data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "At what angle to the x-axis is the tangent to the graph of the function \\( g(x) = x^2 \\ln x \\) inclined at the point \\( x_0 = 1 \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 23149,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $\\sec x+\\tan x=\\frac{22}7$ and that $\\csc x+\\cot x=\\frac mn,$ where $\\frac mn$ is in lowest terms. Find $m+n.$\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 23150,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are three flavors of chocolates in a jar: hazelnut, liquor, and milk. There are 12 chocolates that are not hazelnut, 18 chocolates that are not liquor, and 20 chocolates that are not milk. How many chocolates are there in total in the jar? Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 23151,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( S \\) be a subset of \\(\\{1,2,3, \\ldots, 199,200\\}\\). We say that \\( S \\) is pretty if, for every pair of elements \\( a \\) and \\( b \\) in \\( S \\), the number \\( a - b \\) is not a prime number. What is the maximum number of elements in a pretty subset of \\... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 23152,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each integer 1 through 9 is written on a separate slip of paper and all nine slips are put into a hat. Jack picks one of these slips at random and puts it back. Then Jill picks a slip at random. Which digit is most likely to be the units digit of the sum of Jack's integer and J... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 23153,
"solution": "1. **Understanding the Problem**: Each integer from $1$ to $9$ is placed on a slip of paper and put into a hat. Jack and Jill each draw a slip (with replacement), and we want to find the most likely units digit of the sum of the two integers they draw.\n\n2. **Sett... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\angle BAC = 90^{\\circ}$ and the quadrilateral $ADEF$ is a square with side length 1, find the maximum value of $\\frac{1}{AB} + \\frac{1}{BC} + \\frac{1}{CA}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 + \\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 23154,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nAnton makes custom watches for a jewelry store. Each watch consists of a bracelet, a gemstone, and a clasp.\n\nBracelets are available in silver, gold, and steel. Anton has precious stones: cubic zirconia, emerald, quartz, diamond, and agate - and clasps: classic, butterfly, ... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 23155,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a rhombus \\( ABCD \\), the angle at vertex \\( A \\) is \\( 60^\\circ \\). Point \\( N \\) divides side \\( AB \\) in the ratio \\( AN:BN = 2:1 \\). Find the tangent of angle \\( DNC \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{243}}{17}"
} | {
"split": "train",
"index": 23156,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x > 0$, $y > 0$, and $\\frac{1}{2x+y} + \\frac{4}{x+y} = 2$, find the minimum value of $7x + 5y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7 + 2\\sqrt{6}"
} | {
"split": "train",
"index": 23157,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square with a perimeter of 36 is inscribed in a square with a perimeter of 40. What is the greatest distance between a vertex of the inner square and a vertex of the outer square?\nA) $\\sqrt{101}$\nB) $9\\sqrt{2}$\nC) $8\\sqrt{2}$\nD) $\\sqrt{90}$\nE) $10$ Let's think step b... | math | {
"style": "rule",
"ground_truth": "9\\sqrt{2}"
} | {
"split": "train",
"index": 23158,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain orange orchard has a total of 120 acres, consisting of both flat and hilly land. To estimate the average yield per acre, a stratified sampling method is used to survey a total of 10 acres. If the number of hilly acres sampled is 2 times plus 1 acre more than the flat ... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 23159,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\alpha$ be an acute angle, and $\\cos\\left(\\alpha+ \\frac{\\pi}{6}\\right) = \\frac{3 \\sqrt{10}}{10}$, $\\tan(\\alpha+\\beta) = \\frac{2}{5}$. \n(1) Find the value of $\\sin\\left(2\\alpha+ \\frac{\\pi}{6}\\right)$. \n(2) Find the value of $\\tan\\left(2\\beta- \\fra... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{144}"
} | {
"split": "train",
