",
"index": 4658,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S_n$ be the sum of the reciprocals of the non-zero digits of the integers from $1$ to $10^n$ inclusive. Find the smallest positive integer $n$ for which $S_n$ is an integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 4659,
"solution": "Let $K = \\sum_{i=1}^{9}{\\frac{1}{i}}$. Examining the terms in $S_1$, we see that $S_1 = K + 1$ since each digit $n$ appears once and 1 appears an extra time. Now consider writing out $S_2$. Each term of $K$ will appear 10 times in the units place and 10 times in t... |
deepscaler | [
{
"role": "user",
"content": "There are \\( n \\) pieces of paper, each containing 3 different positive integers no greater than \\( n \\). Any two pieces of paper share exactly one common number. Find the sum of all the numbers written on these pieces of paper. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 4660,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T$ be a right triangle with sides having lengths $3$ , $4$ , and $5$ . A point $P$ is called *awesome* if P is the center of a parallelogram whose vertices all lie on the boundary of $T$ . What is the area of the set of awesome points? Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"split": "train",
"index": 4661,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m$ and $n$ satisfy $mn=4$ and $m+n=5$. What is $|m-n|$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 4662,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the sum of the coefficients of $P(x)$ if $\\left(20 x^{27}+2 x^{2}+1\\right) P(x)=2001 x^{2001}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "87"
} | {
"split": "train",
"index": 4663,
"solution": "The sum of coefficients of $f(x)$ is the value of $f(1)$ for any polynomial $f$. Plugging in 1 to the above equation, $P(1)=\\frac{2001}{23}=87$."
} |
deepscaler | [
{
"role": "user",
"content": "Six men and some number of women stand in a line in random order. Let $p$ be the probability that a group of at least four men stand together in the line, given that every man stands next to at least one other man. Find the least number of women in the line such that $p$ does n... | math | {
"style": "rule",
"ground_truth": "594"
} | {
"split": "train",
"index": 4664,
"solution": "Let $n$ be the number of women present, and let _ be some positive number of women between groups of men. Since the problem states that every man stands next to another man, there cannot be isolated men. Thus, there are five cases to consider, where $(k)$ refers to ... |
deepscaler | [
{
"role": "user",
"content": "China Unicom charges for mobile phone calls with two types of packages: Package $A$ (monthly fee of $15$ yuan, call fee of $0.1 yuan per minute) and Package $B$ (monthly fee of $0$ yuan, call fee of $0.15 yuan per minute). Let $y_{1}$ represent the monthly bill for Package $A$ ... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 4665,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four consecutive positive prime numbers have a sum that is divisible by three. What is the smallest possible value of this sum? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 4666,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(a, b, c \\in (0,1]\\) and \\(\\lambda\\) be a real number such that \\(\\frac{\\sqrt{3}}{\\sqrt{a+b+c}} \\geq 1+\\lambda(1-a)(1-b)(1-c)\\) is always satisfied. Find the maximum value of \\(\\lambda\\). Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "64/27"
} | {
"split": "train",
"index": 4667,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{a}$ and $\\mathbf{b}$ be orthogonal vectors. If $\\operatorname{proj}_{\\mathbf{a}} \\begin{pmatrix} 3 \\\\ -3 \\end{pmatrix} = \\begin{pmatrix} -\\frac{3}{5} \\\\ -\\frac{6}{5} \\end{pmatrix},$ then find $\\operatorname{proj}_{\\mathbf{b}} \\begin{pmatrix} 3 \\\... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} \\frac{18}{5} \\\\ -\\frac{9}{5} \\end{pmatrix}"
} | {
"split": "train",
"index": 4668,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of the digits of 3214 is 24. How many distinct four-digit positive integers are such that the product of their digits equals 12? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 4669,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Vasya has three cans of paint of different colors. In how many different ways can he paint a fence consisting of 10 planks so that any two adjacent planks are different colors and he uses all three colors? Provide a justification for your answer. Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "1530"
} | {
"split": "train",
