data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "The value of $x$ is one-half the value of $y$, and the value of $y$ is one-fifth the value of $z$. If $z$ is 60, 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": "6"
} | {
"split": "train",
"index": 34849,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle Q R S$, point $T$ is on $Q S$ with $\\angle Q R T=\\angle S R T$. Suppose that $Q T=m$ and $T S=n$ for some integers $m$ and $n$ with $n>m$ and for which $n+m$ is a multiple of $n-m$. Suppose also that the perimeter of $\\triangle Q R S$ is $p$ and that the numbe... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 34850,
"solution": "In this solution, we will use two geometric results: (i) The Triangle Inequality This result says that, in $\\triangle A B C$, each of the following inequalities is true: $A B+B C>A C \\quad A C+B C>A B \\quad A B+A C>B C$. This result comes from the fact that the ... |
deepscaler | [
{
"role": "user",
"content": "Compute the sum of the squares of the first 10 base-6 numbers and express your answer in base 6. Specifically, find $1_6^2 + 2_6^2 + 3_6^2 + \\cdots + 10_6^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "231_6"
} | {
"split": "train",
"index": 34851,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the sum of the geometric series $-1 -3-9-27 -81-243-729$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1093"
} | {
"split": "train",
"index": 34852,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Arrange the $n$ consecutive positive integers from 1 to $n$ (where $n > 1$) in a sequence such that the sum of each pair of adjacent terms is a perfect square. Find the minimum value of $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 34853,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( a, b, c \\) be prime numbers such that \\( a^5 \\) divides \\( b^2 - c \\), and \\( b + c \\) is a perfect square. Find the minimum value of \\( abc \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1958"
} | {
"split": "train",
"index": 34854,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The numbers $a,$ $b,$ $c,$ $d$ are equal to 1, 2, 3, 4, in some order. Find the largest possible value of\n\\[ab + bc + cd + da.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 34855,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For positive reals $p$ and $q$, define the remainder when $p$ is divided by $q$ as the smallest nonnegative real $r$ such that $\\frac{p-r}{q}$ is an integer. For an ordered pair $(a, b)$ of positive integers, let $r_{1}$ and $r_{2}$ be the remainder when $a \\sqrt{2}+b \\sqrt{... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 34856,
"solution": "The remainder when we divide $a \\sqrt{2}+b \\sqrt{3}$ by $\\sqrt{2}$ is defined to be the smallest non-negative real $r_{1}$ such that $\\frac{a \\sqrt{2}+b \\sqrt{3}-r_{1}}{\\sqrt{2}}$ is integral. As $\\frac{x}{\\sqrt{2}}$ is integral iff $x$ is an integral mult... |
deepscaler | [
{
"role": "user",
"content": "In a sequence of positive integers that starts with 1, certain numbers are sequentially colored red according to the following rules. First, 1 is colored red. Then, the next 2 even numbers, 2 and 4, are colored red. After 4, the next three consecutive odd numbers, 5, 7, and 9, ... | math | {
"style": "rule",
"ground_truth": "3943"
} | {
"split": "train",
"index": 34857,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a circle of radius $3$ units, find the area of the region consisting of all line segments of length $6$ units that are tangent to the circle at their midpoints. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9\\pi"
} | {
"split": "train",
"index": 34858,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that a travel agency plans to arrange a trip for 900 passengers using two types of buses, A and B, with capacities 36 and 60 passengers respectively, rental costs 1600 yuan and 2400 yuan per bus respectively, and the total number of buses rented does not exceed 21, and th... | math | {
"style": "rule",
"ground_truth": "36800"
} | {
"split": "train",
"index": 34859,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice and Ali each select a positive integer less than 250. Alice's number is a multiple of 25, and Ali's number is a multiple of 30. What is the probability that they selected the same number? Express your answer as a common fraction. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{80}"
} | {
"split": "train",
"index": 34860,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider all ordered pairs $(m, n)$ of positive integers satisfying $59 m - 68 n = mn$ . Find the sum of all the possible values of $n$ in these ordered pairs. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "237"