"index": 23160,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $AB=20$ and $AC=11$. The angle bisector of $\\angle A$ intersects $BC$ at point $D$, and point $M$ is the midpoint of $AD$. Let $P$ be the point of the intersection of $AC$ and $BM$. The ratio of $CP$ to $PA$ can be expressed in the form $\\dfrac{m}{n}$, wher... | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 23161,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five positive consecutive integers starting with $a$ have average $b$. What is the average of 5 consecutive integers that start with $b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "a+4"
} | {
"split": "train",
"index": 23162,
"solution": "1. **Define the sequence and calculate the average $b$:** \n The five consecutive integers starting with $a$ are $a, a+1, a+2, a+3, a+4$. The average of these integers, $b$, is calculated as follows:\n \\[\n b = \\frac{a + (a+1) + (a+2) + (a+3) + (a+4)}{5} =... |
deepscaler | [
{
"role": "user",
"content": "Compute $\\sqrt[4]{5508^{3}+5625^{3}+5742^{3}}$, given that it is an integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "855"
} | {
"split": "train",
"index": 23163,
"solution": "Let $a=5625=75^{2}$ and $b=117$. Then we have $5508^{3}+5265^{3}+5742^{3}=(a-b)^{3}+a^{3}+(a+b)^{3}=3a^{3}+6ab^{2}=3a(a^{2}+2b^{2})$. We have $3a=3^{3} \\cdot 5^{4}$, so $a^{2}+2b^{2}=3^{4} \\cdot(625^{2}+2 \\cdot 19^{2})$ should be 3 times a fourth power. This mea... |
deepscaler | [
{
"role": "user",
"content": "Robin bought a four-scoop ice cream cone having a scoop each of vanilla, chocolate, strawberry and cherry. In how many orders can the four scoops be stacked on the cone if they are stacked one on top of the other? Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 23164,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an equilateral triangle of side 10, divide each side into three equal parts, construct an equilateral triangle on the middle part, and then delete the middle part. Repeat this step for each side of the resulting polygon. Find \\( S^2 \\), where \\( S \\) is the area of th... | math | {
"style": "rule",
"ground_truth": "4800"
} | {
"split": "train",
"index": 23165,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the village of Znoynoe, there are exactly 1000 inhabitants, which exceeds the average population of the villages in the valley by 90 people.\n\nHow many inhabitants are there in the village of Raduzhny, which is also located in the Sunny Valley? Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "900"
} | {
"split": "train",
"index": 23166,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x_{1}, \\ldots, x_{100}$ be defined so that for each $i, x_{i}$ is a (uniformly) random integer between 1 and 6 inclusive. Find the expected number of integers in the set $\\{x_{1}, x_{1}+x_{2}, \\ldots, x_{1}+x_{2}+\\ldots+x_{100}\\}$ that are multiples of 6. Let's think ... | math | {
"style": "rule",
"ground_truth": "\\frac{50}{3}"
} | {
"split": "train",
"index": 23167,
"solution": "Note that for any $i$, the probability that $x_{1}+x_{2}+\\ldots+x_{i}$ is a multiple of 6 is $\\frac{1}{6}$ because exactly 1 value out of 6 possible values of $x_{i}$ works. Because these 100 events are independent, the expected value is $100 \\cdot \\frac{1}{6}=... |
deepscaler | [
{
"role": "user",
"content": "Write the numbers $0, 1, 2, 3, 4, 5, 6, 7$ on the eight vertices of a cube (each vertex has one number, and all numbers must be used once), such that the sum of the two numbers on each edge is a prime number. Then, find the maximum sum of the four numbers on one face of the cub... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 23168,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer that cannot be written as the sum of two nonnegative palindromic integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 23169,
"solution": "We need to first prove that every positive integer $N$ less than 21 can be written as sum of two nonnegative palindromic integers. If $N$ is in the interval $[1,9]$, then it can be written as $0+N$. If $N$ is in the interval $[10,18]$, it can be written as $9+(N-9)... |
deepscaler | [
{
"role": "user",
"content": "Given that one third of the students go home on the school bus, one fifth go home by automobile, one eighth go home on their bicycles, and one tenth go home on scooters, what fractional part of the students walk home? Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "\\frac{29}{120}"
} | {
"split": "train",
