data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "On the island of Liars and Knights, a circular arrangement is called correct if everyone standing in the circle can say that among his two neighbors there is a representative of his tribe. One day, 2019 natives formed a correct arrangement in a circle. A liar approached them an... | math | {
"style": "rule",
"ground_truth": "1346"
} | {
"split": "train",
"index": 7600,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Problems 8, 9 and 10 use the data found in the accompanying paragraph and figures\nFour friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.\n$\\circ$ Art's cookies are trapezoids: \n\n$\\circ$ ... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 7601,
"solution": "To solve this problem, we need to determine the number of cookies Trisha can make from the same amount of dough as Art, given that all cookies have the same thickness and thus the same volume per unit area.\n\n1. **Calculate the area of one of Art's cookies**:\n A... |
deepscaler | [
{
"role": "user",
"content": "In the polar coordinate system, curve $C$: $\\rho =2a\\cos \\theta (a > 0)$, line $l$: $\\rho \\cos \\left( \\theta -\\frac{\\pi }{3} \\right)=\\frac{3}{2}$, $C$ and $l$ have exactly one common point. $O$ is the pole, $A$ and $B$ are two points on $C$, and $\\angle AOB=\\frac{\... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{3}"
} | {
"split": "train",
"index": 7602,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the units digit of $n$ given that $mn = 21^6$ and $m$ has a units digit of 7. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 7603,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, triangles $ABC$ and $CBD$ are isosceles. The perimeter of $\\triangle CBD$ is $19,$ the perimeter of $\\triangle ABC$ is $20,$ and the length of $BD$ is $7.$ What is the length of $AB?$ [asy]\n\nsize(7cm);\ndefaultpen(fontsize(11));\n\npair b = (0, 0);\npair d =... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 7604,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\sin\\alpha + \\cos\\alpha = \\frac{\\sqrt{2}}{3}$, where $\\alpha \\in (0, \\pi)$, calculate the value of $\\sin\\left(\\alpha + \\frac{\\pi}{12}\\right)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{2} + \\sqrt{3}}{6}"
} | {
"split": "train",
"index": 7605,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $0 < α < \\dfrac {π}{2}$, $\\sin α= \\dfrac {4}{5}$, and $\\tan (α-β)=- \\dfrac {1}{3}$, find the value of $\\tan β$ and compute the expression $\\dfrac {\\sin (2β- \\dfrac {π}{2})\\cdot \\sin (β+π)}{ \\sqrt {2}\\cos (β+ \\dfrac {π}{4})}$. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "\\dfrac {6}{5}"
} | {
"split": "train",
"index": 7606,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the surface integral\n\n$$\n\\iint_{\\Sigma}(-x+3 y+4 z) d \\sigma\n$$\n\nwhere $\\Sigma$ is the part of the plane\n\n$$\nx + 2y + 3z = 1\n$$\n\nlocated in the first octant (i.e., $x \\geq 0, y \\geq 0, z \\geq 0$). Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{14}}{18}"
} | {
"split": "train",
"index": 7607,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $z$ is a complex number such that\n\\[\n z + z^{-1} = 2\\sqrt{2},\n\\]\nwhat is the value of\n\\[\n z^{100} + z^{-100} \\, ?\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 7608,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\overrightarrow{a}=(1+\\cos \\omega x,-1)$, $\\overrightarrow{b}=( \\sqrt {3},\\sin \\omega x)$ ($\\omega > 0$), and the function $f(x)= \\overrightarrow{a}\\cdot \\overrightarrow{b}$, with the smallest positive period of $f(x)$ being $2\\pi$. \n(1) Find the expression... | math | {
"style": "rule",
"ground_truth": "\\frac {3 \\sqrt {3}+4}{10}"
} | {
"split": "train",
"index": 7609,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At a party, each man danced with exactly three women and each woman danced with exactly two men. Twelve men attended the party. How many women attended the party? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 7610,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the simplest quadratic radical $\\sqrt{m+1}$ and $\\sqrt{8}$ are of the same type of quadratic radical, the value of $m$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "m = 1"
} | {
"split": "train",
"index": 7611,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\sqrt{5 + x} + \\sqrt{20 - x} = 7$, what is the value of $(5 + x)(20 - x)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 7612,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Maria bakes a $24$-inch by $30$-inch pan of brownies, and the brownies are cut into pieces that measure $3$ inches by $4$ inches. Calculate the total number of pieces of brownies the pan contains. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 7613,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The energy stored by any pair of positive charges is inversely proportional to the distance between them, and directly proportional to their charges. Four identical point charges start at the vertices of a square, and this configuration stores 20 Joules of energy. How much more... | math | {
"style": "rule",
"ground_truth": "5(3\\sqrt{2} - 3)"
} | {
"split": "train",