} | {
"split": "train",
"index": 34861,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the random variable $ξ∼B\\left(5,0.5\\right)$, and $η=5ξ$, find the respective values of $Eη$ and $Dη$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{125}{4}"
} | {
"split": "train",
"index": 34862,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T_n$ be the sum of the reciprocals of the non-zero digits of the integers from $1$ to $5^n$ inclusive. Find the smallest positive integer $n$ for which $T_n$ is an integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "504"
} | {
"split": "train",
"index": 34863,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The greatest common divisor (GCD) of 17 and 51 is , and the least common multiple (LCM) is . The GCD of 6 and 8 is , and the LCM of 8 and 9 is . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 34864,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two lines with slopes 3 and -1 intersect at the point $(3, 1)$. What is the area of the triangle enclosed by these two lines and the horizontal line $y = 8$?\n- **A)** $\\frac{25}{4}$\n- **B)** $\\frac{98}{3}$\n- **C)** $\\frac{50}{3}$\n- **D)** 36\n- **E)** $\\frac{200}{9}$ Le... | math | {
"style": "rule",
"ground_truth": "\\frac{98}{3}"
} | {
"split": "train",
"index": 34865,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the number of ordered pairs of positive integers $(a, b)$ satisfying the equation $\\operatorname{gcd}(a, b) \\cdot a+b^{2}=10000$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "99"
} | {
"split": "train",
"index": 34866,
"solution": "Let $\\operatorname{gcd}(a, b)=d, a=d a^{\\prime}, b=d b^{\\prime}$. Then, $d^{2}\\left(a^{\\prime}+b^{\\prime 2}\\right)=100^{2}$. Consider each divisor $d$ of 100. Then, we need to find the number of solutions in coprime integers to $a^{\\prime}+b^{\\prime 2}=\\f... |
deepscaler | [
{
"role": "user",
"content": "Determine the greatest number m such that the\nsystem \u001a $x^2$ + $y^2$ = 1; | $x^3$ - $y^3$ |+|x-y|= $m^3$ has a solution. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[3]{2}"
} | {
"split": "train",
"index": 34867,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A = (1,0),$ $B = (4,3),$ and $C = (p,q)$ be three points on the parabola $y = -x^2 + 6x - 5,$ where $1 \\le p \\le 4.$ Find the largest possible area of triangle $ABC.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{27}{8}"
} | {
"split": "train",
"index": 34868,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\alpha$ and $\\beta$ be real numbers. Find the minimum value of\n\\[(2 \\cos \\alpha + 5 \\sin \\beta - 8)^2 + (2 \\sin \\alpha + 5 \\cos \\beta - 15)^2.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 34869,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To promote her school's annual Kite Olympics, Genevieve makes a small kite and a large kite for a bulletin board display. For her small kite, Genevieve draws the kite on a grid with the points one inch apart, as shown below.\n\n[asy]\nfor ( int x = 0; x <= 6; ++x )\n{\n\nfor ( ... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 34870,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For all $m, n$ satisfying $1 \\leqslant n \\leqslant m \\leqslant 5$, the number of distinct hyperbolas represented by the polar equation $\\rho = \\frac{1}{1 - c_{m}^{n} \\cos \\theta}$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 34871,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $5 \\times 5$ grid of squares is filled with integers. Call a rectangle corner-odd if its sides are grid lines and the sum of the integers in its four corners is an odd number. What is the maximum possible number of corner-odd rectangles within the grid? Let's think step by s... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 34872,
"solution": "Consider any two rows and the five numbers obtained by adding the two numbers which share a given column. Suppose $a$ of these are odd and $b$ of these are even. The number of corner-odd rectangles with their sides contained in these two rows is $a b$. Since $a+b=5... |
deepscaler | [
{
"role": "user",
"content": "Express $0.6\\overline{03}$ as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{104}{165}"
} | {
"split": "train",
"index": 34873,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger's allowance was $A$ dollars. The cost of his movie ticket was $20\\%$ of the difference between $A$ and the cost of his soda, while the cost of his soda was $5\\%$ of the difference betwe... | math | {
"style": "rule",
"ground_truth": "23\\%"