"index": 23170,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A$ be a set of positive integers satisfying the following : $a.)$ If $n \\in A$ , then $n \\le 2018$ . $b.)$ If $S \\subset A$ such that $|S|=3$ , then there exists $m,n \\in S$ such that $|n-m| \\ge \\sqrt{n}+\\sqrt{m}$ What is the maximum cardinality of $A$... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 23171,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Someone observed that $6! = 8 \\cdot 9 \\cdot 10$. Find the largest positive integer $n$ for which $n!$ can be expressed as the product of $n - 3$ consecutive positive integers.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 23172,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence: $2016$, $2017$, $1$, $-2016$, $-2017$, $…$, the characteristic of this sequence is that starting from the second term, each term is equal to the sum of the term before and after it. Find the sum of the first $2017$ terms of the sequence. Let's think step by st... | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"split": "train",
"index": 23173,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $\\sin (A-B)+ \\sin C= \\sqrt {2}\\sin A$.\n(I) Find the value of angle $B$;\n(II) If $b=2$, find the maximum value of $a^{2}+c^{2}$, and find the values of angles $A$ and $C$ whe... | math | {
"style": "rule",
"ground_truth": "8+4 \\sqrt {2}"
} | {
"split": "train",
"index": 23174,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T$ be a positive integer whose only digits are 0s and 1s. If $X = T \\div 24$ and $X$ is an integer, what is the smallest possible value of $X$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4625"
} | {
"split": "train",
"index": 23175,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a \\star b=ab-2$. Compute the remainder when $(((579 \\star 569) \\star 559) \\star \\cdots \\star 19) \\star 9$ is divided by 100. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 23176,
"solution": "Note that $$(10a+9) \\star (10b+9)=(100ab+90a+90b+81)-2 \\equiv 90(a+b)+79 \\pmod{100}$$ so throughout our process all numbers will end in 9, so we will just track the tens digit. Then the \"new operation\" is $$a \\dagger b \\equiv -(a+b)+7 \\bmod 10$$ where $a$ a... |
deepscaler | [
{
"role": "user",
"content": "Given the function $$f(x)=\\sin(x+ \\frac {\\pi}{6})+2\\sin^{2} \\frac {x}{2}$$.\n(1) Find the equation of the axis of symmetry and the coordinates of the center of symmetry for the function $f(x)$.\n(2) Determine the intervals of monotonicity for the function $f(x)$.\n(3) In t... | math | {
"style": "rule",
"ground_truth": "\\frac {3}{2}"
} | {
"split": "train",
"index": 23177,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the product of the digits in the base 8 representation of $9876_{10}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 23178,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let\n\\[x^5 - x^2 - x - 1 = p_1(x) p_2(x) \\dotsm p_k(x),\\]where each non-constant polynomial $p_i(x)$ is monic with integer coefficients, and cannot be factored further over the integers. Compute $p_1(2) + p_2(2) + \\dots + p_k(2).$ Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 23179,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Bonnie constructs a rectangular prism frame using 12 pieces of wire, each 8 inches long. Meanwhile, Roark uses 2-inch-long pieces of wire to construct a series of unit rectangular prism frames that are not connected to each other. The total volume of Roark's prisms is equal to ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"split": "train",
"index": 23180,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( a \\leq b < c \\) be the side lengths of a right triangle. Find the maximum constant \\( M \\) such that \\( \\frac{1}{a} + \\frac{1}{b} + \\frac{1}{c} \\geq \\frac{M}{a+b+c} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5 + 3 \\sqrt{2}"
} | {
"split": "train",
"index": 23181,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, each of the two circles has center \\(O\\). Also, \\(O P: P Q = 1:2\\). If the radius of the larger circle is 9, what is the area of the shaded region? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72 \\pi"
} | {
"split": "train",
"index": 23182,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x)=x^{3}+3x^{2}+6x$, $f(a)=1$, $f(b)=-9$, the value of $a+b$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 23183,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The reform pilot of basic discipline enrollment, also known as the Strong Foundation Plan, is an enrollment reform project initiated by the Ministry of Education, mainly to select and cultivate students who are willing to serve the country's major strategic needs and have excel... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"split": "train",