"index": 7614,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the measure of angle \\( B \\widehat{A} D \\), given that \\( D \\widehat{A C}=39^{\\circ} \\), \\( A B = A C \\), and \\( A D = B D \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 7615,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A list of seven positive integers has a median of 4 and a mean of 10. What is the maximum possible value of the list's largest element? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 7616,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are enough cuboids with side lengths of 2, 3, and 5. They are neatly arranged in the same direction to completely fill a cube with a side length of 90. The number of cuboids a diagonal of the cube passes through is Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 7617,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given real numbers $a$ and $b$, satisfying $e^{2-a}=a$ and $b\\left(\\ln b-1\\right)=e^{3}$, where $e$ is the base of natural logarithm, the value of $ab$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "e^{3}"
} | {
"split": "train",
"index": 7618,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is planning to offer a discount on the selling price, but the profit margin must not be less than $10\\%$. Find the maximum discount rate that can be offered. Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "12\\%"
} | {
"split": "train",
"index": 7619,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( P \\) be a regular 2006-sided polygon. A diagonal of \\( P \\) is called a \"good edge\" if its endpoints divide the polygon into two parts, each containing an odd number of the polygon's sides. Each side of \\( P \\) is also considered a \"good edge\".\n\nGiven that 20... | math | {
"style": "rule",
"ground_truth": "1003"
} | {
"split": "train",
"index": 7620,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A tractor is dragging a very long pipe on sleds. Gavrila walked along the entire pipe in the direction of the tractor's movement and counted 210 steps. When he walked in the opposite direction, the number of steps was 100. What is the length of the pipe if Gavrila's step is 80 ... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 7621,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the system of equations \\begin{align*}\nxy &= 6 - 2x - 3y,\\\\\nyz &= 6 - 4y - 2z,\\\\\nxz &= 30 - 4x - 3z,\n\\end{align*}find the positive solution of $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 7622,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Liam read for 4 days at an average of 42 pages per day, and for 2 days at an average of 50 pages per day, then read 30 pages on the last day. What is the total number of pages in the book? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "298"
} | {
"split": "train",
"index": 7623,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the even, positive integers less than 62? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "930"
} | {
"split": "train",
"index": 7624,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find\n\\[\n\\cos \\left( 8 \\arccos \\frac{1}{5} \\right).\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-15647}{390625}"
} | {
"split": "train",
"index": 7625,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a 6 by 5 grid, how many 10-step paths are there from $W$ to $X$ that must pass through a point $H$? Assume $W$ is located at the top-left corner, $X$ at the bottom-right corner, and $H$ is three squares to the right and two squares down from $W$. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 7626,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $90r - 44r$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "46r"
} | {
"split": "train",
"index": 7627,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Liz had no money initially, and her friends gave her one-sixth, one-fifth, and one-fourth of their respective amounts, find the fractional part of the group's total money that Liz has. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"split": "train",
"index": 7628,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The field shown has been planted uniformly with wheat. [asy]\ndraw((0,0)--(1/2,sqrt(3)/2)--(3/2,sqrt(3)/2)--(2,0)--(0,0),linewidth(0.8));\nlabel(\"$60^\\circ$\",(0.06,0.1),E);\nlabel(\"$120^\\circ$\",(1/2-0.05,sqrt(3)/2-0.1),E);\nlabel(\"$120^\\circ$\",(3/2+0.05,sqrt(3)/2-0.1),... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{12}"
} | {
"split": "train",
"index": 7629,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The geometric sequence $\\left\\{a_{n}\\right\\}$ has the first term $a_{1}=1536$, and common ratio $q=-\\frac{1}{2}$. Let $\\Pi_{n}$ denote the product of its first $n$ terms $\\left(n \\in \\mathbf{N}^{*}\\right)$. Find the value of $n$ that maximizes $\\Pi_{n}\\$. Let's thin... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 7630,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among all triangles $ABC$, find the maximum value of $\\cos A + \\cos B \\cos C$. 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": 7631,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sphere is inscribed in a right cone with base radius $12$ cm and height $24$ cm, as shown. The radius of the sphere can be expressed as $a\\sqrt{c} - a$ cm. What is the value of $a + c$? [asy]\nimport three; size(120); defaultpen(linewidth(1)); pen dashes = linetype(\"2 2\") ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 7632,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the number of ordered pairs of integers $(x,y)$ with $1\\le x<y\\le 200$ such that $i^x+i^y$ is a real number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4950"
