data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Consider a wardrobe that consists of $6$ red shirts, $7$ green shirts, $8$ blue shirts, $9$ pairs of pants, $10$ green hats, $10$ red hats, and $10$ blue hats. Additionally, you have $5$ ties in each color: green, red, and blue. Every item is distinct. How many outfits can you ... | math | {
"style": "rule",
"ground_truth": "18900"
} | {
"split": "train",
"index": 7000,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Below is a portion of the graph of a quadratic function, $y=p(x)=dx^2 + ex + f$:\n\nThe value of $p(12)$ is an integer. The graph's axis of symmetry is $x = 10.5$, and the graph passes through the point $(3, -5)$. Based on this, what is the value of $p(12)$? Let's think step by... | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 7001,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer $n$ such that $3n$ is a perfect square and $2n$ is a perfect cube? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 7002,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $k$ be a positive real. $A$ and $B$ play the following game: at the start, there are $80$ zeroes arrange around a circle. Each turn, $A$ increases some of these $80$ numbers, such that the total sum added is $1$. Next, $B$ selects ten consecutive numbers with the largest su... | math | {
"style": "rule",
"ground_truth": "1 + 1 + \\frac{1}{2} + \\ldots + \\frac{1}{7}"
} | {
"split": "train",
"index": 7003,
"solution": "\nLet \\( k \\) be a positive real number. \\( A \\) and \\( B \\) play the following game: at the start, there are 80 zeroes arranged around a circle. Each turn, \\( A \\) increases some of these 80 numbers such that the total sum added is 1. Next, \\( B \\) select... |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ $c$ be positive real numbers such that $a + b + c = 1.$ Find the minimum value of\n\\[\\frac{1}{a + 2b} + \\frac{1}{b + 2c} + \\frac{1}{c + 2a}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 7004,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A math teacher requires Noelle to do one homework assignment for each of the first four homework points she wants to earn; for each of the next four homework points, she needs to do two homework assignments; and so on, so that to earn the $n^{\\text{th}}$ homework point, she ha... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 7005,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagram shows a rectangle that has been dissected into nine non-overlapping squares. Given that the width and the height of the rectangle are relatively prime positive integers, find the perimeter of the rectangle.\n[asy]draw((0,0)--(69,0)--(69,61)--(0,61)--(0,0));draw((36,... | math | {
"style": "rule",
"ground_truth": "260"
} | {
"split": "train",
"index": 7006,
"solution": "Call the squares' side lengths from smallest to largest $a_1,\\ldots,a_9$, and let $l,w$ represent the dimensions of the rectangle.\n The picture shows that \\begin{align*} a_1+a_2 &= a_3\\\\ a_1 + a_3 &= a_4\\\\ a_3 + a_4 &= a_5\\\\ a_4 + a_5 &= a_6\\\\ a_2 + a_3 +... |
deepscaler | [
{
"role": "user",
"content": "Suppose $a,$ $b,$ and $c$ are real numbers such that\n\\[\\frac{ac}{a + b} + \\frac{ba}{b + c} + \\frac{cb}{c + a} = -9\\]and\n\\[\\frac{bc}{a + b} + \\frac{ca}{b + c} + \\frac{ab}{c + a} = 10.\\]Compute the value of\n\\[\\frac{b}{a + b} + \\frac{c}{b + c} + \\frac{a}{c + a}.\\... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 7007,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $y = \\lg(-x^2 + x + 2)$ with domain $A$, find the range $B$ for the exponential function $y = a^x$ $(a>0$ and $a \\neq 1)$ where $x \\in A$.\n\n1. If $a=2$, determine $A \\cup B$;\n2. If $A \\cap B = (\\frac{1}{2}, 2)$, find the value of $a$. Let's think ste... | math | {
"style": "rule",
"ground_truth": "a = 2"
} | {
"split": "train",
"index": 7008,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive three-digit integers are divisible by both 11 and 5? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 7009,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fair die is rolled twice in succession, and the numbers facing up are observed and recorded as $x$ and $y$ respectively. \n$(1)$ If the event \"$x+y=8$\" is denoted as event $A$, find the probability of event $A$ occurring; \n$(2)$ If the event \"$x^{2}+y^{2} \\leqslant 12$... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{6}"
} | {
"split": "train",