"index": 23184,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Karen drove continuously from 9:40 a.m. until 1:20 p.m. of the same day and covered a distance of 165 miles. What was her average speed in miles per hour? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 23185,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For real numbers \\( x \\) and \\( y \\) such that \\( x + y = 1 \\), determine the maximum value of the expression \\( A(x, y) = x^4 y + x y^4 + x^3 y + x y^3 + x^2 y + x y^2 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{16}"
} | {
"split": "train",
"index": 23186,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbb{R}$ denote the set of real numbers. Find all functions $f:\\mathbb{R}\\rightarrow\\mathbb{R}$ such that\n\\[f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y)\\]\nfor all $x,y\\in\\mathbb{R}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "f(x)= 0,f(x)= 2-x, f(x)=-x"
} | {
"split": "train",
"index": 23187,
"solution": "\nTo find all functions \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) satisfying \n\n\\[\nf(x^2) + f(xy) = f(x)f(y) + yf(x) + xf(x+y)\n\\]\n\nfor all \\( x, y \\in \\mathbb{R} \\), we will proceed by considering special cases and functional forms.\n\n### Step 1: ... |
deepscaler | [
{
"role": "user",
"content": "Determine the greatest positive integer $ n$ such that in three-dimensional space, there exist n points $ P_{1},P_{2},\\cdots,P_{n},$ among $ n$ points no three points are collinear, and for arbitary $ 1\\leq i < j < k\\leq n$, $ P_{i}P_{j}P_{k}$ isn't obtuse triangle. Let's th... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 23188,
"solution": "\nTo determine the greatest positive integer \\( n \\) such that in three-dimensional space, there exist \\( n \\) points \\( P_{1}, P_{2}, \\cdots, P_{n} \\) where no three points are collinear and for any \\( 1 \\leq i < j < k \\leq n \\), the triangle \\( P_{i}P... |
deepscaler | [
{
"role": "user",
"content": "Sixty students went on a trip to the zoo. Upon returning to school, it turned out that 55 of them forgot gloves at the zoo, 52 forgot scarves, and 50 managed to forget hats. Find the smallest number of the most scatterbrained students - those who lost all three items. Let's thi... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 23189,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the polynomial $x^2 - kx + 24$ has only positive integer roots, find the average of all distinct possibilities for $k$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 23190,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the smallest positive integer $n$, different from 2004, such that there exists a polynomial $f(x)$ with integer coefficients for which the equation $f(x) = 2004$ has at least one integer solution and the equation $f(x) = n$ has at least 2004 different integer solution... | math | {
"style": "rule",
"ground_truth": "(1002!)^2 + 2004"
} | {
"split": "train",
"index": 23191,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=-\\frac{1}{2}x^{2}+x$ with a domain that contains an interval $[m,n]$, and its range on this interval is $[3m,3n]$. Find the value of $m+n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 23192,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A lot of snow has fallen, and the kids decided to make snowmen. They rolled 99 snowballs with masses of 1 kg, 2 kg, 3 kg, ..., up to 99 kg. A snowman consists of three snowballs stacked on top of each other, and one snowball can be placed on another if and only if the mass of t... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 23193,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The solutions to $4x^2 + 3 = 3x - 9$ can be written in the form $x = a \\pm b i,$ where $a$ and $b$ are real numbers. What is $a + b^2$? Express your answer as a fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{207}{64}"
} | {
"split": "train",
"index": 23194,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parabola $C: y^2 = 16x$ with the focus $F$, and the line $l: x = -1$, if a point $A$ lies on $l$ and the line segment $AF$ intersects the parabola $C$ at point $B$ such that $\\overrightarrow{FA} = 5\\overrightarrow{FB}$, then find the length of $|AB|$. Let's think st... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 23195,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Katie and Allie are playing a game. Katie rolls two fair six-sided dice and Allie flips two fair two-sided coins. Katie’s score is equal to the sum of the numbers on the top of the dice. Allie’s score is the product of the values of two coins, where heads is worth $4$ and tai... | math | {
