data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Given the ages of Daisy's four cousins are distinct single-digit positive integers, and the product of two of the ages is $24$ while the product of the other two ages is $35$, find the sum of the ages of Daisy's four cousins. Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 18149,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the market supply of light bulbs, products from Factory A account for 70%, while those from Factory B account for 30%. The pass rate for Factory A's products is 95%, and the pass rate for Factory B's products is 80%. What is the probability of purchasing a qualified light bu... | math | {
"style": "rule",
"ground_truth": "0.665"
} | {
"split": "train",
"index": 18150,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of the eight letters in \"GEOMETRY\" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word \"ANGLE\"? Express your answer as a common fraction. Let's think step... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 18151,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Gwen, Eli, and Kat take turns flipping a coin in their respective order. The first one to flip heads wins. What is the probability that Kat will win? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{7}"
} | {
"split": "train",
"index": 18152,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$ , $AB = 40$ , $BC = 60$ , and $CA = 50$ . The angle bisector of $\\angle A$ intersects the circumcircle of $\\triangle ABC$ at $A$ and $P$ . Find $BP$ .\n\n*Proposed by Eugene Chen* Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 18153,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)=x^{2}+a x+b$ and $g(x)=x^{2}+c x+d$ be two distinct real polynomials such that the $x$-coordinate of the vertex of $f$ is a root of $g$, the $x$-coordinate of the vertex of $g$ is a root of $f$ and both $f$ and $g$ have the same minimum value. If the graphs of the two... | math | {
"style": "rule",
"ground_truth": "-8048"
} | {
"split": "train",
"index": 18154,
"solution": "It is clear, by symmetry, that 2012 is the equidistant from the vertices of the two quadratics. Then it is clear that reflecting $f$ about the line $x=2012$ yields $g$ and vice versa. Thus the average of each pair of roots is 2012 . Thus the sum of the four roots o... |
deepscaler | [
{
"role": "user",
"content": "Given that EF = 40 units, FG = 30 units, and one diagonal EH = 50 units, calculate the perimeter of parallelogram EFGH. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "140"
} | {
"split": "train",
"index": 18155,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of\n$$\\frac{a}{b}+\\frac{b}{c}+\\frac{c}{d}+\\frac{d}{a}.$$\n[i]Israel[/i] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 18156,
"solution": "\nLet \\( a, b, c, \\) and \\( d \\) be positive real numbers such that \\((a+c)(b+d) = ac + bd\\). We are tasked with finding the smallest possible value of \n\\[\nS = \\frac{a}{b} + \\frac{b}{c} + \\frac{c}{d} + \\frac{d}{a}.\n\\]\n\nTo solve this problem, we sta... |
deepscaler | [
{
"role": "user",
"content": "Estimate the population of Nisos in the year 2050. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"split": "train",
"index": 18157,
"solution": "1. **Identify the growth pattern**: The problem states that the population of Nisos triples every 25 years. This means that if the population is $P$ in a certain year, it will be $3P$ 25 years later.\n\n2. **Calculate the population in 2050**: \n - The population... |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}=(1,-2)$ and $\\overrightarrow{b}=(3,4)$, find the projection of vector $\\overrightarrow{a}$ onto the direction of vector $\\overrightarrow{b}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 18158,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all triples of primes $(p,q,r)$ satisfying $3p^{4}-5q^{4}-4r^{2}=26$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(5, 3, 19)"
} | {
"split": "train",
"index": 18159,
"solution": "\nWe are tasked with finding all triples of primes \\((p, q, r)\\) that satisfy the equation:\n\n\\[\n3p^4 - 5q^4 - 4r^2 = 26.\n\\]\n\nWe begin by analyzing the equation with respect to the properties of prime numbers.\n\n1. **Testing Small Primes**:\n - Since \\... |
deepscaler | [
{
"role": "user",