"index": 7010,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p$, $q$, $r$, $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s$, and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q$. Additionally, $p + q + r + s = 201$. Find the value of $pq + rs$. Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "-\\frac{28743}{12}"
} | {
"split": "train",
"index": 7011,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the least positive integer $n$ such that $$\\frac 1{\\sin 30^\\circ\\sin 31^\\circ}+\\frac 1{\\sin 32^\\circ\\sin 33^\\circ}+\\cdots+\\frac 1{\\sin 88^\\circ\\sin 89^\\circ}+\\cos 89^\\circ=\\frac 1{\\sin n^\\circ}.$$ Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "n = 1"
} | {
"split": "train",
"index": 7012,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the squares of the solutions to\n\\[\\left| x^2 - x + \\frac{1}{2010} \\right| = \\frac{1}{2010}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2008}{1005}"
} | {
"split": "train",
"index": 7013,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circles $C_1$ and $C_2$ intersect at points $X$ and $Y$ . Point $A$ is a point on $C_1$ such that the tangent line with respect to $C_1$ passing through $A$ intersects $C_2$ at $B$ and $C$ , with $A$ closer to $B$ than $C$ , such that $2016 \\cdot AB ... | math | {
"style": "rule",
"ground_truth": "2017"
} | {
"split": "train",
"index": 7014,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Starting with $10,000,000$, Esha forms a sequence by alternatively dividing by 2 and multiplying by 3. If she continues this process, what is the form of her sequence after 8 steps? Express your answer in the form $a^b$, where $a$ and $b$ are integers and $a$ is as small as pos... | math | {
"style": "rule",
"ground_truth": "(2^3)(3^4)(5^7)"
} | {
"split": "train",
"index": 7015,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the product of all real numbers that are tripled when added to their reciprocals? 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": 7016,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $\\triangle ABC$ has a right angle at $C$, $\\angle A = 60^\\circ$, and $AC=10$. Find the radius of the incircle of $\\triangle ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5(\\sqrt{3}-1)"
} | {
"split": "train",
"index": 7017,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among the six-digit numbers formed by the digits 0, 1, 2, 3, 4, 5 without repetition, calculate the number of the numbers that are divisible by 2. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "312"
} | {
"split": "train",
"index": 7018,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At the beginning of a trip, the mileage odometer read $56,200$ miles. The driver filled the gas tank with $6$ gallons of gasoline. During the trip, the driver filled his tank again with $12$ gallons of gasoline when the odometer read $56,560$. At the end of the trip, the driver... | math | {
"style": "rule",
"ground_truth": "26.9"
} | {
"split": "train",
"index": 7019,
"solution": "1. **Identify the total distance traveled**: \n The odometer readings at the start and end of the trip are given as $56,200$ miles and $57,060$ miles, respectively. Therefore, the total distance traveled during the trip is:\n \\[\n 57,060 - 56,200 = 860 \\text... |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of the function \\( f(x)=\\cos 4x + 6\\cos 3x + 17\\cos 2x + 30\\cos x \\) for \\( x \\in \\mathbb{R} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-18"
} | {
"split": "train",
"index": 7020,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $f(x)$ is a function defined on $\\mathbb{R}$ with a period of $2$, in the interval $[1,3]$, $f(x)= \\begin{cases}x+ \\frac {a}{x}, & 1\\leqslant x < 2 \\\\ bx-3, & 2\\leqslant x\\leqslant 3\\end{cases}$, and $f( \\frac {7}{2})=f(- \\frac {7}{2})$, find the value of ... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 7021,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\dfrac {\\cos (\\pi-2\\alpha)}{\\sin (\\alpha- \\dfrac {\\pi}{4})}=- \\dfrac { \\sqrt {2}}{2}$, then $\\sin 2\\alpha=$ \\_\\_\\_\\_\\_\\_ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "- \\dfrac {3}{4}"
} | {
"split": "train",
"index": 7022,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among the following propositions, the true one is __________ \n (1) In a plane, the locus of points whose sum of distances from two fixed points $F_{1}$ and $F_{2}$ is a constant is an ellipse; \n (2) If vectors $\\overrightarrow{e_{1}}$, $\\overrightarrow{e_{2}}$, $\\overrig... | math | {
"style": "rule",
"ground_truth": "(3)"
} | {
"split": "train",