} | {
"split": "train",
"index": 34874,
"solution": "1. **Define Variables:**\n Let $m$ represent the cost of the movie ticket and $s$ represent the cost of the soda.\n\n2. **Set Up Equations:**\n According to the problem, we have:\n \\[ m = 0.20(A - s) \\]\n \\[ s = 0.05(A - m) \\]\n\n3. **Convert to Fractio... |
deepscaler | [
{
"role": "user",
"content": "Suppose $a$, $b$, $c$, and $d$ are integers such that:\n- $a - b + c = 7$\n- $b - c + d = 8$\n- $c - d + a = 4$\n- $d - a + b = 3$\n- $a + b + c - d = 10$\nFind the 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": "16"
} | {
"split": "train",
"index": 34875,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( XYZ \\) be an acute-angled triangle. Let \\( s \\) be the side length of the square which has two adjacent vertices on side \\( YZ \\), one vertex on side \\( XY \\), and one vertex on side \\( XZ \\). Let \\( h \\) be the distance from \\( X \\) to the side \\( YZ \\) ... | math | {
"style": "rule",
"ground_truth": "4034"
} | {
"split": "train",
"index": 34876,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest four-digit negative integer congruent to $2 \\pmod{17}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1001"
} | {
"split": "train",
"index": 34877,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $q(x)$ be a monic quartic polynomial such that $q(1) = 3$, $q(2) = 6, $ $q(3) = 11$, and $q(4) = 18$. Find $q(5)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 34878,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From a barrel, 4 liters of wine are drawn, and this is replaced with 4 liters of water. From the resulting mixture, 4 liters are drawn again and replaced with 4 liters of water. This operation is repeated a total of three times, and the final result is that there are 2.5 liters... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 34879,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)=x^{2}+x^{4}+x^{6}+x^{8}+\\cdots$, for all real $x$ such that the sum converges. For how many real numbers $x$ does $f(x)=x$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 34880,
"solution": "Clearly $x=0$ works. Otherwise, we want $x=x^{2} /\\left(1-x^{2}\\right)$, or $x^{2}+x-1=0$. Discard the negative root (since the sum doesn't converge there), but $(-1+\\sqrt{5}) / 2$ works, for a total of 2 values."
} |
deepscaler | [
{
"role": "user",
"content": "If the direction vector of line $l$ is $\\overrightarrow{e}=(-1,\\sqrt{3})$, calculate the inclination angle of line $l$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"split": "train",
"index": 34881,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the equation of the directrix of the parabola $y = \\frac{x^2 - 6x + 5}{12}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "y = -\\frac{10}{3}"
} | {
"split": "train",
"index": 34882,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given two vectors $\\overrightarrow {a}$ and $\\overrightarrow {b}$ in a plane, satisfying $|\\overrightarrow {a}|=1$, $|\\overrightarrow {b}|=2$, and the dot product $(\\overrightarrow {a}+ \\overrightarrow {b})\\cdot (\\overrightarrow {a}-2\\overrightarrow {b})=-7$, find the ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"split": "train",
"index": 34883,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the eve of the Spring Festival, a convenience store sold a batch of goods that were purchased at 12 yuan per piece at a price of 20 yuan per piece, selling 240 pieces per day. After a period of sales, it was found that if the price per piece was increased by 0.5 yuan, then 1... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 34884,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the system of equations:\n\n\\[\n8x - 6y = a,\n\\]\n\\[\n12y - 18x = b.\n\\]\nIf there's a solution $(x, y)$ where both $x$ and $y$ are nonzero, determine $\\frac{a}{b}$, assuming $b$ is nonzero. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "-\\frac{4}{9}"
} | {
"split": "train",
"index": 34885,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the degree measure of the smaller angle between the hour hand and the minute hand of a clock at exactly 2:30 p.m. on a 12-hour analog clock? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "105^\\circ"
} | {
"split": "train",
"index": 34886,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Choose two different numbers from the set of numbers {1, 2, ..., 8, 9}, and find the probability that their product is an odd number. (Express the result as a numerical value). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{18}"
} | {
"split": "train",