"content": "A fair coin is flipped every second and the results are recorded with 1 meaning heads and 0 meaning tails. What is the probability that the sequence 10101 occurs before the first occurrence of the sequence 010101? Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "\\frac{21}{32}"
} | {
"split": "train",
"index": 18160,
"solution": "Call it a win if we reach 10101, a loss if we reach 010101. Let $x$ be the probability of winning if the first flip is a 1, let $y$ be the probability of winning if the first flip is a 0. Then the probability of winning is $(x+y) / 2$ since the first flip is 1 or 0... |
deepscaler | [
{
"role": "user",
"content": "A three-digit number is composed of three different non-zero digits in base ten. When divided by the sum of these three digits, the smallest quotient value is what? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10.5"
} | {
"split": "train",
"index": 18161,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five identical white pieces and ten identical black pieces are arranged in a row. It is required that the right neighbor of each white piece must be a black piece. The number of different arrangements is . Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "252"
} | {
"split": "train",
"index": 18162,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rebecca has twenty-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 resis... | math | {
"style": "rule",
"ground_truth": "1015080877"
} | {
"split": "train",
"index": 18163,
"solution": "This is the same problem as in OEIS A153588. It is helpful to see (or guess) that neither the numerator or the denominator of the final resistance exceed the $(n+1)$-th Fibonacci number, which in this case is $F_{25}=75025$, using concepts on the line of continued ... |
deepscaler | [
{
"role": "user",
"content": "Simplify $(r^2 + 3r - 2) - (r^2 + 7r - 5)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4r+3"
} | {
"split": "train",
"index": 18164,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $x$ in the following expressions: \n(1) $8x^3 = 27$; (2) $(x-2)^2 = 3$. 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": 18165,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x_1$, $x_2$, ..., $x_7$ be natural numbers, and $x_1 < x_2 < x_3 < \\ldots < x_6 < x_7$, also $x_1 + x_2 + \\ldots + x_7 = 159$, then the maximum value of $x_1 + x_2 + x_3$ is. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "61"
} | {
"split": "train",
"index": 18166,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Isabella's house has $3$ bedrooms. Each bedroom is $12$ feet long, $10$ feet wide, and $8$ feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy $60$ square feet in each bedroom. How many square feet of walls must ... | math | {
"style": "rule",
"ground_truth": "876"
} | {
"split": "train",
"index": 18167,
"solution": "1. **Calculate the area of the walls in one bedroom**: \n Each bedroom has four walls, two pairs of opposite walls. Each pair consists of:\n - Two walls of dimensions $12$ feet (length) by $8$ feet (height).\n - Two walls of dimensions $10$ feet (width) by $8... |
deepscaler | [
{
"role": "user",
"content": "Consider the lines \\begin{align*}\ny&=3x+5 \\\\ 2y&=4x+5 \\\\ 3y&=9x-2 \\\\ 2y&=x-3 \\\\ 4y&=x-5.\n\\end{align*}Let's say that a pair of lines is $\\emph{good}$ if the two lines are either parallel or perpendicular to each other. Among all the pairs of lines shown, how many p... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 18168,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Little kids were eating candies. Each ate 11 candies less than the rest combined but still more than one candy. How many candies were eaten in total? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 18169,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Car $A$ departs from Station $J$ towards Station $Y$, while cars $B$ and $C$ depart from Station $Y$ towards Station $J$ simultaneously, and move in opposite directions towards car $A$. Car $A$ meets car $B$ first, then 20 minutes later it meets car $C$. Given the speeds of car... | math | {
"style": "rule",
"ground_truth": "425"
} | {
"split": "train",
"index": 18170,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number $r$ can be expressed as a four-place decimal $0.abcd,$ where $a, b, c,$ and $d$ represent digits, any of which could be zero. It is desired to approximate $r$ by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to $r... | math | {