} | {
"split": "train",
"index": 7633,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Numbers from 1 to 9 are arranged in the cells of a \\(3 \\times 3\\) table such that the sum of the numbers on one diagonal is 7, and the sum on the other diagonal is 21. What is the sum of the numbers in the five shaded cells?\n Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 7634,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( S = \\{ 1, 2, \\cdots, 2005 \\} \\). Find the smallest number \\( n \\) such that in any subset of \\( n \\) pairwise coprime numbers from \\( S \\), there is at least one prime number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 7635,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The fraction $\\frac{1}{5}$ is written as an infinite binary fraction. How many ones are there among the first 2022 digits after the binary point in this representation? (12 points) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1010"
} | {
"split": "train",
"index": 7636,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the maximum number of checkers that can be placed on an $8 \\times 8$ board so that each one is being attacked? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 7637,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\frac{60}{2^3\\cdot5^8}$ is expressed as a decimal, how many non-zero digits are to the right of the decimal point? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 7638,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The coefficient of the $x^3$ term in the expansion of $(2-\\sqrt{x})^8$ is $1120x^3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 7639,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest value of \\(a\\) for which the sum of the squares of the roots of the equation \\(x^{2}-3ax+a^{2}=0\\) is equal to \\(0.28\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-0.2"
} | {
"split": "train",
"index": 7640,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Light of a blue laser (wavelength $\\lambda=475 \\, \\text{nm}$ ) goes through a narrow slit which has width $d$ . After the light emerges from the slit, it is visible on a screen that is $ \\text {2.013 m} $ away from the slit. The distance between the center of the screen... | math | {
"style": "rule",
"ground_truth": "1250"
} | {
"split": "train",
"index": 7641,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the universal set $U = \\{-1, 0, 1, 2\\}$, and $P = \\{x \\in \\mathbb{Z} \\,|\\, -\\sqrt{2} < x < \\sqrt{2}\\}$, determine the complement of $P$ in $U$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\{2\\}"
} | {
"split": "train",
"index": 7642,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Complete the conversion between the following number systems: $101101_{(2)} = \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_{(10)}\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_{(7)}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "63_{(7)}"
} | {
"split": "train",
"index": 7643,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If\n\\[\n\\sum_{n=1}^{\\infty}\\frac{\\frac11 + \\frac12 + \\dots + \\frac 1n}{\\binom{n+100}{100}} = \\frac pq\n\\]\nfor relatively prime positive integers $p,q$ , find $p+q$ .\n\n*Proposed by Michael Kural* Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "9901"
} | {
"split": "train",
"index": 7644,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, with $C= \\frac{\\pi}{3}$, $b=8$, and the area of $\\triangle ABC$ is $10\\sqrt{3}$.\n\n$(1).$ Find the value of $c$; $(2).$ Find the value of $\\cos(B-C)$. Let's ... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{14}"
} | {
"split": "train",
"index": 7645,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sweater costs 160 yuan, it was first marked up by 10% and then marked down by 10%. Calculate the current price compared to the original. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.99"
} | {
"split": "train",
"index": 7646,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of $y=ax^2+bx+c$ is given below, where $a$, $b$, and $c$ are integers. Find $a$.\n\n[asy]\nsize(140);\nLabel f;\n\nf.p=fontsize(4);\n\nxaxis(-3,3,Ticks(f, 1.0));\n\nyaxis(-4,4,Ticks(f, 1.0));\n\nreal f(real x)\n\n{\n\nreturn x^2+2x-1;\n\n}\n\ndraw(graph(f,-2.7,.7),li... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 7647,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the least positive integer $n$ satisfying the following statement: for eash pair of positive integers $a$ and $b$ such that $36$ divides $a+b$ and $n$ divides $ab$ it follows that $36$ divides both $a$ and $b$ . Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "1296"
} | {
"split": "train",