"style": "rule",
"ground_truth": "25/72"
} | {
"split": "train",
"index": 23196,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square and a regular octagon have equal perimeters. If the area of the square is 16, what is the area of the octagon?\nA) $8 + 4\\sqrt{2}$\nB) $4 + 2\\sqrt{2}$\nC) $16 + 8\\sqrt{2}$\nD) $4\\sqrt{2}$\nE) $8\\sqrt{2}$ Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "8 + 4\\sqrt{2}"
} | {
"split": "train",
"index": 23197,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Diana and Apollo each roll a standard die obtaining a number at random from $1$ to $6$. What is the probability that Diana's number is larger than Apollo's number? 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": 23198,
"solution": "To find the probability that Diana's number is larger than Apollo's number, we consider all possible outcomes when both Diana and Apollo roll a standard six-sided die.\n\n1. **List the possible outcomes for each roll of Apollo:**\n - If Apollo rolls a 1, Diana ca... |
deepscaler | [
{
"role": "user",
"content": "Emily and John each solved three-quarters of the homework problems individually and the remaining one-quarter together. Emily correctly answered 70% of the problems she solved alone, achieving an overall accuracy of 76% on her homework. John had an 85% success rate with the pro... | math | {
"style": "rule",
"ground_truth": "87.25\\%"
} | {
"split": "train",
"index": 23199,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many positive integers $n \\leq 100$ is it true that $10 n$ has exactly three times as many positive divisors as $n$ has? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 23200,
"solution": "Let $n=2^{a} 5^{b} c$, where $2,5 \\nmid c$. Then, the ratio of the number of divisors of $10 n$ to the number of divisors of $n$ is $\\frac{a+2}{a+1} \\frac{b+2}{b+1}=3$. Solving for $b$, we find that $b=\\frac{1-a}{2 a+1}$. This forces $(a, b)=(0,1),(1,0)$. There... |
deepscaler | [
{
"role": "user",
"content": "The lengths of the three sides of a triangle are 11, 15, and \\( k \\) \\((k \\in Z^{+})\\). The number of values of \\( k \\) that make the triangle an obtuse triangle is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 23201,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the total number of digits used when the first 2500 positive even integers are written? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9449"
} | {
"split": "train",
"index": 23202,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sin\\alpha + \\cos\\alpha = \\frac{1}{5}$, and $0 \\leq \\alpha < \\pi$, find the value of $\\tan\\alpha$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "- \\frac {4}{3}"
} | {
"split": "train",
"index": 23203,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The first term of a sequence is 934. Each subsequent term is equal to the sum of the digits of the previous term multiplied by 13. Find the 2019th term of the sequence. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "130"
} | {
"split": "train",
"index": 23204,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a square with a side length of $1$, if two points are randomly selected with equal probability from the center and the vertices of the square, what is the probability that the distance between these two points is $\\frac{\\sqrt{2}}{2}$? Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"split": "train",
"index": 23205,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct triangles can be formed by selecting three vertices from a set of points arranged in a 2×4 grid (2 rows and 4 columns)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 23206,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The tangent value of the angle between the slant height and the base is when the lateral area of the cone with volume $\\frac{\\pi}{6}$ is minimum. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 23207,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x_{1}=y_{1}=x_{2}=y_{2}=1$, then for $n \\geq 3$ let $x_{n}=x_{n-1} y_{n-2}+x_{n-2} y_{n-1}$ and $y_{n}=y_{n-1} y_{n-2}- x_{n-1} x_{n-2}$. What are the last two digits of $\\left|x_{2012}\\right|$ ? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 23208,
"solution": "Let $z_{n}=y_{n}+x_{n} i$. Then the recursion implies that: $$\\begin{aligned} & z_{1}=z_{2}=1+i \\\\ & z_{n}=z_{n-1} z_{n-2} \\end{aligned}$$ This implies that $$z_{n}=\\left(z_{1}\\right)^{F_{n}}$$ where $F_{n}$ is the $n^{\\text {th }}$ Fibonacci number $\\left(... |
deepscaler | [
{