"style": "rule",
"ground_truth": "417"
} | {
"split": "train",
"index": 18171,
"solution": "The nearest fractions to $\\frac 27$ with numerator $1$ are $\\frac 13, \\frac 14$; and with numerator $2$ are $\\frac 26, \\frac 28 = \\frac 13, \\frac 14$ anyway. For $\\frac 27$ to be the best approximation for $r$, the decimal must be closer to $\\frac 27 \\app... |
deepscaler | [
{
"role": "user",
"content": "Find the sum of $327_8$ and $73_8$ in base $8$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "422_8"
} | {
"split": "train",
"index": 18172,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that in a class test, $15\\%$ of the students scored $60$ points, $50\\%$ scored $75$ points, $20\\%$ scored $85$ points, and the rest scored $95$ points, calculate the difference between the mean and median score of the students' scores on this test. Let's think step by ... | math | {
"style": "rule",
"ground_truth": "2.75"
} | {
"split": "train",
"index": 18173,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $656_7=3ab_{10}$, where $a$ and $b$ represent base-10 digits. Find $\\frac{a\\cdot b}{15}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 18174,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many different ways can a chess king move from square $e1$ to square $h5$, if it is only allowed to move one square to the right, upward, or diagonally right-upward? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "129"
} | {
"split": "train",
"index": 18175,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be an odd integer with exactly 11 positive divisors. Find the number of positive divisors of $8n^3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "124"
} | {
"split": "train",
"index": 18176,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On an algebra test, there were $5x$ problems. Lucky Lacy missed $x$ of them. What percent of the problems did she get correct? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80 \\%"
} | {
"split": "train",
"index": 18177,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Trapezoid $ABCD$ has $AD||BC$, $BD = 1$, $\\angle DBA = 23^{\\circ}$, and $\\angle BDC = 46^{\\circ}$. The ratio $BC: AD$ is $9: 5$. What is $CD$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"split": "train",
"index": 18178,
"solution": "\n#### Step 1: Extend Lines and Identify Angles\nExtend $\\overline{AB}$ and $\\overline{DC}$ to meet at point $E$. Since $\\angle DBA = 23^\\circ$ and $\\angle BDC = 46^\\circ$, we can calculate $\\angle EDB$ and $\\angle DBE$:\n- $\\angle EDB = 180^\\circ - \\ang... |
deepscaler | [
{
"role": "user",
"content": "Observe the table below: \n1, \n2,3 \n4,5,6,7 \n8,9,10,11,12,13,14,15, \n... \nQuestion: (1) What is the last number of the $n$th row in this table? \n(2) What is the sum of all numbers in the $n$th row? \n(3) Which row and position is the number 2008? Let's think step ... | math | {
"style": "rule",
"ground_truth": "985"
} | {
"split": "train",
"index": 18179,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The perimeter of a square with side length $x$ units is equal to the circumference of a circle with radius 2 units. What is the value of $x$? Express your answer as a decimal to the nearest hundredth. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "3.14"
} | {
"split": "train",
"index": 18180,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A public bus departs on schedule at 6:30, 7:00, and 7:30. Student Xiao Ming arrives at the station between 6:50 and 7:30 to catch the bus, and his time of arrival is random. The probability that his waiting time is no more than 10 minutes is ______. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 18181,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the sum $$\\lceil\\sqrt{5}\\rceil + \\lceil\\sqrt{6}\\rceil + \\lceil\\sqrt{7}\\rceil + \\cdots + \\lceil\\sqrt{39}\\rceil$$\nNote: For a real number $x,$ $\\lceil x \\rceil$ denotes the smallest integer that is greater than or equal to $x.$ Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "175"
} | {
"split": "train",