"index": 7648,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ali Baba and the thief are dividing a hoard consisting of 100 gold coins, distributed in 10 piles of 10 coins each. Ali Baba chooses 4 piles, places a cup near each of them, and puts a certain number of coins (at least one, but not the entire pile) into each cup. The thief must... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 7649,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse with a chord passing through the focus and perpendicular to the major axis of length $\\sqrt{2}$, and a distance from the focus to the corresponding directrix of $1$, determine the eccentricity of the ellipse. Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 7650,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "James is standing at the point $(0,1)$ on the coordinate plane and wants to eat a hamburger. For each integer $n \\geq 0$, the point $(n, 0)$ has a hamburger with $n$ patties. There is also a wall at $y=2.1$ which James cannot cross. In each move, James can go either up, right,... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{3}"
} | {
"split": "train",
"index": 7651,
"solution": "Note that we desire to compute the number of times James moves to the right before moving down to the line $y=0$. Note also that we can describe James's current state based on whether his $y$-coordinate is 0 or 1 and whether or not the other vertically adjacent poin... |
deepscaler | [
{
"role": "user",
"content": "There are $5$ different books to be arranged in a row. Book A must be adjacent to Book B, while Book C cannot be adjacent to Book D. The number of different arrangement methods is $\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 7652,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line is parameterized by\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} 2 \\\\ 3 \\end{pmatrix} + t \\begin{pmatrix} -1 \\\\ 5 \\end{pmatrix}.\\]A second line is parameterized by\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} 0 \\\\ 7 \\end{p... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 6 \\\\ -17 \\end{pmatrix}"
} | {
"split": "train",
"index": 7653,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Henry decides one morning to do a workout, and he walks $\\frac{3}{4}$ of the way from his home to his gym. The gym is $2$ kilometers away from Henry's home. At that point, he changes his mind and walks $\\frac{3}{4}$ of the way from where he is back toward home. When he reache... | math | {
"style": "rule",
"ground_truth": "1 \\frac{1}{5}"
} | {
"split": "train",
"index": 7654,
"solution": "1. **Define the sequence of positions**: Let $A$ be the point closer to Henry’s home, and $B$ be the point closer to the gym. Define $(a_n)$ to be the position of Henry after $2n$ walks (even steps, returning towards home), and $(b_n)$ to be the position of Henry af... |
deepscaler | [
{
"role": "user",
"content": "The average age of the $6$ people in Room A is $40$. The average age of the $4$ people in Room B is $25$. If the two groups are combined, what is the average age of all the people? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 7655,
"solution": "1. **Calculate the total age of people in Room A**: \n Since the average age of the 6 people in Room A is 40, the total age for Room A is calculated by multiplying the average age by the number of people:\n \\[\n \\text{Total age in Room A} = 6 \\times 40 = 2... |
deepscaler | [
{
"role": "user",
"content": "Determine the values of $x$ and $y$ given that 15 is the arithmetic mean of the set $\\{8, x, 21, y, 14, 11\\}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 7656,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many lattice points lie on the hyperbola $x^2 - y^2 = 3^4 \\cdot 17^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 7657,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{O A} \\perp \\overrightarrow{O B}$, and $|\\overrightarrow{O A}|=|\\overrightarrow{O B}|=24$. Find the minimum value of $|t \\overrightarrow{A B}-\\overrightarrow{A O}|+\\left|\\frac{5}{12} \\overrightarrow{B O}-(1-t) \\overrightarrow{B A}\\right... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 7658,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of different monic quadratic polynomials (i.e., with the leading coefficient equal to 1) with integer coefficients such that they have two different roots which are powers of 5 with natural exponents, and their coefficients do not exceed in absolute value $125^{... | math | {
"style": "rule",
"ground_truth": "5112"
} | {
"split": "train",
"index": 7659,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $a + b = 6$, $c = 2$, and $\\cos C = \\frac{7}{9}$:\n1. Find the values of $a$ and $b$.\n2. Calculate the area of triangle $ABC$. Let's think step by step... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 7660,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the function $f(x)=\\sin \\left( \\frac{x+\\varphi}{3}\\right)$ is an even function, determine the value of $\\varphi$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\pi}{2}"
} | {
"split": "train",
"index": 7661,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, where $AB = 50$, $BC = 36$, and $AC = 42$. A line $CX$ from $C$ is perpendicular to $AB$ and intersects $AB$ at point $X$. Find the ratio of the area of $\\triangle BCX$ to the area of $\\triangle ACX$. Express your answer as a simplified common fraction. Let... | math | {
"style": "rule",
"ground_truth": "\\frac{6}{7}"
} | {
"split": "train",
"index": 7662,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the equation, $\\frac15+\\frac{5}{x}=\\frac{12}{x}+\\frac{1}{12}$, 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": "60"
