data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Given a function $f(x)$ $(x \\in \\mathbb{R})$ that satisfies the equation $f(-x) = 8 - f(4 + x)$, and another function $g(x) = \\frac{4x + 3}{x - 2}$. If the graph of $f(x)$ has 168 intersection points with the graph of $g(x)$, denoted as $P_i(x_i, y_i)$ $(i = 1,2, \\dots, 168... | math | {
"style": "rule",
"ground_truth": "1008"
} | {
"split": "train",
"index": 6200,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $(2 + 1)(2^2 + 1^2)(2^4 + 1^4)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "255"
} | {
"split": "train",
"index": 6201,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area of the shaded region in the figure below? Round your answer to the nearest square centimeter.\n\n[asy]\ndraw((0,0) -- (3,0) -- (3,3) -- (0,3)--cycle) ; draw((3,0)-- (12,0) -- (12,9) -- (3, 9)--cycle);\n\nlabel ( \"3 cm\", (0,1.5), W); label ( \"3 cm\", (1.5 ,0)... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 6202,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sum of the first $n$ terms of the arithmetic sequences $\\{a_n\\}$ and $\\{b_n\\}$ are $(S_n)$ and $(T_n)$, respectively. If for any positive integer $n$, $\\frac{S_n}{T_n}=\\frac{2n-5}{3n-5}$, determine the value of $\\frac{a_7}{b_2+b_8}+\\frac{a_3}{b_4+b_6}$. L... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{22}"
} | {
"split": "train",
"index": 6203,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $x^{2}+y^{2}=4$, find the minimum value of $\\sqrt{2-y}+\\sqrt{5-2x}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 6204,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $|\\overrightarrow{a}|=1$, $|\\overrightarrow{b}|=2$, and $\\overrightarrow{a}⊥(\\overrightarrow{a}+\\overrightarrow{b})$, find the projection of $\\overrightarrow{a}$ onto $\\overrightarrow{b}$. 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": 6205,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five years from now, Billy's age will be twice Joe's current age. Currently, the sum of their ages is 60. How old is Billy right now? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "38\\frac{1}{3}"
} | {
"split": "train",
"index": 6206,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be an equilateral triangle with side length 1. Points $D, E, F$ lie inside triangle $A B C$ such that $A, E, F$ are collinear, $B, F, D$ are collinear, $C, D, E$ are collinear, and triangle $D E F$ is equilateral. Suppose that there exists a unique equilateral trian... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{1+\\sqrt[3]{2}}"
} | {
"split": "train",
"index": 6207,
"solution": "First, note that point $X$ can be constructed from intersection of $\\odot(D O F)$ and side $\\overline{B C}$. Thus, if there is a unique equilateral triangle, then we must have that $\\odot(D O F)$ is tangent to $\\overline{B C}$. Furthermore, $\\odot(D O F)$ is ta... |
deepscaler | [
{
"role": "user",
"content": "In a school, 40 students are enrolled in both the literature and science classes. Ten students received an A in literature and 18 received an A in science, including six who received an A in both subjects. Determine how many students did not receive an A in either subject. Let'... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 6208,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The repeating decimal \\( 0.\\dot{x}y\\dot{3} = \\frac{a}{27} \\), where \\( x \\) and \\( y \\) are distinct digits. Find the integer \\( a \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 6209,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six horizontal lines and five vertical lines are drawn in a plane. If a specific point, say (3, 4), exists in the coordinate plane, in how many ways can four lines be chosen such that a rectangular region enclosing the point (3, 4) is formed? Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 6210,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A ticket contains six digits \\(a, b, c, d, e, f\\). This ticket is said to be \"lucky\" if \\(a + b + c = d + e + f\\). How many lucky tickets are there (including the ticket 000000)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "55252"