"index": 18182,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular triangular pyramid \\(SABC\\) is given, with the edge of its base equal to 1. Medians of the lateral faces are drawn from the vertices \\(A\\) and \\(B\\) of the base \\(ABC\\), and these medians do not intersect. It is known that the edges of a certain cube lie on th... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{2}"
} | {
"split": "train",
"index": 18183,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of every even positive integer less than 233 not divisible by 10. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10812"
} | {
"split": "train",
"index": 18184,
"solution": "We find the sum of all positive even integers less than 233 and then subtract all the positive integers less than 233 that are divisible by 10. $2 + 4 + \\ldots + 232 = 2(1 + 2 + \\ldots + 116) = 116 \\cdot 117 = 13572$. The sum of all positive integers less than 2... |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of the polynomial f(x) = 7x^7 + 6x^6 + 5x^5 + 4x^4 + 3x^3 + 2x^2 + x using the Qin Jiushao algorithm when x = 3. Find the value of V₄. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "789"
} | {
"split": "train",
"index": 18185,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least possible value of $(xy-1)^2+(x+y)^2$ for real numbers $x$ and $y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 18186,
"solution": "1. **Expand the given expression**:\n \\[\n (xy-1)^2 + (x+y)^2 = (x^2y^2 - 2xy + 1) + (x^2 + 2xy + y^2)\n \\]\n Combine like terms:\n \\[\n x^2y^2 + x^2 + 2xy + y^2 - 2xy + 1 = x^2y^2 + x^2 + y^2 + 1\n \\]\n\n2. **Factor the expression**:\n Notice t... |
deepscaler | [
{
"role": "user",
"content": "What fraction of the large $12$ by $18$ rectangular region is shaded? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"split": "train",
"index": 18187,
"solution": "1. **Identify the total area of the large rectangle**: \n The large rectangle has dimensions $12$ by $18$. Therefore, its area is calculated as:\n \\[\n \\text{Area} = 12 \\times 18 = 216 \\text{ square units}\n \\]\n\n2. **Determine the area of the quarter... |
deepscaler | [
{
"role": "user",
"content": "A cone is inscribed in a regular quadrilateral pyramid. Find the ratio of the total surface area of the cone to the lateral surface area of the cone, given that the side length of the pyramid's base is 4 and the angle between the pyramid's height and the plane of its lateral fa... | math | {
"style": "rule",
"ground_truth": "\\frac{3 + \\sqrt{3}}{3}"
} | {
"split": "train",
"index": 18188,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $45_{10} + 28_{10}$ in base 4? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1021_4"
} | {
"split": "train",
"index": 18189,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the matrix $\\mathbf{M}$ that performs the transformation which sends square $ABCD$ to square $A'B'C'D'$? (In particular, $A$ goes to $A',$ and so on.)\n\n[asy]\n\nsize(200);\nimport graph;\n\npair Z=(0,0), A=(2,3), B=(-3,2), C=(-4,1), D=(-1,-4);\n\nLabel f; \nf.p=font... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 1 & -1 \\\\ 1 & \\phantom -1 \\end{pmatrix}"
} | {
"split": "train",
"index": 18190,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice has six magical pies in her pocket - two that make you grow and the rest make you shrink. When Alice met Mary Ann, she blindly took three pies out of her pocket and gave them to Mary. Find the probability that one of the girls has no growth pies. Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "0.4"
} | {
"split": "train",
"index": 18191,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a/b$ be the probability that a randomly chosen positive divisor of $12^{2007}$ is also a divisor of $12^{2000}$ , where $a$ and $b$ are relatively prime positive integers. Find the remainder when $a+b$ is divided by $2007$ . Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "79"
} | {
"split": "train",
"index": 18192,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Bill buys a stock for $100. On the first day, the stock decreases by $25\\%$, on the second day it increases by $35\\%$ from its value at the end of the first day, and on the third day, it decreases again by $15\\%$. What is the overall percentage change in the stock's value ov... | math | {
"style": "rule",
"ground_truth": "-13.9375\\%"
} | {
"split": "train",