"index": 7023,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the areas of all triangles whose vertices are also vertices of a $1$ by $1$ by $1$ cube is $m + \\sqrt{n} + \\sqrt{p},$ where $m, n,$ and $p$ are integers. Find $m + n + p.$\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "348"
} | {
"split": "train",
"index": 7024,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An ellipse has foci at $(0, 2)$ and $(3, 0)$. It has two $x$-intercepts, one of which is the origin. What is the other one? Enter your answer as an ordered pair. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left(\\tfrac{15}{4},0\\right)"
} | {
"split": "train",
"index": 7025,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose a sequence of positive numbers $\\left\\{a_{n}\\right\\}$ satisfies: $a_{0} = 1, a_{n} = a_{n+1} + a_{n+2},$ for $n = 0, 1, 2, \\ldots$. Find $a_{1}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5} - 1}{2}"
} | {
"split": "train",
"index": 7026,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A very bizarre weighted coin comes up heads with probability $\\frac12$, tails with probability $\\frac13$, and rests on its edge with probability $\\frac16$. If it comes up heads, I win 1 dollar. If it comes up tails, I win 3 dollars. But if it lands on its edge, I lose 5 d... | math | {
"style": "rule",
"ground_truth": "\\$\\dfrac23 \\approx \\$0.67"
} | {
"split": "train",
"index": 7027,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = 4x + c$ and $g(x) = cx + 2.$ If $f(g(x)) = 12x + d,$ then find $d.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 7028,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers $n$ satisfy $(n+3)(n-7) \\le 0$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 7029,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence of real numbers \\( a_{1}, a_{2}, a_{3}, \\ldots \\) that satisfy\n1) \\( a_{1}=\\frac{1}{2} \\), and\n2) \\( a_{1} + a_{2} + \\ldots + a_{k} = k^{2} a_{k} \\), for \\( k \\geq 2 \\).\nDetermine the value of \\( a_{100} \\). Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{10100}"
} | {
"split": "train",
"index": 7030,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four different natural numbers, of which one is 1, have the following properties: the sum of any two of them is a multiple of 2, the sum of any three of them is a multiple of 3, and the sum of all four numbers is a multiple of 4. What is the minimum possible sum of these four n... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 7031,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given points $A(-2, 0)$ and $B(0, 2)$, $k$ is a constant, and $M$, $N$ are two distinct points on the circle ${x^2} + {y^2} + kx = 0$. $P$ is a moving point on the circle ${x^2} + {y^2} + kx = 0$. If $M$ and $N$ are symmetric about the line $x - y - 1 = 0$, find the maximum are... | math | {
"style": "rule",
"ground_truth": "3 + \\sqrt{2}"
} | {
"split": "train",
"index": 7032,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ where all terms are positive integers, let $S_n$ denote the sum of the first $n$ terms. If $a_{n+1}=\\begin{cases} \\frac{a_n}{2},a_n \\text{ is even} \\\\\\\\ 3a_n+1,a_n \\text{ is odd} \\end{cases}$ and $a_1=5$, calculate $S_{2015}$. Let's think s... | math | {
"style": "rule",
"ground_truth": "4725"
} | {
"split": "train",
"index": 7033,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The hypotenuse of an isosceles right triangle is $4\\sqrt{2}$ units. How many square units are in the area of the triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 7034,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "You flip a fair coin which results in heads ( $\\text{H}$ ) or tails ( $\\text{T}$ ) with equal probability. What is the probability that you see the consecutive sequence $\\text{THH}$ before the sequence $\\text{HHH}$ ? Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{8}"
} | {
"split": "train",
"index": 7035,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Any type of nature use affects at least one of the natural resources, including lithogenic base, soil, water, air, plant world, and animal world. Types that affect the same set of resources belong to the same type. Research has shown that types of nature use developed in the la... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 7036,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the positive solution to\n\\[\\sqrt[3]{x + \\sqrt[3]{x + \\sqrt[3]{x + \\dotsb}}} = \\sqrt[3]{x \\sqrt[3]{x \\sqrt[3]{x \\dotsm}}}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3 + \\sqrt{5}}{2}"
} | {
"split": "train",