} | {
"split": "train",
"index": 6211,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle ABC$ be a right triangle with right angle at $C.$ Let $D$ and $E$ be points on $\\overline{AB}$ with $D$ between $A$ and $E$ such that $\\overline{CD}$ and $\\overline{CE}$ trisect $\\angle C.$ If $\\frac{DE}{BE} = \\frac{8}{15},$ then $\\tan B$ can be written a... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 6212,
"solution": "Without loss of generality, set $CB = 1$. Then, by the Angle Bisector Theorem on triangle $DCB$, we have $CD = \\frac{8}{15}$. We apply the Law of Cosines to triangle $DCB$ to get $1 + \\frac{64}{225} - \\frac{8}{15} = BD^{2}$, which we can simplify to get $BD = \\f... |
deepscaler | [
{
"role": "user",
"content": "Given the quadratic function \\( y = x^{2} - \\frac{2n+1}{n(n+2)} x + \\frac{n+1}{n(n+2)^{2}} \\), the length of the segment intercepted on the \\( x \\)-axis is \\( d_n \\). Find the value of \\( \\sum_{n=1}^{100} d_n \\). Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "\\frac{7625}{10302}"
} | {
"split": "train",
"index": 6213,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "All vertices of a regular tetrahedron \\( A B C D \\) are located on one side of the plane \\( \\alpha \\). It turns out that the projections of the vertices of the tetrahedron onto the plane \\( \\alpha \\) are the vertices of a certain square. Find the value of \\(A B^{2}\\),... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 6214,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point \\((x, y)\\) is randomly picked from inside the rectangle with vertices \\((0,0)\\), \\((5,0)\\), \\((5,2)\\), and \\((0,2)\\). What is the probability that \\(x < 2y\\)? 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": 6215,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\cos x + \\cos y = \\frac{1}{2}$ and $\\sin x + \\sin y = \\frac{1}{3}$, find the value of $\\cos (x - y)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{59}{72}"
} | {
"split": "train",
"index": 6216,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of the eight vertices of a rectangular prism is truncated, and each face of the prism is divided by a diagonal cut into two triangles. Calculate the total number of edges of the altered shape. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 6217,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the system of equations\n\\begin{align*}\n4x+2y &= c, \\\\\n6y - 12x &= d,\n\\end{align*}\nwhere \\(d \\neq 0\\), find the value of \\(\\frac{c}{d}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{3}"
} | {
"split": "train",
"index": 6218,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The coach of the math training team needs to photocopy a set of materials for 23 team members. The on-campus copy shop charges 1.5 yuan per page for the first 300 pages and 1 yuan per page for any additional pages. The cost of photocopying these 23 sets of materials together is... | math | {
"style": "rule",
"ground_truth": "950"
} | {
"split": "train",
"index": 6219,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cube has side length 1. Find the product of the lengths of the diagonals of this cube (a diagonal is a line between two vertices that is not an edge). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "576"
} | {
"split": "train",
"index": 6220,
"solution": "There are 12 diagonals that go along a face and 4 that go through the center of the cube, so the answer is $\\sqrt{2}^{12} \\cdot \\sqrt{3}^{4}=576$."