"index": 34887,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The minimum value of the function $y=|x-1|$ is 0, the minimum value of the function $y=|x-1|+|x-2|$ is 1, and the minimum value of the function $y=|x-1|+|x-2|+|x-3|$ is 2. What is the minimum value of the function $y=|x-1|+|x-2|+\\ldots+|x-10|$? Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 34888,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of positive integers $n$ satisfying:\n\n\n\n- $n<10^6$\n- $n$ is divisible by 7\n- $n$ does not contain any of the digits 2,3,4,5,6,7,8.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "104"
} | {
"split": "train",
"index": 34889,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $a$ and $b$ are real numbers such that the line $y=a x+b$ intersects the graph of $y=x^{2}$ at two distinct points $A$ and $B$. If the coordinates of the midpoint of $A B$ are $(5,101)$, compute $a+b$. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"split": "train",
"index": 34890,
"solution": "Solution 1: Let $A=\\left(r, r^{2}\\right)$ and $B=\\left(s, s^{2}\\right)$. Since $r$ and $s$ are roots of $x^{2}-a x-b$ with midpoint 5, $r+s=10=a$ (where the last equality follows by Vieta's formula). Now, as $-r s=b$ (Vieta's formula), observe that $$202=r^{2}+... |
deepscaler | [
{
"role": "user",
"content": "In right triangle $XYZ$ with $\\angle Z = 90^\\circ$, the length $XY = 15$ and the length $XZ = 8$. Find $\\sin Y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{161}}{15}"
} | {
"split": "train",
"index": 34891,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "BoatsRUs built 7 canoes in January of this year and then each subsequent calendar month they built twice the number of canoes they had built the previous month. How many total canoes were built by BoatsRUs by the end of May of this year? Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "217"
} | {
"split": "train",
"index": 34892,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a finite sequence $S=(a_1,a_2,\\ldots ,a_n)$ of $n$ real numbers, let $A(S)$ be the sequence \n$\\left(\\frac{a_1+a_2}{2},\\frac{a_2+a_3}{2},\\ldots ,\\frac{a_{n-1}+a_n}{2}\\right)$\nof $n-1$ real numbers. Define $A^1(S)=A(S)$ and, for each integer $m$, $2\\le m\\le n-1$,... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}-1"
} | {
"split": "train",
"index": 34893,
"solution": "1. **Understanding the Sequence Transformation**:\n Given a sequence $S = (a_1, a_2, \\ldots, a_n)$, the sequence $A(S)$ is defined as:\n \\[\n A(S) = \\left(\\frac{a_1 + a_2}{2}, \\frac{a_2 + a_3}{2}, \\ldots, \\frac{a_{n-1} + a_n}{2}\\right)\n \\]\n Thi... |
deepscaler | [
{
"role": "user",
"content": "There is a set of natural numbers (it is known that there are at least seven numbers) such that the sum of every seven of them is less than 15, and the sum of all numbers in the set is 100. What is the smallest number of numbers that can be in the set? Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 34894,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify first, then evaluate: $(1-\\frac{m}{{m+3}})÷\\frac{{{m^2}-9}}{{{m^2}+6m+9}}$, where $m=\\sqrt{3}+3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 34895,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( x_{i}=\\frac{i}{101} \\), find the value of \\( S=\\sum_{i=1}^{101} \\frac{x_{i}^{3}}{3 x_{i}^{2}-3 x_{i}+1} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 34896,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $5a+2b=0$ and $a$ is two less than $b$, what is $7b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 34897,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( x, y, z \\in \\mathbb{Z}_{+} \\) with \\( x \\leq y \\leq z \\), how many sets of solutions satisfy the equation \\(\\frac{1}{x}+\\frac{1}{y}+\\frac{1}{z}=\\frac{1}{2}\\) ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 34898,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence of positive terms $\\{a_n\\}$ with the sum of the first $n$ terms denoted as $S_n$, it satisfies the equation $2S_n = a_n^2 + a_n$ for all natural numbers $n$. Define a new sequence $\\{c_n\\}$ where $c_n = (-1)^n \\frac{2a_n + 1}{2S_n}$. Find the sum of the fi... | math | {
"style": "rule",
"ground_truth": "- \\frac{2016}{2017}"
} | {
"split": "train",
"index": 34899,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To the eight-digit number 20222023, append one digit to the left and one digit to the right so that the resulting ten-digit number is divisible by 72. Determine all possible solutions. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3202220232"
} | {
"split": "train",