"role": "user",
"content": "Calculate the arc lengths of curves given by the equations in polar coordinates.\n\n$$\n\\rho = 3 \\varphi, \\quad 0 \\leq \\varphi \\leq \\frac{4}{3}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10}{3} + \\frac{3}{2} \\ln 3"
} | {
"split": "train",
"index": 23209,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest integer value of $x$ for which $5-4x>17$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 23210,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a moving large circle $\\odot O$ tangent externally to a fixed small circle $\\odot O_{1}$ with radius 3 at point $P$, $AB$ is the common external tangent of the two circles with $A$ and $B$ as the points of tangency. A line $l$ parallel to $AB$ is tangent to $\\odot O_{1... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 23211,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a city, there are 10,000 bicycles with all possible numbers from 1 to 10,000. What is the probability that the number of the first bicycle encountered does not contain the digit 8? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.6561"
} | {
"split": "train",
"index": 23212,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "David is taking a true/false exam with $9$ questions. Unfortunately, he doesn’t know the answer to any of the questions, but he does know that exactly $5$ of the answers are True. In accordance with this, David guesses the answers to all $9$ questions, making sure that ex... | math | {
"style": "rule",
"ground_truth": "9/14"
} | {
"split": "train",
"index": 23213,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a hexagon \\( A B C D E F \\) with an area of 60 that is inscribed in a circle \\( \\odot O \\), where \\( AB = BC, CD = DE, \\) and \\( EF = AF \\). What is the area of \\( \\triangle B D F \\)? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 23214,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ten digit positive integers with distinct digits are multiples of $11111$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3456"
} | {
"split": "train",
"index": 23215,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$n$ coins are simultaneously flipped. The probability that two or fewer of them show tails is $\\frac{1}{4}$. Find $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "n = 5"
} | {
"split": "train",
"index": 23216,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify first, then evaluate: $\\left(\\frac{a+1}{2a-2}-\\frac{5}{2{a}^{2}-2}-\\frac{a+3}{2a+2}\\right)÷\\frac{{a}^{2}}{{a}^{2}-1}$, where the value of $a$ is chosen as an appropriate integer from the solution set of the inequality system $\\left\\{\\begin{array}{l}{a-\\sqrt{5... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{8}"
} | {
"split": "train",
"index": 23217,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $XYZ$, where $XY = 8$, $YZ = 12$, and $XZ = 14$. Points $D$ and $E$ are selected on $\\overline{XY}$ and $\\overline{XZ}$ respectively, such that $XD = 3$ and $XE = 9$. Calculate the area of triangle $XDE$. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "\\frac{405 \\sqrt{17}}{112}"
} | {
"split": "train",
"index": 23218,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a function defined on $\\mathbb{R}$, $f(x)=A\\sin (\\omega x+\\varphi)$ where $A > 0$, $\\omega > 0$, and $|\\varphi| \\leqslant \\frac {\\pi}{2}$, the minimum value of the function is $-2$, and the distance between two adjacent axes of symmetry is $\\frac {\\pi}{2}$. Aft... | math | {
"style": "rule",
"ground_truth": "- \\frac { \\sqrt {741}}{32}- \\frac {3}{32}"
} | {
"split": "train",
"index": 23219,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways are there to win tic-tac-toe in $\\mathbb{R}^{n}$? (That is, how many lines pass through three of the lattice points $(a_{1}, \\ldots, a_{n})$ in $\\mathbb{R}^{n}$ with each coordinate $a_{i}$ in $\\{1,2,3\\}$? Express your answer in terms of $n$. Let's think step... | math | {
"style": "rule",
"ground_truth": "\\left(5^{n}-3^{n}\\right) / 2"
} | {
"split": "train",
"index": 23220,
"solution": "A line consists of three points. Each coordinate can do one of three things passing from the first point to the last point: increase by 1 each time, stay the same, or decrease by 1 each time. There are three ways to stay the same (three coordinates), one way to inc... |
deepscaler | [
{
"role": "user",