"index": 18193,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that four A's, four B's, four C's, and four D's are to be placed in a 4 × 4 grid so that each row and column contains one of each letter, and A is placed in the upper right corner, calculate the number of possible arrangements. Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "216"
} | {
"split": "train",
"index": 18194,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that all edges of a tetrahedron have a length of $\\sqrt{2}$, and the four vertices are on the same sphere, calculate the surface area of this sphere. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 \\pi"
} | {
"split": "train",
"index": 18195,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A state requires that all boat licenses consist of the letter A or M followed by any five digits. What is the number of groups of letters and numbers available for boat licenses? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "200000"
} | {
"split": "train",
"index": 18196,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given positive real numbers $a$ and $b$ satisfying $a+b=2$, the minimum value of $\\dfrac{1}{a}+\\dfrac{2}{b}$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{3+2 \\sqrt{2}}{2}"
} | {
"split": "train",
"index": 18197,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive value of $x$ such that $x + 3956$ results in a palindrome? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 18198,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(1) Simplify: $\\dfrac{\\sin(\\pi -\\alpha)\\cos(\\pi +\\alpha)\\sin(\\dfrac{\\pi}{2}+\\alpha)}{\\sin(-\\alpha)\\sin(\\dfrac{3\\pi}{2}+\\alpha)}$.\n\n(2) Given $\\alpha \\in (\\dfrac{\\pi}{2}, \\pi)$, and $\\sin(\\pi -\\alpha) + \\cos \\alpha = \\dfrac{7}{13}$, find $\\tan \\al... | math | {
"style": "rule",
"ground_truth": "-\\dfrac{12}{5}"
} | {
"split": "train",
"index": 18199,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\triangle ABC$ have $\\angle ABC=67^{\\circ}$ . Point $X$ is chosen such that $AB = XC$ , $\\angle{XAC}=32^\\circ$ , and $\\angle{XCA}=35^\\circ$ . Compute $\\angle{BAC}$ in degrees.\n\n*Proposed by Raina Yang* Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 18200,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate: $\\sqrt{25}-\\left(-1\\right)^{2}+|2-\\sqrt{5}|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2+\\sqrt{5}"
} | {
"split": "train",
"index": 18201,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all polynomials of the form $$P_n(x)=n!x^n+a_{n-1}x^{n-1}+\\dots+a_1x+(-1)^n(n+1)$$ with integer coefficients, having $n$ real roots $x_1,\\dots,x_n$ satisfying $k \\leq x_k \\leq k+1$ for $k=1, \\dots,n$. Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "P_1(x) = x - 2"
} | {
"split": "train",
"index": 18202,
"solution": "\nTo find the polynomials of the form \n\\[\nP_n(x) = n!x^n + a_{n-1}x^{n-1} + \\cdots + a_1x + (-1)^n(n+1)\n\\]\nwith integer coefficients, having \\( n \\) real roots \\( x_1, x_2, \\ldots, x_n \\) satisfying \\( k \\leq x_k \\leq k+1 \\) for \\( k = 1, \\ldots, ... |
deepscaler | [
{
"role": "user",
"content": "If the radius of a circle is increased by $1$ unit, the ratio of the new circumference to the new diameter is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\pi"
} | {
"split": "train",
"index": 18203,
"solution": "1. **Define the original and new dimensions of the circle:**\n - Let the original radius of the circle be $r$.\n - The original diameter is therefore $2r$.\n - The original circumference is $2\\pi r$.\n\n2. **Calculate the new dimensions after increasing the ... |
deepscaler | [
{
"role": "user",
"content": "János, a secretary of a rural cooperative, travels to Budapest weekly. His wife leaves home at 4 o'clock to meet him at the station, arriving at exactly the same time as the train. They are home by 5 o'clock. One day, the train arrived earlier, unbeknownst to his wife, so she e... | math | {
"style": "rule",
"ground_truth": "3.5"
} | {
"split": "train",
"index": 18204,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $a=1$, $B=45^{\\circ}$, $S_{\\triangle ABC}=2$, calculate the diameter of the circumcircle of $\\triangle ABC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5\\sqrt{2}"
} | {
"split": "train",
"index": 18205,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "As a result of measuring the four sides and one of the diagonals of a certain quadrilateral, the following numbers were obtained: $1 ; 2 ; 2.8 ; 5 ; 7.5$. What is the length of the measured diagonal? Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "2.8"