"index": 34900,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The altitude of an equilateral triangle is $\\sqrt{12}$ units. What is the area and the perimeter of the triangle, expressed in simplest radical form? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 34901,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A tetrahedron \\( P-ABC \\) has edge lengths \\( PA = BC = \\sqrt{6} \\), \\( PB = AC = \\sqrt{8} \\), and \\( PC = AB = \\sqrt{10} \\). Find the radius of the circumsphere of this tetrahedron. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 34902,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square with sides 8 inches is shown. If $Q$ is a point such that the segments $\\overline{QA}$, $\\overline{QB}$, $\\overline{QC}$ are equal in length, and segment $\\overline{QC}$ is perpendicular to segment $\\overline{HD}$, find the area, in square inches, of triangle $AQB... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 34903,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice starts with the number 0. She can apply 100 operations on her number. In each operation, she can either add 1 to her number, or square her number. After applying all operations, her score is the minimum distance from her number to any perfect square. What is the maximum s... | math | {
"style": "rule",
"ground_truth": "94"
} | {
"split": "train",
"index": 34904,
"solution": "Note that after applying the squaring operation, Alice's number will be a perfect square, so she can maximize her score by having a large number of adding operations at the end. However, her scores needs to be large enough that the many additions do not bring her c... |
deepscaler | [
{
"role": "user",
"content": "If $f(x)=3x+2$ for all real $x$, then the statement:\n\"$|f(x)+4|<a$ whenever $|x+2|<b$ and $a>0$ and $b>0$\"\nis true when Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "b \\le a/3"
} | {
"split": "train",
"index": 34905,
"solution": "1. **Substitute $f(x)$ into the inequality**: Given $f(x) = 3x + 2$, we substitute this into the inequality $|f(x) + 4| < a$:\n \\[\n |3x + 2 + 4| = |3x + 6|.\n \\]\n Simplifying further, we have:\n \\[\n |3x + 6| = |3(x + 2)| = 3|x + 2|.\n \\]\n\n2. ... |
deepscaler | [
{
"role": "user",
"content": "Let $w_1, w_2, \\dots, w_n$ be complex numbers. A line $L$ in the complex plane is called a mean line for the points $w_1, w_2, \\dots, w_n$ if $L$ contains points (complex numbers) $z_1, z_2, \\dots, z_n$ such that \\[\\sum_{k = 1}^n (z_k - w_k) = 0.\\] For the numbers $w_1 = ... | math | {
"style": "rule",
"ground_truth": "163"
} | {
"split": "train",
"index": 34906,
"solution": "The mean line for $w_1, . . ., w_5$ must pass through the mean (the center of mass) of these points, which, if we graph them on the complex plane, is $(\\frac{3}{5}, \\frac{504i}{5})$. Since we now have two points, namely that one and $(0, 3i)$, we can simply find ... |
deepscaler | [
{
"role": "user",
"content": "Given a set of $n$ positive integers in which the difference between any two elements is either divisible by 5 or divisible by 25, find the maximum value of $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 34907,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Expand $-(4-c)(c+2(4-c) + c^2)$. What is the sum of the coefficients of the expanded form? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-24"
} | {
"split": "train",
"index": 34908,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nHow many ways can you arrange 15 dominoes (after removing all dominoes with five or six pips) in a single line according to the usual rules of the game, considering arrangements from left to right and right to left as different? As always, the dominoes must be placed such tha... | math | {
"style": "rule",
"ground_truth": "126760"
} | {
"split": "train",
"index": 34909,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the maximum number of rooks that can be placed in an \\(8 \\times 8 \\times 8\\) cube so that they do not attack each other? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 34910,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $x$: $0.05x + 0.07(30 + x) = 15.4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "110.8333333"
} | {
"split": "train",
"index": 34911,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different $4\\times 4$ arrays whose entries are all 1's and -1's have the property that the sum of the entries in each row is 0 and the sum of the entries in each column is 0? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 34912,
"solution": "We can think about it as shading a $4 \\times 4$ array so that there are exactly two shaded unit squares in each row and each column. Then the answer is $\\boxed{90}$."