"index": 7037,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $\\lfloor x\\rfloor$ is defined as the largest integer less than or equal to $x$. For example, $\\lfloor 5.67\\rfloor = 5$, $\\lfloor -\\tfrac 14\\rfloor = -1$, and $\\lfloor 8\\rfloor = 8$.\n\nWhat is the range of the function $$f(x) = \\lfloor x\\rfloor - x~?$$Ex... | math | {
"style": "rule",
"ground_truth": "(-1,0]"
} | {
"split": "train",
"index": 7038,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagram shows a semicircle with diameter $20$ and the circle with greatest diameter that fits inside the semicircle. The area of the shaded region is $N\\pi$ , where $N$ is a positive integer. Find $N$ . Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 7039,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find \\( n > m \\geq 1 \\) such that the last two digits of the decimal representation of \\( 42^{n} \\) are the same as those of \\( 42^{m} \\), and such that the sum \\( m+n \\) is minimal. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 7040,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In isosceles $\\triangle ABC$, $\\angle BAC=120^{\\circ}$, $AB=AC=2$, $\\overrightarrow{BC}=2 \\overrightarrow{BD}$, $\\overrightarrow{AC}=3 \\overrightarrow{AE}$. Calculate the value of $\\overrightarrow{AD}\\cdot \\overrightarrow{BE}$. Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "-\\dfrac {2}{3}"
} | {
"split": "train",
"index": 7041,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a circle with center $O$, the measure of $\\angle TIQ$ is $45^\\circ$ and the radius $OT$ is 12 cm. Find the number of centimeters in the length of arc $TQ$. Express your answer in terms of $\\pi$. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "6\\pi"
} | {
"split": "train",
"index": 7042,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the x-coordinate of point Q, given that point P has coordinates $(\\frac{3}{5}, \\frac{4}{5})$, point Q is in the third quadrant with $|OQ| = 1$ and $\\angle POQ = \\frac{3\\pi}{4}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{7\\sqrt{2}}{10}"
} | {
"split": "train",
"index": 7043,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence of natural numbers $\\left\\{x_{n}\\right\\}$ is constructed according to the following rules:\n$$\nx_{1}=a, x_{2}=b, x_{n+2}=x_{n}+x_{n+1}, \\text{ for } n \\geq 1.\n$$\n\nIt is known that some term in the sequence is 1000. What is the smallest possible value of $a+... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 7044,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ellipse $5x^2 - ky^2 = 5$ has one of its foci at $(0, 2)$. Find the value of $k$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 7045,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ike and Mike go into a sandwich shop with a total of $30.00 to spend. Sandwiches cost $4.50 each and soft drinks cost $1.00 each. Ike and Mike plan to buy as many sandwiches as they can, and use any remaining money to buy soft drinks. Counting both sandwiches and soft drinks, h... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 7046,
"solution": "1. **Define the variables**: Let $s$ be the number of sandwiches and $d$ be the number of soft drinks. The total amount of money Ike and Mike have is $30.00$.\n\n2. **Set up the equation**: The cost of each sandwich is $4.50$ and each soft drink costs $1.00$. The to... |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $(xOy)$, the parametric equations of the curve $C$ are given by $\\begin{cases} x=3\\cos \\alpha \\\\ y=\\sin \\alpha \\end{cases}$ ($\\alpha$ is the parameter). In the polar coordinate system with the origin as the pole and the positive $x$-a... | math | {
"style": "rule",
"ground_truth": "\\dfrac{18\\sqrt{2}}{5}"
} | {
"split": "train",
"index": 7047,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangle is divided into 40 identical squares. The rectangle contains more than one row of squares. Andrew coloured all the squares in the middle row. How many squares did he not colour? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 7048,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many nonnegative integers can be expressed as \\[b_6\\cdot4^6+b_5\\cdot4^5+b_4\\cdot4^4+b_3\\cdot4^3+b_2\\cdot4^2+b_1\\cdot4^1+b_0\\cdot4^0,\\] where $b_i \\in \\{-1,0,1\\}$ for $0 \\leq i \\leq 6$? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "5462"
} | {
"split": "train",
"index": 7049,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The asymptotes of a hyperbola are $y = x + 1$ and $y = 3 - x.$ Also, the hyperbola passes through $(3,3).$ Find the distance between the foci of the hyperbola. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{6}"
} | {