"content": "For positive real numbers $a,$ $b,$ $c,$ and $d,$ compute the maximum value of\n\\[\\frac{abcd(a + b + c + d)}{(a + b)^2 (c + d)^2}.\\] 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": 23221,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Of the thirteen members of the volunteer group, Hannah selects herself, Tom Morris, Jerry Hsu, Thelma Paterson, and Louise Bueller to teach the September classes. When she is done, she decides that it's not necessary to balance the number of female and male teachers with the p... | math | {
"style": "rule",
"ground_truth": "1261"
} | {
"split": "train",
"index": 23222,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the quadrilateral pyramid \\(P-ABCD\\), given that \\(AB\\) is parallel to \\(CD\\), \\(AB\\) is perpendicular to \\(AD\\), \\(AB=4\\), \\(AD=2\\sqrt{2}\\), \\(CD=2\\), and \\(PA\\) is perpendicular to the plane \\(ABCD\\), with \\(PA=4\\). Let \\(Q\\) be a point on line seg... | math | {
"style": "rule",
"ground_truth": "7/12"
} | {
"split": "train",
"index": 23223,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence starts with 800,000; each subsequent term is obtained by dividing the previous term by 3. 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": "800000"
} | {
"split": "train",
"index": 23224,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many triangles are there with all sides being integers and the longest side being 11? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 23225,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function \\( f \\) satisfies the equation \\((x-1) f(x) + f\\left(\\frac{1}{x}\\right) = \\frac{1}{x-1}\\) for each value of \\( x \\) not equal to 0 and 1. Find \\( f\\left(\\frac{2016}{2017}\\right) \\). Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "2017"
} | {
"split": "train",
"index": 23226,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the $y$-intercepts of the following system of equations:\n1. $2x - 3y = 6$\n2. $x + 4y = -8$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 23227,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x_1$ , $x_2$ , and $x_3$ be the roots of the polynomial $x^3+3x+1$ . There are relatively prime positive integers $m$ and $n$ such that $\\tfrac{m}{n}=\\tfrac{x_1^2}{(5x_2+1)(5x_3+1)}+\\tfrac{x_2^2}{(5x_1+1)(5x_3+1)}+\\tfrac{x_3^2}{(5x_1+1)(5x_2+1)}$ . Find $m... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 23228,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x$ is a real number and $\\lceil x \\rceil = 11,$ how many possible values are there for $\\lceil x^2 \\rceil$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 23229,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A small class of nine boys are to change their seating arrangement by drawing their new seat numbers from a box. After the seat change, what is the probability that there is only one pair of boys who have switched seats with each other and only three boys who have unchanged sea... | math | {
"style": "rule",
"ground_truth": "1/32"
} | {
"split": "train",
"index": 23230,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number $n$ is a prime number between 20 and 30. If you divide $n$ by 8, the remainder is 5. What is the value of $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 23231,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cube with side length $1$ is sliced by a plane that passes through two diagonally opposite vertices $A$ and $C$ and the midpoints $B$ and $D$ of two opposite edges not containing $A$ or $C$, as shown. What is the area of quadrilateral $ABCD$? Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{2}"
} | {
"split": "train",
"index": 23232,
"solution": "1. **Identify the Shape of Quadrilateral $ABCD$**:\n - The cube has side length $1$.\n - Vertices $A$ and $C$ are diagonally opposite in the cube, making $AC$ a space diagonal.\n - Points $B$ and $D$ are midpoints of two opposite edges not containing $A$ or $... |
deepscaler | [
{
"role": "user",
"content": "Six small circles, each of radius 4 units, are tangent to a large circle. Each small circle is also tangent to its two neighboring small circles. Additionally, all small circles are tangent to a horizontal line that bisects the large circle. What is the diameter of the large ci... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 23233,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From an external point \\(A\\), a tangent \\(AB\\) and a secant \\(ACD\\) are drawn to a circle. Find the area of triangle \\(CBD\\), given that the ratio \\(AC : AB = 2 : 3\\) and the area of triangle \\(ABC\\) is 20. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 23234,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": " $K, L, M$ and $N$ are points on sides $AB, BC, CD$ and $DA$ , respectively, of the unit square $ABCD$ such that $KM$ is parallel to $BC$ and $LN$ is parallel to $AB$ . The perimeter of triangle $KLB$ is equal to $1$ . What is the area of triangle $MND$ ? Let... | math | {