} |
deepscaler | [
{
"role": "user",
"content": "Let $ ABP, BCQ, CAR$ be three non-overlapping triangles erected outside of acute triangle $ ABC$. Let $ M$ be the midpoint of segment $ AP$. Given that $ \\angle PAB \\equal{} \\angle CQB \\equal{} 45^\\circ$, $ \\angle ABP \\equal{} \\angle QBC \\equal{} 75^\\circ$, $ \\angle ... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 34913,
"solution": "\nLet \\( ABP, BCQ, CAR \\) be three non-overlapping triangles erected outside of acute triangle \\( ABC \\). Let \\( M \\) be the midpoint of segment \\( AP \\). Given that \\( \\angle PAB = \\angle CQB = 45^\\circ \\), \\( \\angle ABP = \\angle QBC = 75^\\circ \\... |
deepscaler | [
{
"role": "user",
"content": "We shuffle a deck of French playing cards and then draw the cards one by one. In which position is it most likely to draw the second ace? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 34914,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of \\[\\cot(\\cot^{-1}5 + \\cot^{-1}11 + \\cot^{-1}17 + \\cot^{-1}23).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{97}{40}"
} | {
"split": "train",
"index": 34915,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the nearest integer to $(3+\\sqrt2)^6$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7414"
} | {
"split": "train",
"index": 34916,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What fraction of the entire wall is painted red if Matilda paints half of her section red and Ellie paints one third of her section red? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{12}"
} | {
"split": "train",
"index": 34917,
"solution": "Matilda and Ellie each take $\\frac{1}{2}$ of the wall. Matilda paints $\\frac{1}{2}$ of her half, or $\\frac{1}{2} \\times \\frac{1}{2} = \\frac{1}{4}$ of the entire wall. Ellie paints $\\frac{1}{3}$ of her half, or $\\frac{1}{3} \\times \\frac{1}{2} = \\frac{1}{6... |
deepscaler | [
{
"role": "user",
"content": "Let $S_n$ be the sum of the first $n$ terms of the difference sequence $\\{a_n\\}$, given that $a_2 + a_{12} = 24$ and $S_{11} = 121$. \n(1) Find the general term formula for $\\{a_n\\}$. \n(2) Let $b_n = \\frac {1}{a_{n+1}a_{n+2}}$, and $T_n = b_1 + b_2 + \\ldots + b_n$. If ... | math | {
"style": "rule",
"ground_truth": "m = \\frac {3}{7}"
} | {
"split": "train",
"index": 34918,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cylinder has a radius of 5 cm and a height of 10 cm. What is the longest segment, in centimeters, that would fit inside the cylinder? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10\\sqrt{2}"
} | {
"split": "train",
"index": 34919,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{a}$ and $\\mathbf{b}$ be unit vectors such that $\\mathbf{a} + 2 \\mathbf{b}$ and $5 \\mathbf{a} - 4 \\mathbf{b}$ are orthogonal. Find the angle between $\\mathbf{a}$ and $\\mathbf{b},$ in degrees.\n\nNote: A unit vector is a vector of magnitude 1. Let's think st... | math | {
"style": "rule",
"ground_truth": "60^\\circ"
} | {
"split": "train",
"index": 34920,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, it is known that $AB=2$, $AC=3$, and $A=60^{\\circ}$.\n$(1)$ Find the length of $BC$;\n$(2)$ Find the value of $\\sin 2C$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{3}}{7}"
} | {
"split": "train",
"index": 34921,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = 4\\sin^2 x + \\sin\\left(2x + \\frac{\\pi}{6}\\right) - 2$,\n$(1)$ Determine the interval over which $f(x)$ is strictly decreasing;\n$(2)$ Find the maximum value of $f(x)$ on the interval $[0, \\frac{\\pi}{2}]$ and determine the value(s) of $x$ at whi... | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{12}"
} | {
"split": "train",
"index": 34922,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation $x^2 + 2x = i$ has two complex solutions. Determine the product of their real parts. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1-\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 34923,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any $h = 2^{r}$ ($r$ is a non-negative integer), find all $k \\in \\mathbb{N}$ which satisfy the following condition: There exists an odd natural number $m > 1$ and $n \\in \\mathbb{N}$, such that $k \\mid m^{h} - 1, m \\mid n^{\\frac{m^{h}-1}{k}} + 1$. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "2^{r+1}"
} | {
"split": "train",
"index": 34924,