"split": "train",
"index": 7050,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the expansion of $(1+x){(x-\\frac{2}{x})}^{3}$, calculate the coefficient of $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 7051,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the arithmetic sequence $\\left\\{a_{n}\\right\\}$, if $\\frac{a_{11}}{a_{10}} < -1$ and the sum of its first $n$ terms $S_{n}$ has a maximum value, then when $S_{n}$ takes the smallest positive value, $n = (\\quad$ ). Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 7052,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a positive geometric sequence $\\{a_{n}\\}$, it is known that $a_{1}a_{2}a_{3}=4$, $a_{4}a_{5}a_{6}=8$, and $a_{n}a_{n+1}a_{n+2}=128$. Find the value of $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 7053,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $F(a, b, c, d) = a^b + c \\times d$, what is the value of $x$ such that $F(2, x, 4, 11) = 300$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 7054,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)=2\\sin x\\cos x-2\\cos ^{2}(x+\\frac{π}{4})$. \n$(1)$ Find the intervals where $f(x)$ is monotonically increasing and its center of symmetry. \n$(2)$ Given $x\\in (0,\\frac{π}{2})$, if $f(x+\\frac{π}{6})=\\frac{3}{5}$, find the value of $\\cos 2x$. Let's think step ... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{3}-3}{10}"
} | {
"split": "train",
"index": 7055,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence 1,000,000; 500,000; 250,000 and so on, is made by repeatedly dividing by 2. What is the last integer in this sequence? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15625"
} | {
"split": "train",
"index": 7056,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Twenty-four 4-inch wide square posts are evenly spaced with 5 feet between adjacent posts to enclose a square field, as shown. What is the outer perimeter, in feet, of the fence? Express your answer as a mixed number. [asy]\nunitsize(2mm);\ndefaultpen(linewidth(.7pt));\ndotfact... | math | {
"style": "rule",
"ground_truth": "129\\frac{1}{3}"
} | {
"split": "train",
"index": 7057,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If a number is a multiple of 4 or contains the digit 4, we say this number is a \"4-inclusive number\", such as 20, 34. Arrange all \"4-inclusive numbers\" in the range \\[0, 100\\] in ascending order to form a sequence. What is the sum of all items in this sequence? Let's thin... | math | {
"style": "rule",
"ground_truth": "1883"
} | {
"split": "train",
"index": 7058,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the $2019$ th strictly positive integer $n$ such that $\\binom{2n}{n}$ is not divisible by $5$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "37805"
} | {
"split": "train",
"index": 7059,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If I have a $5\\times 5$ chess board, in how many ways can I place five distinct pawns on the board such that each column and row of the board contains no more than one pawn? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14400"
} | {
"split": "train",
"index": 7060,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $45 \\cdot 68 \\cdot 99 \\equiv m \\pmod{25}$, where $0 \\leq m < 25$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 7061,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence of functions $\\, \\{f_n(x) \\} \\,$ is defined recursively as follows: \\begin{align*} f_1(x) &= \\sqrt {x^2 + 48}, \\quad \\text{and} \\\\ f_{n + 1}(x) &= \\sqrt {x^2 + 6f_n(x)} \\quad \\text{for } n \\geq 1. \\end{align*} (Recall that $\\sqrt {\\makebox[5mm]{}}$ ... | math | {
"style": "rule",
"ground_truth": "x = 4"
} | {
"split": "train",
"index": 7062,
"solution": "We define $f_0(x) = 8$ . Then the recursive relation holds for $n=0$ , as well.\nSince $f_n (x) \\ge 0$ for all nonnegative integers $n$ , it suffices to consider nonnegative values of $x$ .\nWe claim that the following set of relations hold true for all natural nu... |
deepscaler | [
{
"role": "user",
"content": "Each of six, standard, six-sided dice is rolled once. What is the probability that there is exactly one pair and one triplet (three dice showing the same value), and the remaining dice show different values? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "\\frac{25}{162}"
} | {
"split": "train",
"index": 7063,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A digit was crossed out from a six-digit number, resulting in a five-digit number. When this five-digit number was subtracted from the original six-digit number, the result was 654321. Find the original six-digit number. Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "727023"