"index": 4670,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\{ z_n \\}_{n \\ge 1}$ be a sequence of complex numbers, whose odd terms are real, even terms are purely imaginary, and for every positive integer $k$, $|z_k z_{k+1}|=2^k$. Denote $f_n=|z_1+z_2+\\cdots+z_n|,$ for $n=1,2,\\cdots$\n(1) Find the minimum of $f_{2020}$.\n(2) F... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 4671,
"solution": "\nLet \\(\\{ z_n \\}_{n \\ge 1}\\) be a sequence of complex numbers, whose odd terms are real, even terms are purely imaginary, and for every positive integer \\(k\\), \\(|z_k z_{k+1}|=2^k\\). Denote \\(f_n=|z_1+z_2+\\cdots+z_n|,\\) for \\(n=1,2,\\cdots\\).\n\n1. To... |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overset{→}{a}=(\\cos x,-1+\\sin x)$ and $\\overset{→}{b}=(2\\cos x,\\sin x)$,\n\n(1) Express $\\overset{→}{a}·\\overset{→}{b}$ in terms of $\\sin x$.\n\n(2) Find the maximum value of $\\overset{→}{a}·\\overset{→}{b}$ and the corresponding value of $x$. Let's th... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{4}"
} | {
"split": "train",
"index": 4672,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The cost of 1 piece of gum is 2 cents. What is the cost of 500 pieces of gum, in cents and in dollars? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10.00"
} | {
"split": "train",
"index": 4673,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a \\bowtie b = a + \\sqrt{b + \\sqrt{b + \\sqrt{b + \\ldots}}}$. If $5 \\bowtie x = 12$, find the value of $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 4674,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The cost of five pencils and one pen is $\\$2.50$, and the cost of one pencil and two pens is $\\$1.85$. What is the cost of two pencils and one pen? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1.45"
} | {
"split": "train",
"index": 4675,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify and then evaluate the expression:\n$$( \\frac {x}{x-1}- \\frac {x}{x^{2}-1})÷ \\frac {x^{2}-x}{x^{2}-2x+1}$$\nwhere $$x= \\sqrt {2}-1$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1- \\frac { \\sqrt {2}}{2}"
} | {
"split": "train",
"index": 4676,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We make colored cubes according to the following specifications:\n\n1. Each face of the cube is divided into two triangles by a diagonal, such that every drawn diagonal connects with two other such diagonals at each end point.\n2. Each face's two triangles must be of different ... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 4677,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "All natural numbers from 1 to 2017 inclusive were written in a row. How many times was the digit 7 written? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "602"
} | {
"split": "train",
"index": 4678,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive odd integers greater than 1 and less than $200$ are square-free? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 4679,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $C$ be the circle with equation $x^2-6y-3=-y^2-4x$. If $(a,b)$ is the center of $C$ and $r$ is its radius, what is the value of $a+b+r$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 4680,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The center of the circle with equation $x^2+y^2=8x-6y-20$ is the point $(x,y)$. What is $x+y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 4681,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a house at the center of a circular field. From it, 6 straight roads radiate, dividing the field into 6 equal sectors. Two geologists start their journey from the house, each choosing a road at random and traveling at a speed of 4 km/h. Determine the probability that t... | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"split": "train",
"index": 4682,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sphere with a radius of \\(\\sqrt{3}\\) has a cylindrical hole drilled through it; the axis of the cylinder passes through the center of the sphere, and the diameter of the base of the cylinder is equal to the radius of the sphere. Find the volume of the remaining part of the... | math | {
"style": "rule",
"ground_truth": "\\frac{9 \\pi}{2}"
} | {
"split": "train",
"index": 4683,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Without using any tables, find the exact value of the product:\n\\[ P = \\cos \\frac{\\pi}{15} \\cos \\frac{2\\pi}{15} \\cos \\frac{3\\pi}{15} \\cos \\frac{4\\pi}{15} \\cos \\frac{5\\pi}{15} \\cos \\frac{6\\pi}{15} \\cos \\frac{7\\pi}{15}. \\] Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "1/128"
} | {
"split": "train",