} | {
"split": "train",
"index": 7663,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the sum \\[\\frac{1}{3^1} + \\frac{2}{3^2} + \\frac{3}{3^3} + \\cdots + \\frac{k}{3^k} + \\cdots \\] 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": 7664,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $4x\\equiv 8\\pmod{20}$ and $3x\\equiv 16\\pmod{20}$, then what is the remainder when $x^2$ is divided by $20$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 7665,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(1) Given $\\sin\\left( \\alpha +\\frac{\\pi }{4} \\right)=\\frac{\\sqrt{2}}{10}$, with $\\alpha\\in(0,\\pi)$, find the value of $\\cos \\alpha$;\n\n(2) Evaluate: $\\left( \\tan {10^{\\circ} }-\\sqrt{3} \\right)\\sin {40^{\\circ} }$. Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 7666,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest possible radius of a circle contained in a 4-dimensional hypercube of side length 1? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 7667,
"solution": "The largest possible radius is $\\frac{\\sqrt{2}}{2}$.\nIt will be convenient to solve\nthe problem for a hypercube of side length 2 instead, in which case\nwe are trying to show that the largest radius is $\\sqrt{2}$.\n\nChoose coordinates so that the interior of t... |
deepscaler | [
{
"role": "user",
"content": "Let \\(\\alpha\\) and \\(\\beta\\) be the two real roots of the quadratic equation \\(x^{2} - 2kx + k + 20 = 0\\). Find the minimum value of \\((\\alpha+1)^{2} + (\\beta+1)^{2}\\), and determine the value of \\(k\\) for which this minimum value is achieved. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 7668,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular table shown below, the number $1$ is written in the upper-left hand corner, and every number is the sum of the any numbers directly to its left and above. The table extends infinitely downwards and to the right.\n\\[ \n \\begin{array}{cccccc}\n 1 & 1 &... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 7669,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The elevator buttons in Harvard's Science Center form a $3 \\times 2$ grid of identical buttons, and each button lights up when pressed. One day, a student is in the elevator when all the other lights in the elevator malfunction, so that only the buttons which are lit can be se... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 7670,
"solution": "We first note that there are $2^{6}-1=63$ possibilities for lights in total. We now count the number of duplicates we need to subtract by casework on the number of buttons lit. To do this, we do casework on the size of the minimal \"bounding box\" of the lights: - I... |
deepscaler | [
{
"role": "user",
"content": "Find $537_{8} - 261_{8}$. Verify by adding the result to $261_{8}$ and checking if it matches $537_{8}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "256_8"
} | {
"split": "train",
"index": 7671,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To control her blood pressure, Jill's grandmother takes one half of a pill every other day. If one supply of medicine contains $60$ pills, then the supply of medicine would last approximately Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8\\text{ months}"
} | {
"split": "train",
"index": 7672,
"solution": "1. **Determine the rate of consumption**: Jill's grandmother takes one half of a pill every other day. This means that in two days, she consumes one half of a pill.\n\n2. **Calculate the number of days per pill**: Since she takes half a pill every two days, she will... |
deepscaler | [
{
"role": "user",
"content": "Find the value of the expression \\(\\cos ^{4} \\frac{7 \\pi}{24}+\\sin ^{4} \\frac{11 \\pi}{24}+\\sin ^{4} \\frac{17 \\pi}{24}+\\cos ^{4} \\frac{13 \\pi}{24}\\). 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": 7673,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For $n \\geq 1$ , let $a_n$ be the number beginning with $n$ $9$ 's followed by $744$ ; eg., $a_4=9999744$ . Define $$ f(n)=\\text{max}\\{m\\in \\mathbb{N} \\mid2^m ~ \\text{divides} ~ a_n \\} $$ , for $n\\geq 1$ . Find $f(1)+f(2)+f(3)+ \\cdots + f(10)$ . Let's thi... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 7674,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many three-digit numbers are there in which the hundreds digit is greater than both the tens digit and the units digit? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "285"
} | {
"split": "train",
"index": 7675,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Joe's quiz scores were 88, 92, 95, 81, and 90, and then he took one more quiz and scored 87. What was his mean score after all six quizzes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "88.83"
} | {
"split": "train",
"index": 7676,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sarah intended to multiply a two-digit number and a three-digit number, but she left out the multiplication sign and simply placed the two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sara... | math | {
"style": "rule",
"ground_truth": "126"
} | {
"split": "train",
"index": 7677,
"solution": "As shown above, we have $1000x+y=9xy$, so $1000/y=9-1/x$. $1000/y$ must be just a little bit smaller than 9, so we find $y=112$, $x=14$, and the solution is $\\boxed{126}$."