} |
deepscaler | [
{
"role": "user",
"content": "Four balls of radius $1$ are mutually tangent, three resting on the floor and the fourth resting on the others.\nA tetrahedron, each of whose edges have length $s$, is circumscribed around the balls. Then $s$ equals Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "2+2\\sqrt{6}"
} | {
"split": "train",
"index": 6221,
"solution": "To solve this problem, we need to find the edge length $s$ of the tetrahedron that circumscribes four mutually tangent balls of radius $1$. The configuration described is such that three balls are resting on the floor and the fourth ball is resting on top of these t... |
deepscaler | [
{
"role": "user",
"content": "Carl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the c... | math | {
"style": "rule",
"ground_truth": "336"
} | {
"split": "train",
"index": 6222,
"solution": "1. **Determine the number of posts on each side**: \n Let $x$ be the number of posts on each of the shorter sides, including the corners. Since the longer side has twice as many posts as the shorter side, it has $2x$ posts, including the corners.\n\n2. **Account f... |
deepscaler | [
{
"role": "user",
"content": "In the drawing, 5 lines intersect at a single point. One of the resulting angles is $34^\\circ$. What is the sum of the four angles shaded in gray, in degrees? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "146"
} | {
"split": "train",
"index": 6223,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 6 locked suitcases and 6 keys for them. However, it is unknown which key opens which suitcase. What is the minimum number of attempts needed to ensure that all suitcases are opened? How many attempts are needed if there are 10 suitcases and 10 keys? Let's think step b... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 6224,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $101^{4} - 4 \\cdot 101^{3} + 6 \\cdot 101^2 - 4 \\cdot 101 + 1$? (No calculators!) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100000000"
} | {
"split": "train",
"index": 6225,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "During intervals, students played table tennis. Any two students played against each other no more than one game. At the end of the week, it turned out that Petya played half, Kolya played a third, and Vasya played a fifth of all the games played during the week. How many games... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 6226,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given non-zero vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfy $|\\overrightarrow{a}|=2|\\overrightarrow{b}|$, and $(\\overrightarrow{a}-\\overrightarrow{b})\\bot \\overrightarrow{b}$, then the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$ is ___... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 6227,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $a$ so that the lines described by\n\\[\\begin{pmatrix} 2 \\\\ -1 \\\\ 0 \\end{pmatrix} + t \\begin{pmatrix} a \\\\ -2 \\\\ 1 \\end{pmatrix}\\]and\n\\[\\begin{pmatrix} 1 \\\\ -3/2 \\\\ -5 \\end{pmatrix} + u \\begin{pmatrix} 1 \\\\ 3/2 \\\\ 2 \\end{pmatrix}\\]a... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 6228,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{m}=(\\sin x, -1)$ and $\\overrightarrow{n}=(\\sqrt{3}\\cos x, -\\frac{1}{2})$, let $f(x)=(\\overrightarrow{m}+\\overrightarrow{n})\\cdot \\overrightarrow{m}$.\n(1) Find the analytic expression for $f(x)$ and its intervals of monotonic increase;\n... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 6229,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow {a}, \\overrightarrow {b}$ that satisfy $\\overrightarrow {a}\\cdot ( \\overrightarrow {a}+ \\overrightarrow {b})=5$, and $|\\overrightarrow {a}|=2$, $|\\overrightarrow {b}|=1$, find the angle between vectors $\\overrightarrow {a}$ and $\\overrig... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 6230,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the quadratic $x^2+4mx+m$ has exactly one real root, find the positive value of $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac14"
} | {
"split": "train",
"index": 6231,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( f(x) = x - \\frac{x^3}{2} + \\frac{x^5}{2 \\cdot 4} - \\frac{x^7}{2 \\cdot 4 \\cdot 6} + \\cdots \\), and \\( g(x) = 1 + \\frac{x^2}{2^2} + \\frac{x^4}{2^2 \\cdot 4^2} + \\frac{x^6}{2^2 \\cdot 4^2 \\cdot 6^2} + \\cdots \\). Find \\( \\int_{0}^{\\infty} f(x) g(x) \\, dx ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{e}"
} | {
"split": "train",