} | {
"split": "train",
"index": 18206,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( k_{1} \\) and \\( k_{2} \\) be two circles with the same center, with \\( k_{2} \\) inside \\( k_{1} \\). Let \\( A \\) be a point on \\( k_{1} \\) and \\( B \\) a point on \\( k_{2} \\) such that \\( AB \\) is tangent to \\( k_{2} \\). Let \\( C \\) be the second inter... | math | {
"style": "rule",
"ground_truth": "5/3"
} | {
"split": "train",
"index": 18207,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Exactly at noon, Anna Kuzminichna looked out the window and saw Klava, the rural store salesperson, going on break. At two minutes past noon, Anna Kuzminichna looked out the window again, and there was still no one in front of the closed store. Klava was gone for exactly 10 min... | math | {
"style": "rule",
"ground_truth": "0.75"
} | {
"split": "train",
"index": 18208,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If Alice is walking north at a speed of 4 miles per hour and Claire is walking south at a speed of 6 miles per hour, determine the time it will take for Claire to meet Alice, given that Claire is currently 5 miles north of Alice. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 18209,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Phil rolls 6 fair 6-sided dice. What is the probability that at least two dice show the same number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{319}{324}"
} | {
"split": "train",
"index": 18210,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two parabolas are the graphs of the equations $y=2x^2-7x+1$ and $y=8x^2+5x+1$. Give all points where they intersect. List the points in order of increasing $x$-coordinate, separated by semicolons. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(0, 1)"
} | {
"split": "train",
"index": 18211,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of ways that 2010 can be written as a sum of one or more positive integers in non-decreasing order such that the difference between the last term and the first term is at most 1. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2010"
} | {
"split": "train",
"index": 18212,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integer side lengths are possible to complete a triangle in which the other sides measure 6 units and 3 units? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 18213,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When a right triangle is rotated about one leg, the volume of the cone produced is $800\\pi \\;\\textrm{ cm}^3$. When the triangle is rotated about the other leg, the volume of the cone produced is $1920\\pi \\;\\textrm{ cm}^3$. What is the length (in cm) of the hypotenuse of t... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 18214,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the minimum of the following function defined in the interval $45^{\\circ}<x<90^{\\circ}$:\n\n$$\ny=\\tan x+\\frac{\\tan x}{\\sin \\left(2 x-90^{\\circ}\\right)}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3}"
} | {
"split": "train",
"index": 18215,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be the plane passing through the origin with normal vector $\\begin{pmatrix} 1 \\\\ 1 \\\\ -1 \\end{pmatrix}.$ Find the matrix $\\mathbf{R}$ such that for any vector $\\mathbf{v},$ $\\mathbf{R} \\mathbf{v}$ is the reflection of $\\mathbf{v}$ through plane $P.$ Let's th... | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} \\frac{1}{3} & -\\frac{2}{3} & \\frac{2}{3} \\\\ -\\frac{2}{3} & \\frac{1}{3} & \\frac{2}{3} \\\\ \\frac{2}{3} & \\frac{2}{3} & \\frac{1}{3} \\end{pmatrix}"
} | {
"split": "train",
"index": 18216,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $C:\\frac{{x}^{2}}{{a}^{2}}-\\frac{{y}^{2}}{{b}^{2}}=1(a > 0,b > 0)$, the line $l$ passing through point $P(3,6)$ intersects $C$ at points $A$ and $B$, and the midpoint of $AB$ is $N(12,15)$. Determine the eccentricity of the hyperbola $C$. Let's think step ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 18217,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many multiples of 15 are there between 40 and 240? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 18218,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$, $b$, $c$, $d$, and $e$ be positive integers with $a+b+c+d+e=2010$ and let $M$ be the largest of the sum $a+b$, $b+c$, $c+d$ and $d+e$. What is the smallest possible value of $M$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "671"