"style": "rule",
"ground_truth": "1/4"
} | {
"split": "train",
"index": 23235,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\frac{m}{n}=\\frac{4}{3}$ and $\\frac{r}{t}=\\frac{9}{14}$, the value of $\\frac{3mr-nt}{4nt-7mr}$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{11}{14}"
} | {
"split": "train",
"index": 23236,
"solution": "1. **Start by expressing $m$ and $n$ in terms of each other using the given ratio:**\n \\[\n \\frac{m}{n} = \\frac{4}{3} \\implies 3m = 4n \\implies m = \\frac{4n}{3}\n \\]\n\n2. **Similarly, express $r$ and $t$ in terms of each other using the given ratio:**... |
deepscaler | [
{
"role": "user",
"content": "In regular octagon $ABCDEFGH$, $M$ and $N$ are midpoints of $\\overline{BC}$ and $\\overline{FG}$ respectively. Compute $[ABMO]/[EDCMO]$. ($[ABCD]$ denotes the area of polygon $ABCD$.) [asy]\npair A,B,C,D,E,F,G,H;\nF=(0,0); E=(2,0); D=(2+sqrt(2),sqrt(2)); C=(2+sqrt(2),2+sqrt(2... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 23237,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many solutions does the equation $\\tan x = \\tan (\\tan x)$ have in the interval $0 \\le x \\le \\tan^{-1} 1000$? Assume $\\tan \\theta > \\theta$ for $0 < \\theta < \\frac{\\pi}{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "318"
} | {
"split": "train",
"index": 23238,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $-9, a_1, a_2, -1$ form an arithmetic sequence and $-9, b_1, b_2, b_3, -1$ form a geometric sequence, find the value of $b_2(a_2 - a_1)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"split": "train",
"index": 23239,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangle has integer length and width, and a perimeter of 120 units. Determine the number of square units in the greatest possible area, given that one of the dimensions must be a prime number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "899"
} | {
"split": "train",
"index": 23240,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $m$ and $n$ are positive integers that satisfy the equation $3m^{3}=5n^{5}$, what is the smallest possible value for $m+n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "720"
} | {
"split": "train",
"index": 23241,
"solution": "Since $3m^{3}$ is a multiple of 3, then $5n^{5}$ is a multiple of 3. Since 5 is not a multiple of 3 and 3 is a prime number, then $n^{5}$ is a multiple of 3. Since $n^{5}$ is a multiple of 3 and 3 is a prime number, then $n$ is a multiple of 3, which means that $5n... |
deepscaler | [
{
"role": "user",
"content": "What is the value of $(-1)^1+(-1)^2+\\cdots+(-1)^{2007}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 23242,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The surface area of a particular sphere is $324\\pi\\text{ cm}^2$. What is the volume, in cubic centimeters, of the sphere? 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": "972\\pi"
} | {
"split": "train",
"index": 23243,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area of the triangle bounded by the lines $y=x,$ $y=-x,$ and $y=6$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 23244,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)=x+3$ and $g(x)=3x+5$. Find $f(g(4))-g(f(4))$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 23245,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider an arithmetic sequence $\\{a_n\\}$ with the sum of its first $n$ terms denoted as $S_n$. Given that $a_1=9$, $a_2$ is an integer, and $S_n \\leq S_5$, find the sum of the first 9 terms of the sequence $\\{\\frac{1}{a_n a_{n+1}}\\}$. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{9}"
} | {
"split": "train",
"index": 23246,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $AC = \\sqrt{7}$, $BC = 2$, and $\\angle B = 60^\\circ$. What is the area of $\\triangle ABC$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 23247,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For the largest possible $n$, can you create two bi-infinite sequences $A$ and $B$ such that any segment of length $n$ from sequence $B$ is contained in $A$, where $A$ has a period of 1995, but $B$ does not have this property (it is either not periodic or has a different period... | math | {
"style": "rule",
"ground_truth": "1995"
} | {
"split": "train",
"index": 23248,
"solution": ""
} |
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