"index": 6232,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five identical right-angled triangles can be arranged so that their larger acute angles touch to form a star. It is also possible to form a different star by arranging more of these triangles so that their smaller acute angles touch. How many triangles are needed to form the se... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 6233,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A store owner bought 2000 markers at $0.20 each. To make a minimum profit of $200, if he sells the markers for $0.50 each, calculate the number of markers he must sell at least to achieve or exceed this profit. Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "1200"
} | {
"split": "train",
"index": 6234,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cooperative receives apple and grape juice in identical containers and produces an apple-grape drink in identical cans. One container of apple juice is enough for exactly 6 cans of the drink, and one container of grape juice is enough for exactly 10 cans. When the recipe of t... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 6235,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $\\angle C = 4\\angle A$, $a = 36$, and $c = 60$. Determine the length of side $b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 6236,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that in acute triangle ABC, the sides opposite to angles A, B, C are a, b, c respectively, and bsinA = acos(B - $ \\frac{\\pi}{6}$).\n(1) Find the value of angle B.\n(2) If b = $\\sqrt{13}$, a = 4, and D is a point on AC such that S<sub>△ABD</sub> = 2$\\sqrt{3}$, find the... | math | {
"style": "rule",
"ground_truth": "\\frac{2 \\sqrt{13}}{3}"
} | {
"split": "train",
"index": 6237,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal{S}$ be the set $\\{1, 2, 3, \\dots, 12\\}$. Let $n$ be the number of sets of two non-empty disjoint subsets of $\\mathcal{S}$. Calculate the remainder when $n$ is divided by 500. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 6238,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the line $y=kx+b$ is tangent to the graph of the function $f\\left(x\\right)=\\frac{1}{2}x^{2}+\\ln x$, calculate 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": 6239,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the angle between the vectors $\\mathbf{a}$ and $\\mathbf{b}$ is $43^\\circ,$ what is the angle between the vectors $-\\mathbf{a}$ and $\\mathbf{b}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "137^\\circ"
} | {
"split": "train",
"index": 6240,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We denote by $\\mathbb{R}^\\plus{}$ the set of all positive real numbers.\n\nFind all functions $f: \\mathbb R^ \\plus{} \\rightarrow\\mathbb R^ \\plus{}$ which have the property:\n\\[f(x)f(y)\\equal{}2f(x\\plus{}yf(x))\\]\nfor all positive real numbers $x$ and $y$.\n\n[i] Let'... | math | {
"style": "rule",
"ground_truth": "f(x) = 2"
} | {
"split": "train",
"index": 6241,
"solution": "\nTo solve the problem, we need to find all functions \\( f: \\mathbb{R}^{+} \\rightarrow \\mathbb{R}^{+} \\) such that for all positive real numbers \\( x \\) and \\( y \\), the functional equation holds:\n\n\\[\nf(x)f(y) = 2f(x + yf(x)).\n\\]\n\n### Step 1: Analyz... |
deepscaler | [
{
"role": "user",
"content": "If \\(\\frac{5+6+7+8}{4} = \\frac{2014+2015+2016+2017}{N}\\), calculate the value of \\(N\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1240"
} | {
"split": "train",
"index": 6242,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $x$ so that the vectors $\\begin{pmatrix} 2 \\\\ 5 \\end{pmatrix}$ and $\\begin{pmatrix} x \\\\ -3 \\end{pmatrix}$ are orthogonal. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{15}{2}"
} | {
"split": "train",
"index": 6243,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An angle is drawn on a set of equally spaced parallel lines as shown. The ratio of the area of shaded region $C$ to the area of shaded region $B$ is 11/5. Find the ratio of shaded region $D$ to the area of shaded region $A.$\n[asy] defaultpen(linewidth(0.7)+fontsize(10)); for(i... | math | {
"style": "rule",
"ground_truth": "408"
} | {
"split": "train",
"index": 6244,
"solution": "Let the distances from the apex to the parallel lines be $x$ and $y$ and the distance between the intersections be $a,b.$ We know the area ratio means $\\frac{(x+4a)(y+4b)-(x+3a)(y+3b)}{(x+2a)(y+2b)-(x+a)(y+b)} =\\frac{5}{11}$ which simplifying yields $ab = 3ay+3bx.... |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive number that is both prime and a palindrome, and is exactly $8$ less than a perfect square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 6245,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $z$ be a complex number such that\n\\[|z^2 + 4| = |z(z + 2i)|.\\]Find the smallest possible value of $|z + i|.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 6246,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $7\\times 1$ board is completely covered by $m\\times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let $N$ be the number of tilings of the $7\\times 1$ boa... | math | {