} |
deepscaler | [
{
"role": "user",
"content": "The integers \\( m \\) and \\( n \\) satisfy the equation \\( 3^{m} \\times n = 7! + 8! + 9! \\). What is the smallest possible value for \\( n \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "560"
} | {
"split": "train",
"index": 7678,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An 8 by 8 grid of numbers obeys the following pattern: 1) The first row and first column consist of all 1s. 2) The entry in the $i$th row and $j$th column equals the sum of the numbers in the $(i-1)$ by $(j-1)$ sub-grid with row less than $i$ and column less than $j$. What is t... | math | {
"style": "rule",
"ground_truth": "2508"
} | {
"split": "train",
"index": 7679,
"solution": "Let $x_{i, j}$ be the number in the $i$th row and the $j$th column. Then if $i, j \\geq 2, x_{i+1, j+1}-x_{i+1, j}-x_{i, j+1}+x_{i, j}$ only counts the term $x_{i, j}$ since every other term is added and subtracted the same number of times. Thus $x_{i+1, j+1}=x_{i+1... |
deepscaler | [
{
"role": "user",
"content": "Quadratic equations of the form $ax^2 + bx + c = 0$ have real roots with coefficients $a$, $b$, and $c$ selected from the set of integers $\\{1, 2, 3, 4\\}$, where $a \\neq b$. Calculate the total number of such equations. Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 7680,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two distinct positive integers from 1 to 60 inclusive are chosen. Let the sum of the integers equal $S$ and the product equal $P$. What is the probability that $P+S$ is one less than a multiple of 7? Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "\\frac{148}{590}"
} | {
"split": "train",
"index": 7681,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\{a\\_n\\}$ that satisfies the condition: when $n \\geqslant 2$ and $n \\in \\mathbb{N}^+$, we have $a\\_n + a\\_{n-1} = (-1)^n \\times 3$. Calculate the sum of the first 200 terms of the sequence $\\{a\\_n\\}$. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"split": "train",
"index": 7682,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cara is sitting at a circular table with her seven friends. Two of her friends, Alice and Bob, insist on sitting together but not next to Cara. How many different possible pairs of people could Cara be sitting between? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 7683,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When a single number is added to each member of the sequence 20, 50, 100, the sequence becomes expressible as $x, a x, a^{2} x$. Find $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"split": "train",
"index": 7684,
"solution": "$\\frac{5}{3}$."