} | {
"split": "train",
"index": 7064,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given there are $50$ students in a class, a systematic sampling method is used to select $10$ students from these $50$. The $50$ students are randomly numbered from $1$ to $50$ and then evenly divided into $10$ groups in numerical order, with each group containing $5$ students.... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 7065,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the line $y=x-2$ intersects the hyperbola $C: \\frac{{x}^{2}}{{a}^{2}}-\\frac{{y}^{2}}{{b}^{2}}=1 (a>0, b>0)$ at the intersection points $A$ and $B (not coincident)$. The perpendicular bisector of line segment $AB$ passes through the point $(4,0)$. The eccentricity o... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 7066,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different lines pass through at least two points in this 3-by-3 grid of lattice points shown?\n\n[asy]\nsize(30);\ndot((0,0));\ndot((1,0));\ndot((2,0));\ndot((0,1));\ndot((1,1));\ndot((2,1));\ndot((0,2));\ndot((1,2));\ndot((2,2));\n[/asy] Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 7067,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $40$ feet high screen is put on a vertical wall $10$ feet above your eye-level. How far should you stand to maximize the angle subtended by the screen (from top to bottom) at your eye? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10\\sqrt{5}"
} | {
"split": "train",
"index": 7068,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiaoming has several 1-yuan, 2-yuan, and 5-yuan banknotes. He wants to use no more than 10 banknotes to buy a kite priced at 18 yuan, and he must use at least two different denominations. How many different payment methods are possible? Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 7069,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number halfway between $\\dfrac{1}{8}$ and $\\dfrac{1}{3}$ is\nA) $\\dfrac{11}{48}$\nB) $\\dfrac{11}{24}$\nC) $\\dfrac{5}{24}$\nD) $\\dfrac{1}{4}$\nE) $\\dfrac{1}{5}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{11}{48}"
} | {
"split": "train",
"index": 7070,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Svitlana writes the number 147 on a blackboard. Then, at any point, if the number on the blackboard is $n$, she can perform one of the following three operations: - if $n$ is even, she can replace $n$ with $\\frac{n}{2}$; - if $n$ is odd, she can replace $n$ with $\\frac{n+255}... | math | {
"style": "rule",
"ground_truth": "163"
} | {
"split": "train",
"index": 7071,
"solution": "The answer is $163=\\sum_{i=0}^{4}\\binom{8}{i}$. This is because we can obtain any integer less than $2^{8}$ with less than or equal to 4 ones in its binary representation. Note that $147=2^{7}+2^{4}+2^{1}+2^{0}$. We work in binary. Firstly, no operation can increa... |
deepscaler | [
{
"role": "user",
"content": "The ratio of girls to boys in Ms. Snow's math class is 3:2. If there is a total of 45 students, how many girls are in Ms. Snow's math class? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 7072,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The side lengths \\(a, b, c\\) of triangle \\(\\triangle ABC\\) satisfy the conditions:\n1. \\(a, b, c\\) are all integers;\n2. \\(a, b, c\\) form a geometric sequence;\n3. At least one of \\(a\\) or \\(c\\) is equal to 100. \n\nFind all possible sets of the triplet \\((a, b, c... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 7073,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The domain of the function \\( f(x) \\) is \\( (0,1) \\), and the function is defined as follows:\n\\[ \nf(x)=\\begin{cases}\nx, & \\text{if } x \\text{ is an irrational number}, \\\\\n\\frac{p+1}{q}, & \\text{if } x=\\frac{p}{q}, \\; p, q \\in \\mathbf{N}^{*}, \\; (p, q) = 1, ... | math | {
"style": "rule",
"ground_truth": "16/17"
} | {
"split": "train",
"index": 7074,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive divisors does the number $360$ have? Also, calculate the sum of all positive divisors of $360$ that are greater than $30$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1003"
} | {
"split": "train",
"index": 7075,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number that becomes a perfect square either by adding 5 or by subtracting 11. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 7076,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider integers c and d where c consists of 1986 nines, and d consists of 1986 sixes. What is the sum of the digits of the resulting number in base 10 when these numbers are added? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9931"