"style": "rule",
"ground_truth": "106"
} | {
"split": "train",
"index": 6247,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We define $N$ as the set of natural numbers $n<10^6$ with the following property:\n\nThere exists an integer exponent $k$ with $1\\le k \\le 43$ , such that $2012|n^k-1$ .\n\nFind $|N|$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1988"
} | {
"split": "train",
"index": 6248,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a fixed point $C(2,0)$ and a line $l: x=8$ on a plane, $P$ is a moving point on the plane, $PQ \\perp l$, with the foot of the perpendicular being $Q$, and $\\left( \\overrightarrow{PC}+\\frac{1}{2}\\overrightarrow{PQ} \\right)\\cdot \\left( \\overrightarrow{PC}-\\frac{1}... | math | {
"style": "rule",
"ground_truth": "12-4\\sqrt{3}"
} | {
"split": "train",
"index": 6249,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using the six digits 0, 1, 2, 3, 4, 5, we can form _ _ _ _ _ different five-digit numbers that do not have repeating digits and are divisible by 5. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "216"
} | {
"split": "train",
"index": 6250,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that a polynomial of the form $p(x)=x^{2010} \\pm x^{2009} \\pm \\cdots \\pm x \\pm 1$ has no real roots. What is the maximum possible number of coefficients of -1 in $p$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1005"
} | {
"split": "train",
"index": 6251,
"solution": "Let $p(x)$ be a polynomial with the maximum number of minus signs. $p(x)$ cannot have more than 1005 minus signs, otherwise $p(1)<0$ and $p(2) \\geq 2^{2010}-2^{2009}-\\ldots-2-1=$ 1, which implies, by the Intermediate Value Theorem, that $p$ must have a root greate... |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\log_82$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac13"
} | {
"split": "train",
"index": 6252,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\( |z| = 2 \\) and \\( u = \\left|z^{2} - z + 1\\right| \\), find the minimum value of \\( u \\) (where \\( z \\in \\mathbf{C} \\)). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2} \\sqrt{3}"
} | {
"split": "train",
"index": 6253,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integers $n$ between 1 and 20 (inclusive) is $\\frac{n}{42}$ a repeating decimal? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 6254,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a rectangular sheet of paper, a picture in the shape of a \"cross\" is drawn from two rectangles $ABCD$ and $EFGH$, with sides parallel to the edges of the sheet. It is known that $AB=9$, $BC=5$, $EF=3$, and $FG=10$. Find the area of the quadrilateral $AFCH$. Let's think ste... | math | {
"style": "rule",
"ground_truth": "52.5"
} | {
"split": "train",
"index": 6255,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a parabola $C: y^2 = 3x$ with focus $F$, find the length of segment $AB$ where the line passing through $F$ at a $30^\\circ$ angle intersects the parabola at points $A$ and $B$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 6256,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, two side lengths are $2$ and $3$, and the cosine value of the included angle is $\\frac{1}{3}$. Find the radius of the circumscribed circle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9\\sqrt{2}}{8}"
} | {
"split": "train",
"index": 6257,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the $y$-coordinate of the point on the $y$-axis that is equidistant from points $A( -2, 0)$ and $B(-1,4)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{8}"
} | {
"split": "train",
"index": 6258,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $10$ birds on the ground. For any $5$ of them, there are at least $4$ birds on a circle. Determine the least possible number of birds on the circle with the most birds. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 6259,
"solution": "\nGiven that there are 10 birds on the ground and for any 5 of them, there are at least 4 birds on a circle, we need to determine the least possible number of birds on the circle with the most birds.\n\nTo solve this, consider the following steps:\n\n1. **Initial As... |
deepscaler | [
{
"role": "user",
"content": "If the three points $$\\begin{aligned} & (1, a, b) \\\\ & (a, 2, b) \\\\ & (a, b, 3) \\end{aligned}$$ are collinear (in 3-space), what is the value of $a+b$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 6260,
"solution": "The first two points are distinct (otherwise we would have $a=1$ and $a=2$ simultaneously), and they both lie on the plane $z=b$, so the whole line is in this plane and $b=3$. Reasoning similarly with the last two points gives $a=1$, so $a+b=4$."