} |
deepscaler | [
{
"role": "user",
"content": "A and B play a guessing game where A first thinks of a number denoted as $a$, and then B guesses the number A thought of, denoting B's guess as $b$. Both $a$ and $b$ belong to the set $\\{0,1,2,…,9\\}$. If $|a-b| \\leqslant 1$, then A and B are considered to have a \"telepathic... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{25}"
} | {
"split": "train",
"index": 7685,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Star lists the whole numbers $1$ through $30$ once and Emilio copies those numbers replacing each occurrence of the digit $3$ by the digit $2$, find the difference between Star's and Emilio's total sums. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 7686,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a triangle with sides of lengths 3, 4, and 5. Three circles are constructed with radii of 1, centered at each vertex of the triangle. Find the total area of the portions of the circles that lie within the triangle. Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"split": "train",
"index": 7687,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Convert the point $(2 \\sqrt{3}, 6, -4)$ in rectangular coordinates to spherical coordinates. Enter your answer in the form $(\\rho,\\theta,\\phi),$ where $\\rho > 0,$ $0 \\le \\theta < 2 \\pi,$ and $0 \\le \\phi \\le \\pi.$ Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "\\left( 8, \\frac{\\pi}{3}, \\frac{2 \\pi}{3} \\right)"
} | {
"split": "train",
"index": 7688,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive number $x$ satisfies the inequality $\\sqrt{x} < 2x$ if and only if 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": 7689,
"solution": "To solve the inequality $\\sqrt{x} < 2x$, we start by squaring both sides, assuming $x \\geq 0$ since we are dealing with a square root and $x$ is given as positive:\n\n1. Squaring both sides:\n \\[\n (\\sqrt{x})^2 < (2x)^2\n \\]\n \\[\n x < 4x^2\n \\]\n... |
deepscaler | [
{
"role": "user",
"content": "Given that $M(2,5)$ is the midpoint of $\\overline{AB}$ and $A(3,1)$ is one endpoint, what is the product of the coordinates of point $B$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 7690,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the $2007$ roots of $(x-1)^{2007}+2(x-2)^{2006}+3(x-3)^{2005}+\\cdots+2006(x-2006)^2+2007(x-2007)$.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2005"
} | {
"split": "train",
"index": 7691,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the convex quadrilateral \\(ABCD\\), \\(E\\) is the intersection of the diagonals. The areas of triangles \\(ADE, BCE, CDE\\) are \\(12 \\, \\text{cm}^2, 45 \\, \\text{cm}^2, 18 \\, \\text{cm}^2\\), respectively, and the length of side \\(AB\\) is \\(7 \\, \\text{cm}\\).\n\n... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 7692,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At the end of $1997$, the desert area in a certain region was $9\\times 10^{5}hm^{2}$ (note: $hm^{2}$ is the unit of area, representing hectares). Geologists started continuous observations from $1998$ to understand the changes in the desert area of this region. The observation... | math | {
"style": "rule",
"ground_truth": "2021"
} | {
"split": "train",
"index": 7693,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular octagon $ABCDEFGH$ has its sides' midpoints connected to form a smaller octagon inside it. Determine the fraction of the area of the larger octagon $ABCDEFGH$ that is enclosed by this smaller octagon. Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 7694,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Tio Mané has two boxes, one with seven distinct balls numbered from 1 to 7 and another with eight distinct balls numbered with all prime numbers less than 20. He draws one ball from each box.\n\nCalculate the probability that the product is odd. What is the probability that the... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 7695,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle has sides of length $7$ and $23$. What is the smallest whole number greater than the perimeter of any triangle with these side lengths? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 7696,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $a$ is a multiple of $3$ and $b$ is a multiple of $6$. Which of the following statements must be true?\n\nA. $b$ is a multiple of $3$.\nB. $a-b$ is a multiple of $3$.\nC. $a-b$ is a multiple of $6$.\nD. $a-b$ is a multiple of $2$.\n\nList the choices in your answ... | math | {
"style": "rule",
"ground_truth": "\\text{A, B}"
} | {
"split": "train",
"index": 7697,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $z_1,$ $z_2,$ $\\dots,$ $z_{20}$ be the twenty (complex) roots of the equation\n\\[z^{20} - 4z^{19} + 9z^{18} - 16z^{17} + \\dots + 441 = 0.\\]Calculate $\\cot \\left( \\sum_{k = 1}^{20} \\operatorname{arccot} z_k \\right).$ Note that the addition formula for cotangent is ... | math | {
"style": "rule",
"ground_truth": "\\frac{241}{220}"
} | {
"split": "train",
"index": 7698,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define $\\operatorname{gcd}(a, b)$ as the greatest common divisor of integers $a$ and $b$. Given that $n$ is the smallest positive integer greater than 1000 that satisfies:\n$$\n\\begin{array}{l}\n\\operatorname{gcd}(63, n+120) = 21, \\\\\n\\operatorname{gcd}(n+63, 120) = 60\n\... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 7699,
"solution": ""
} |
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