} | {
"split": "train",
"index": 7077,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a hyperbola with the equation $x^2 - y^2 = 9$. A line passing through the left focus $F_1$ of the hyperbola intersects the left branch of the hyperbola at points $P$ and $Q$. Let $F_2$ be the right focus of the hyperbola. If the length of segment $PQ$ is 7, then calcul... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 7078,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many sequences of zeros and ones of length 20 have all the zeros consecutive, or all the ones consecutive, or both? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "382"
} | {
"split": "train",
"index": 7079,
"solution": "1. **Count sequences with all 1's consecutive**: \n - If there are $k$ consecutive 1's, then we treat these $k$ 1's as a single block. This block, along with the remaining $20-k$ zeros, can be arranged in $\\binom{20-k+1}{1}$ ways (since there are $20-k+1$ positio... |
deepscaler | [
{
"role": "user",
"content": "In bag A, there are 3 white balls and 2 red balls, while in bag B, there are 2 white balls and 4 red balls. If a bag is randomly chosen first, and then 2 balls are randomly drawn from that bag, the probability that the second ball drawn is white given that the first ball drawn ... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{32}"
} | {
"split": "train",
"index": 7080,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}=(\\sqrt{2}\\cos \\omega x,1)$ and $\\overrightarrow{b}=(2\\sin (\\omega x+ \\frac{\\pi}{4}),-1)$ where $\\frac{1}{4}\\leqslant \\omega\\leqslant \\frac{3}{2}$, and the function $f(x)= \\overrightarrow{a}\\cdot \\overrightarrow{b}$, and the gra... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{10}+2}{9}"
} | {
"split": "train",
"index": 7081,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A = (3, \\theta_1)$ and $B = (9, \\theta_2)$ in polar coordinates. If $\\theta_1 - \\theta_2 = \\frac{\\pi}{2},$ then find the distance $AB.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 \\sqrt{10}"
} | {
"split": "train",
"index": 7082,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\\\(\\sin \\theta + \\cos \\theta = \\frac{3}{4}\\\\), where \\\\(\\theta\\\\) is an angle of a triangle, the value of \\\\(\\sin \\theta - \\cos \\theta\\\\) is \\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{23}}{4}"
} | {
"split": "train",
"index": 7083,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Starting with a pair of adult rabbits, the number of pairs of adult rabbits after one year can be calculated using the recurrence relation of the Fibonacci sequence, and the result is given by $F(13). Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "233"
} | {
"split": "train",
"index": 7084,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\tbinom{n}{k}=\\tfrac{n!}{k!(n-k)!}$ , the value of $$ \\sum_{n=3}^{10}\\frac{\\binom{n}{2}}{\\binom{n}{3}\\binom{n+1}{3}} $$ can be written in the form $\\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $m+n$ . Let's thi... | math | {
"style": "rule",
"ground_truth": "329"
} | {
"split": "train",
"index": 7085,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A factory estimates that the total demand for a particular product in the first $x$ months starting from the beginning of 2016, denoted as $f(x)$ (in units of 'tai'), is approximately related to the month $x$ as follows: $f(x)=x(x+1)(35-2x)$, where $x \\in \\mathbb{N}^*$ and $x... | math | {
"style": "rule",
"ground_truth": "171"
} | {
"split": "train",
"index": 7086,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kelvin the Frog is playing the game of Survival. He starts with two fair coins. Every minute, he flips all his coins one by one, and throws a coin away if it shows tails. The game ends when he has no coins left, and Kelvin's score is the *square* of the number of minutes elapse... | math | {
"style": "rule",
"ground_truth": "\\frac{64}{9}"
} | {
"split": "train",
"index": 7087,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $a, b, c$, and $d$ are pairwise distinct positive perfect squares such that $a^{b}=c^{d}$. Compute the smallest possible value of $a+b+c+d$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "305"
} | {
"split": "train",
"index": 7088,
"solution": "Note that if $a$ and $c$ are divisible by more than one distinct prime, then we can just take the prime powers of a specific prime. Thus, assume $a$ and $c$ are powers of a prime $p$. Assume $a=4^{x}$ and $c=4^{y}$. Then $x b=y d$. Because $b$ and $d$ are squares, t... |