} |
deepscaler | [
{
"role": "user",
"content": "In a new shooting competition, ten clay targets are set up in four hanging columns with four targets in column $A$, three in column $B$, two in column $C$, and one in column $D$. A shooter must continue following the sequence:\n1) The shooter selects one of the columns.\n2) The... | math | {
"style": "rule",
"ground_truth": "12600"
} | {
"split": "train",
"index": 6261,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of a number $M$ and six less than $M$ is $-5$. What is the sum of all possible values of $M$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 6262,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I planned to work 20 hours a week for 12 weeks this summer to earn $3000 to buy a used car. Unfortunately, due to unforeseen events, I wasn't able to work any hours during the first three weeks of the summer. How many hours per week do I need to work for the remaining summer to... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 6263,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fruit store sells a type of fruit. The purchase price is $30$ yuan per kilogram. According to a survey, when the selling price is $40$ yuan per kilogram, the store can sell an average of $500$ kilograms per day. For every $1$ yuan increase in the selling price, the store sell... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 6264,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the angle between the unit vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ is obtuse, the minimum value of $|\\overrightarrow{b} - t\\overrightarrow{a}| (t \\in \\mathbb{R})$ is $\\frac{\\sqrt{3}}{2}$, and $(\\overrightarrow{c} - \\overrightarrow{a}) \\cdot (... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3} + 1}{2}"
} | {
"split": "train",
"index": 6265,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If two distinct members of the set $\\{ 3, 7, 21, 27, 35, 42, 51 \\}$ are randomly selected and multiplied, what is the probability that the product is a multiple of 63? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{7}"
} | {
"split": "train",
"index": 6266,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $X$ and $Y$ are digits in base $d > 8$ such that $\\overline{XY}_d + \\overline{XX}_d = 234_d$. Find $X_d - Y_d$ in base $d$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 6267,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given every 8-digit whole number is a possible telephone number except those that begin with 0 or 1, or end in 9, determine the fraction of these telephone numbers that begin with 9 and end with 0. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{72}"
} | {
"split": "train",
"index": 6268,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $f$ satisfies the condition $$ f (x + 1) = \\frac{1 + f (x)}{1 - f (x)} $$ for all real $x$ , for which the function is defined. Determine $f(2012)$ , if we known that $f(1000)=2012$ . Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "2012"
} | {
"split": "train",
"index": 6269,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $b = 8$ and $n = 15$, calculate the number of positive factors of $b^n$ where both $b$ and $n$ are positive integers, with $n$ being 15. Determine if this choice of $b$ and $n$ maximizes the number of factors compared to similar calculations with other bases less tha... | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 6270,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sanitation workers plan to plant 7 trees in a row on one side of a road, choosing only from plane trees and willow trees. What is the number of planting methods where no two adjacent trees are both willows? Let's think step by step and put only the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 6271,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum value of the expression \\( (\\sin 2x + \\sin 3y + \\sin 4z)(\\cos 2x + \\cos 3y + \\cos 4z) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"split": "train",
"index": 6272,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the base five product of the numbers $203_{5}$ and $14_{5}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3402_5"
} | {
"split": "train",
"index": 6273,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Place the numbers 1, 2, 3, 4, 5, 6, 7, and 8 on the eight vertices of a cube such that the sum of any three numbers on a face is at least 10. Find the minimum sum of the four numbers on any face. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 6274,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Using equal-length toothpicks to form a rectangular diagram as shown, if the length of the rectangle is 20 toothpicks long and the width is 10 toothpicks long, how many toothpicks are used? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "430"