"index": 4684,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A$ and $B$ be the endpoints of a semicircular arc of radius $4$. This arc is divided into nine congruent arcs by eight equally spaced points $C_1$, $C_2$, $\\dots$, $C_8$. Draw all chords of the form $\\overline{AC_i}$ or $\\overline{BC_i}$. Find the product of the lengths... | math | {
"style": "rule",
"ground_truth": "38654705664"
} | {
"split": "train",
"index": 4685,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $2.5-0.32.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2.18"
} | {
"split": "train",
"index": 4686,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\sin (α- \\frac {π}{6})= \\frac {2}{3}$, $α∈(π, \\frac {3π}{2})$, $\\cos ( \\frac {π}{3}+β)= \\frac {5}{13}$, $β∈(0,π)$, find the value of $\\cos (β-α)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac{10+12 \\sqrt{5}}{39}"
} | {
"split": "train",
"index": 4687,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\frac{1}{2}x^{2}-a\\ln x+b$ where $a\\in R$.\n(I) If the equation of the tangent line to the curve $y=f(x)$ at $x=1$ is $3x-y-3=0$, find the values of the real numbers $a$ and $b$.\n(II) If $x=1$ is the extreme point of the function $f(x)$, find the va... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 4688,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a circle, there are 2018 points. Each of these points is labeled with an integer. Each number is greater than the sum of the two numbers that immediately precede it in a clockwise direction.\n\nDetermine the maximum possible number of positive numbers that can be among the 2... | math | {
"style": "rule",
"ground_truth": "1008"
} | {
"split": "train",
"index": 4689,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the product of $0.\\overline{6}$ and 6. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 4690,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nEgorov decided to open a savings account to buy a car worth 900,000 rubles. The initial deposit is 300,000 rubles. Every month, Egorov plans to add 15,000 rubles to his account. The bank offers a monthly interest rate of $12\\%$ per annum. The interest earned each month is ad... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 4691,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a+b=3$ and $a^3+b^3=81$, find $ab$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 4692,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\\\(f(x)\\\\) be defined on \\\\((-∞,+∞)\\\\) and satisfy \\\\(f(2-x)=f(2+x)\\\\) and \\\\(f(7-x)=f(7+x)\\\\). If in the closed interval \\\\([0,7]\\\\), only \\\\(f(1)=f(3)=0\\\\), then the number of roots of the equation \\\\(f(x)=0\\\\) in the closed interval \\\\([-200... | math | {
"style": "rule",
"ground_truth": "802"
} | {
"split": "train",
"index": 4693,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the product of the solutions of the equation $-35=-x^2-2x?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-35"
} | {
"split": "train",
"index": 4694,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a function $f(x)$ defined on $\\mathbb{R}$ that is an odd function, and the period of the function $f(2x+1)$ is 5, if $f(1) = 5$, calculate the value of $f(2009) + f(2010)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 4695,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Eight students from a university are preparing to carpool for a trip. There are two students from each grade level—freshmen, sophomores, juniors, and seniors—divided into two cars, Car A and Car B, with each car seating four students. The seating arrangement of the four student... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 4696,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a parabola $C: y^2 = 2px (p > 0)$ that passes through the point $(1, -2)$, a line $l$ through focus $F$ intersects the parabola $C$ at points $A$ and $B$. If $Q$ is the point $(-\\frac{7}{2}, 0)$ and $BQ \\perp BF$, find the value of $|BF| - |AF|$. Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"split": "train",
"index": 4697,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\frac{x-y}{x+y}=5$, what is the value of $\\frac{2x+3y}{3x-2y}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 4698,
"solution": "Since $\\frac{x-y}{x+y}=5$, then $x-y=5(x+y)$. This means that $x-y=5x+5y$ and so $0=4x+6y$ or $2x+3y=0$. Therefore, $\\frac{2x+3y}{3x-2y}=\\frac{0}{3x-2y}=0$."
} |
deepscaler | [
{
"role": "user",
"content": "Find the square root of $\\dfrac{10!}{210}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72\\sqrt{5}"
} | {
"split": "train",
"index": 4699,
"solution": ""
} |
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