"solution": "\nFor any \\( h = 2^{r} \\) (where \\( r \\) is a non-negative integer), we need to find all \\( k \\in \\mathbb{N} \\) which satisfy the following condition: There exists an odd natural number \\( m > 1 \\) and \\( n \\in \\mathbb{N} \\), such that \\( k \\mid m^{... |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$ with a common difference $d = -2$, and $a_1 + a_4 + a_7 + \\ldots + a_{97} = 50$, find the value of $a_3 + a_6 + a_9 + \\ldots + a_{99}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-66"
} | {
"split": "train",
"index": 34925,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate \\(3 \\cdot 15 + 20 \\div 4 + 1\\).\n\nThen add parentheses to the expression so that the result is:\n\n1. The largest possible integer,\n2. The smallest possible integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 34926,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Pile up 2019 stones into one pile. First, person A splits this pile into two piles and writes the product of the number of stones in each pile on the blackboard. Then, person A selects one pile from the two and splits it into two more piles, again writing the product of the num... | math | {
"style": "rule",
"ground_truth": "2037171"
} | {
"split": "train",
"index": 34927,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Distinct points $P$, $Q$, $R$, $S$ lie on the circle $x^{2}+y^{2}=25$ and have integer coordinates. The distances $PQ$ and $RS$ are irrational numbers. What is the greatest possible value of the ratio $\\frac{PQ}{RS}$? Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 34928,
"solution": "1. **Identify Possible Points**: The circle given by the equation $x^2 + y^2 = 25$ has a radius of 5. We need to find points $(x, y)$ with integer coordinates that lie on this circle. Solving $x^2 + y^2 = 25$ for integer solutions, we find the points $(\\pm 3, \\pm... |
deepscaler | [
{
"role": "user",
"content": "What is the product of the prime numbers less than 20? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9699690"
} | {
"split": "train",
"index": 34929,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The square of a three-digit number ends with three identical digits different from zero. Write the smallest such three-digit number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "462"
} | {
"split": "train",
"index": 34930,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "500"
} | {
"split": "train",
"index": 34931,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point is randomly thrown on the segment [12, 17] and let $k$ be the resulting value. Find the probability that the roots of the equation $\\left(k^{2}+k-90\\right) x^{2}+(3 k-8) x+2=0$ satisfy the condition $x_{1} \\leq 2 x_{2}$. Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "2/3"
} | {
"split": "train",
"index": 34932,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rebecca has four resistors, each with resistance 1 ohm . Every minute, she chooses any two resistors with resistance of $a$ and $b$ ohms respectively, and combine them into one by one of the following methods: - Connect them in series, which produces a resistor with resistance ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 34933,
"solution": "Let $R_{n}$ be the set of all possible resistances using exactly $n$ 1-ohm circuit segments (without shorting any of them), then we get $R_{n}=\\bigcup_{i=1}^{n-1}\\left(\\left\\{a+b \\mid a \\in R_{i}, b \\in R_{n-i}\\right\\} \\cup\\left\\{\\left.\\frac{a b}{a+b}... |
deepscaler | [
{
"role": "user",
"content": "If $A$ and $B$ are nonzero digits, then the number of digits (not necessarily different) in the sum of the three whole numbers is\n$\\begin{array}{cccc} 9 & 8 & 7 & 6 \\\\ & A & 3 & 2 \\\\ & B & 1 \\\\ \\hline \\end{array}$ Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 34934,
"solution": "1. **Identify the numbers to be summed**: The problem presents a sum of three numbers arranged in a column, which are:\n - The first number: $9876$\n - The second number: $A32$ (where $A$ is a digit)\n - The third number: $B1$ (where $B$ is a digit)\n\n2. **D... |
deepscaler | [
{
"role": "user",