} | {
"split": "train",
"index": 6275,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $g(x) = 2x^2 - 3$ and $h(x) = 4x^3 +1$, what is the value of $g(h(-1))$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 6276,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the greatest common divisor of $45$ and $4410$ and the least common multiple of $45$ and $4410$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4455"
} | {
"split": "train",
"index": 6277,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two thousand points are given on a circle. Label one of the points $1$. From this point, count $2$ points in the clockwise direction and label this point $2$. From the point labeled $2$, count $3$ points in the clockwise direction and label this point $3$. (See figure.) Continu... | math | {
"style": "rule",
"ground_truth": "118"
} | {
"split": "train",
"index": 6278,
"solution": "The label $1993$ will occur on the $\\frac12(1993)(1994) \\pmod{2000}$th point around the circle. (Starting from 1) A number $n$ will only occupy the same point on the circle if $\\frac12(n)(n + 1)\\equiv \\frac12(1993)(1994) \\pmod{2000}$.\n Simplifying this expres... |
deepscaler | [
{
"role": "user",
"content": "Let $a, b, c, d, e$ be nonnegative integers such that $625 a+250 b+100 c+40 d+16 e=15^{3}$. What is the maximum possible value of $a+b+c+d+e$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "153"
} | {
"split": "train",
"index": 6279,
"solution": "The intuition is that as much should be in $e$ as possible. But divisibility obstructions like $16 \\nmid 15^{3}$ are in our way. However, the way the coefficients $5^{4}>5^{3} \\cdot 2>\\cdots$ are set up, we can at least easily avoid having $a, b, c, d$ too large ... |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbb{R}$ denote the set of the reals. Find all $f : \\mathbb{R} \\to \\mathbb{R}$ such that\n$$ f(x)f(y) = xf(f(y-x)) + xf(2x) + f(x^2) $$\nfor all real $x, y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "f(x)=0,f(x)=3x \\text{ }\\forall x"
} | {
"split": "train",
"index": 6280,
"solution": "\nTo solve the functional equation for \\( f : \\mathbb{R} \\to \\mathbb{R} \\),\n\n\\[\nf(x)f(y) = xf(f(y-x)) + xf(2x) + f(x^2),\n\\]\n\nfor all real \\( x, y \\), we proceed as follows:\n\n1. **Substitute \\( y = 0 \\):** \n Considering \\( y = 0 \\), the equat... |
deepscaler | [
{
"role": "user",
"content": "A sphere intersects the $xy$-plane in a circle centered at $(3,5,0)$ with radius 2. The sphere also intersects the $yz$-plane in a circle centered at $(0,5,-8),$ with radius $r.$ Find $r.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{59}"
} | {
"split": "train",
"index": 6281,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I am playing a walking game with myself. On move 1, I do nothing, but on move $n$ where $2 \\le n \\le 25$, I take one step forward if $n$ is prime and two steps backwards if the number is composite. After all 25 moves, I stop and walk back to my original starting point. How... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 6282,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $AB = 16$, $AC = 24$, $BC = 19$, and $AD$ is an angle bisector. Find the ratio of the area of triangle $ABD$ to the area of triangle $ACD$. (Express your answer as a fraction in lowest terms.) Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 6283,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $x^2+px+q$ with $p$ and $q$ greater than zero has its minimum value when: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x=\\frac{-p}{2}"
} | {
"split": "train",
"index": 6284,
"solution": "1. **Identify the form of the function**: The given function is $x^2 + px + q$, which is a quadratic function in the standard form $ax^2 + bx + c$ where $a = 1$, $b = p$, and $c = q$.\n\n2. **Determine the vertex**: The vertex of a parabola represented by the quadra... |
deepscaler | [
{
"role": "user",
"content": "Find the product of all positive integral values of $n$ such that $n^2-35n+306= p$ for some prime number $p$. Note that there is at least one such $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "304"
} | {
"split": "train",
"index": 6285,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What number, when divided by 2, gives the same result as when 2 is subtracted from it? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 6286,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the positive difference between $\\frac{6^2 + 6^2}{6}$ and $\\frac{6^2 \\times 6^2}{6}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "204"