deepscaler | [
{
"role": "user",
"content": "Let $f : \\mathbb{R} \\to \\mathbb{R}$ be a function such that $f(1) = 1$ and\n\\[f(x^2 - y^2) = (x - y) (f(x) + f(y))\\]for all real numbers $x$ and $y.$\n\nLet $n$ be the number of possible values of $f(2),$ and let $s$ be the sum of all possible values of $f(2).$ Find $n \\... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 7089,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive divisors do 9240 and 13860 have in common? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 7090,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A scientist begins an experiment with a cell culture that starts with some integer number of identical cells. After the first second, one of the cells dies, and every two seconds from there another cell will die (so one cell dies every odd-numbered second from the starting time... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"split": "train",
"index": 7091,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $2$ boys for every $3$ girls in Ms. Johnson's math class. If there are $30$ students in her class, what percent of them are boys? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40\\%"
} | {
"split": "train",
"index": 7092,
"solution": "1. **Understanding the Ratio**: The problem states that there are 2 boys for every 3 girls in the class. This sets up a ratio of boys to total students. For every 5 students (2 boys + 3 girls), 2 are boys.\n\n2. **Calculating the Fraction of Boys**: The fraction of ... |
deepscaler | [
{
"role": "user",
"content": "Isabella and Evan are cousins. The 10 letters from their names are placed on identical cards so that each of 10 cards contains one letter. Without replacement, two cards are selected at random from the 10 cards. What is the probability that one letter is from each cousin's name... | math | {
"style": "rule",
"ground_truth": "\\frac{16}{45}"
} | {
"split": "train",
"index": 7093,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given circle $C$: $x^{2}+y^{2}+8x+ay-5=0$ passes through the focus of parabola $E$: $x^{2}=4y$. The length of the chord formed by the intersection of the directrix of parabola $E$ and circle $C$ is $\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "4 \\sqrt{6}"
} | {
"split": "train",
"index": 7094,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Moor has $2016$ white rabbit candies. He and his $n$ friends split the candies equally amongst themselves, and they find that they each have an integer number of candies. Given that $n$ is a positive integer (Moor has at least $1$ friend), how many possible values of $... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 7095,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a parabola C: $y^2=2px$ $(p>0)$ whose focus F coincides with the right focus of the ellipse $\\frac{x^2}{4}+\\frac{y^2}{3}=1$, and a line l passing through point F intersects the parabola at points A and B.\n(1) Find the equation of the parabola C.\n(2) If the line l inte... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 7096,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the largest natural number $r$ with the property that among any five subsets with $500$ elements of the set $\\{1,2,\\ldots,1000\\}$ there exist two of them which share at least $r$ elements. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 7097,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( A B C D E F \\) be a regular hexagon, and let \\( P \\) be a point inside quadrilateral \\( A B C D \\). If the area of triangle \\( P B C \\) is 20, and the area of triangle \\( P A D \\) is 23, compute the area of hexagon \\( A B C D E F \\). Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "189"
} | {
"split": "train",
"index": 7098,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $k$ be an integer greater than 1. Suppose $a_0 > 0$, and define \\[a_{n+1} = a_n + \\frac{1}{\\sqrt[k]{a_n}}\\] for $n > 0$. Evaluate \\[\\lim_{n \\to \\infty} \\frac{a_n^{k+1}}{n^k}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left( \\frac{k+1}{k} \\right)^k"
} | {
"split": "train",
"index": 7099,
"solution": "\\textbf{First solution:} We start with some easy upper and lower bounds on $a_n$. We write $O(f(n))$ and $\\Omega(f(n))$ for functions $g(n)$ such that $f(n)/g(n)$ and $g(n)/f(n)$, respectively, are bounded above. Since $a_n$ is a nondecreasing sequence, $a_{n+1}-a... |
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