"content": "A confectionery factory received 5 rolls of ribbon, each 50 meters long, for packing cakes. How many cuts are needed to obtain pieces of ribbon that are 2 meters each?\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 34935,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the third year of high school, the class organized a fun sports competition. After multiple rounds of competition, Class A and Class B entered the finals. There were three events in the finals, with the winner of each event receiving 2 points and the loser receiving -1 point... | math | {
"style": "rule",
"ground_truth": "0.9"
} | {
"split": "train",
"index": 34936,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of digits in the value of $2^{15} \\times 5^{10} \\times 3^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 34937,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$ABCDE$ is inscribed in a circle with $AB = BC = CD = DE = 4$ and $AE = 1.$ Compute $(1 - \\cos \\angle B)(1 - \\cos \\angle ACE).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{64}"
} | {
"split": "train",
"index": 34938,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sequence $\\{a\\_n\\}$ is an arithmetic progression, and the sequence $\\{b\\_n\\}$ satisfies $b\\_n=a\\_n a_{n+1} \\cdot a_{n+2} (n \\in \\mathbb{N}^*)$, let $S\\_n$ be the sum of the first $n$ terms of $\\{b\\_n\\}$. If $a_{12}=\\frac{3}{8} a_{5} > 0$, find the... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 34939,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Positive real numbers \\( x, y, z \\) satisfy: \\( x^{4} + y^{4} + z^{4} = 1 \\). Find the minimum value of the algebraic expression \\( \\frac{x^{3}}{1-x^{8}} + \\frac{y^{3}}{1-y^{8}} + \\frac{z^{3}}{1-z^{8}} \\). Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "\\frac{9 \\sqrt[4]{3}}{8}"
} | {
"split": "train",
"index": 34940,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The first digit on the left of a six-digit number is 1. If this digit is moved to the last place, the resulting number is 64 times greater than the original number. Find the original number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "142857"
} | {
"split": "train",
"index": 34941,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sum of the coefficients of the expansion of $(2x-1)^{n}$ is less than the sum of the binomial coefficients of the expansion of $(\\sqrt{x}+\\frac{1}{2\\sqrt[4]{x}})^{2n}$ by $255$. \n$(1)$ Find all the rational terms of $x$ in the expansion of $(\\sqrt{x}+\\frac... | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 34942,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of two natural numbers is $17402$. One of the two numbers is divisible by $10$. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "14238"
} | {
"split": "train",
"index": 34943,
"solution": "1. **Identify the properties of the numbers**: We are given two natural numbers whose sum is $17{,}402$. One of these numbers is divisible by $10$, and if the units digit of this number is erased, the other number is obtained. This implies that the number divisible... |
deepscaler | [
{
"role": "user",
"content": "Ten points are given in the plane, and no three points are collinear. Four distinct segments connecting pairs of these points are chosen at random, all with the same probability. What is the probability that three of the chosen segments will form a triangle? Let's think step by... | math | {
"style": "rule",
"ground_truth": "16/473"
} | {
"split": "train",
"index": 34944,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express $\\frac{0.\\overline{666}}{1.\\overline{333}}$ as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 34945,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For \\(50 \\le n \\le 150\\), how many integers \\(n\\) are there such that \\(\\frac{n}{n+1}\\) is a repeating decimal and \\(n+1\\) is not divisible by 3? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 34946,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $z$ be a complex number such that $|z - 5 - i| = 5.$ Find the minimum value of\n\\[|z - 1 + 2i|^2 + |z - 9 - 4i|^2.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 34947,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_{n}\\}$ with the sum of the first $n$ terms denoted as $S_{n}$, satisfying $S_{n} = 2^{n} + r$ (where $r$ is a constant). Define $b_{n} = 2\\left(1 + \\log_{2} a_{n}\\right)$ for $n \\in \\mathbf{N}^{*}$.\n\n(1) Find the sum of the first $n$ ter... | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{2}}{4}"
} | {
"split": "train",
"index": 34948,
"solution": ""
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.