} | {
"split": "train",
"index": 6287,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A person receives an annuity at the end of each year for 15 years as follows: $1000 \\mathrm{K}$ annually for the first five years, $1200 \\mathrm{K}$ annually for the next five years, and $1400 \\mathrm{K}$ annually for the last five years. If they received $1400 \\mathrm{K}$ ... | math | {
"style": "rule",
"ground_truth": "807.95"
} | {
"split": "train",
"index": 6288,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest value x such that, given any point inside an equilateral triangle of side 1, we can always choose two points on the sides of the triangle, collinear with the given point and a distance x apart. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 6289,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the quadratic function $f(x)=3x^{2}-2x$, the sum of the first $n$ terms of the sequence $\\{a_{n}\\}$ is $S_{n}$, and the point $(n,S_{n})$ (where $n \\in \\mathbb{N}^{*}$) is on the graph of the function $y=f(x)$.\n\n$(1)$ Find the general formula for the sequence $\\{a_... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 6290,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the probability of spinning the spinner pictured and getting a prime number? Express your answer as a common fraction. [asy]\nimport olympiad; defaultpen(linewidth(0.8)); size(100); dotfactor=4;\ndraw(Circle((0,0),1));\nstring[] labels = {\"3\",\"6\",\"1\",\"4\",\"5\",\... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 6291,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A trapezoid has one base equal to twice its height, $x$, and the other base is three times as long as the height. Write the expression for the area of the trapezoid as a common fraction in terms of the height $x$. Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "\\dfrac{5x^2}{2}"
} | {
"split": "train",
"index": 6292,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point $P$ is chosen in the interior of $\\triangle ABC$ such that when lines are drawn through $P$ parallel to the sides of $\\triangle ABC$, the resulting smaller triangles $t_{1}$, $t_{2}$, and $t_{3}$ in the figure, have areas $4$, $9$, and $49$, respectively. Find the are... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 6293,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $g(x)$ be the function defined on $-2 \\leq x \\leq 2$ by the formula $$g(x) = 2 - \\sqrt{4-x^2}.$$ This is a vertically stretched version of the previously given function. If a graph of $x=g(y)$ is overlaid on the graph of $y=g(x)$, then one fully enclosed region is formed... | math | {
"style": "rule",
"ground_truth": "2.28"
} | {
"split": "train",
"index": 6294,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the book \"The Nine Chapters on the Mathematical Art,\" a right square pyramid, whose base is a rectangle and has a side edge perpendicular to the base, is called a \"yangma.\" Given a right square pyramid $M-ABCD$, where side edge $MA$ is perpendicular to the base $ABCD$, a... | math | {
"style": "rule",
"ground_truth": "36\\pi - 16\\sqrt{2}\\pi"
} | {
"split": "train",
"index": 6295,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the base \\(AC\\) of an isosceles triangle \\(ABC (AB = BC)\\), point \\(M\\) is marked. It is known that \\(AM = 7\\), \\(MB = 3\\), \\(\\angle BMC = 60^\\circ\\). Find the length of segment \\(AC\\). Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 6296,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the slope of line $\\overline{CD}$ where circles given by $x^2 + y^2 - 6x + 4y - 5 = 0$ and $x^2 + y^2 - 10x + 16y + 24 = 0$ intersect at points $C$ and $D$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 6297,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of $xy = 4$ is a hyperbola. Find the distance between the foci of this hyperbola. 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": 6298,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We call a number greater than $25$, [i] semi-prime[/i] if it is the sum of some two different prime numbers. What is the greatest number of consecutive natural numbers that can be [i]semi-prime[/i]? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 6299,
"solution": "\nTo solve this problem, we need to find the greatest number of consecutive natural numbers greater than 25 that can each be represented as the sum of two different prime numbers, which we refer to as semi-prime.\n\nFirst, let's identify a few prime numbers as poten... |
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