} | {
"split": "train",
"index": 18219,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Euler's formula states that for a convex polyhedron with $V$ vertices, $E$ edges, and $F$ faces, $V-E+F=2$. A particular convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its $V$ vertices, $T$ triangular faces and $P$ pentagonal faces ... | math | {
"style": "rule",
"ground_truth": "250"
} | {
"split": "train",
"index": 18220,
"solution": "We know that $V-E = -30 \\implies V = E-30$ based off the problem condition. Furthermore, if we draw out a few pentagons as well as triangles on each of side of the pentagons, it's clear that each vertex has 4 edges connected to it, with two triangles and two penta... |
deepscaler | [
{
"role": "user",
"content": "Two students were asked to add two positive integers. Alice subtracted the two numbers by mistake and obtained 3. Bob mistakenly multiplied the same two integers and got 63. What was the correct sum of the two integers? Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 18221,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given circles $P$, $Q$, and $R$ each have radius 2, circle $P$ and $Q$ are tangent to each other, and circle $R$ is tangent to the midpoint of $\\overline{PQ}$. Calculate the area inside circle $R$ but outside circle $P$ and circle $Q$. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "2\\pi"
} | {
"split": "train",
"index": 18222,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 6 boxes, each containing a key that cannot be used interchangeably. If one key is placed in each box and all the boxes are locked, and it is required that after breaking open one box, the remaining 5 boxes can still be opened with the keys, then the number of ways to ... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 18223,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the graph of $$f(x)=-\\cos^{2} \\frac {ω}{2}x+ \\frac { \\sqrt {3}}{2}\\sinωx$$ has a distance of $$\\frac {π}{2}(ω＞0)$$ between two adjacent axes of symmetry. \n(Ⅰ) Find the intervals where $f(x)$ is strictly decreasing; \n(Ⅱ) In triangle ABC, a, b, and c are the ... | math | {
"style": "rule",
"ground_truth": "\\sqrt {13}"
} | {
"split": "train",
"index": 18224,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two circles \\( C_{1} \\) and \\( C_{2} \\) have their centers at the point \\( (3, 4) \\) and touch a third circle, \\( C_{3} \\). The center of \\( C_{3} \\) is at the point \\( (0, 0) \\) and its radius is 2. Find the sum of the radii of the two circles \\( C_{1} \\) and \\(... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 18225,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( S_{n} = 1 + \\frac{1}{2} + \\cdots + \\frac{1}{n} \\) for \\( n = 1, 2, \\cdots \\). Find the smallest positive integer \\( n \\) such that \\( S_{n} > 10 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12367"
} | {
"split": "train",
"index": 18226,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(For science students) In the expansion of $(x^2 - 3x + 2)^4$, the coefficient of the $x^2$ term is __________ (Answer with a number). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "248"
} | {
"split": "train",
"index": 18227,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the integers \\( 1, 2, 3, \\ldots, 40 \\), find the greatest possible sum of the positive differences between the integers in twenty pairs, where the positive difference is either 1 or 3. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "58"
} | {
"split": "train",
"index": 18228,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that \\(p\\) and \\(q\\) are two different prime numbers and that \\(n=p^{2} q^{2}\\). What is the number of possible values of \\(n\\) with \\(n<1000\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 18229,
"solution": "We note that \\(n=p^{2} q^{2}=(p q)^{2}\\). Since \\(n<1000\\), then \\((p q)^{2}<1000\\) and so \\(p q<\\sqrt{1000} \\approx 31.6\\). Finding the number of possible values of \\(n\\) is thus equivalent to finding the number of positive integers \\(m\\) with \\(1 \... |
deepscaler | [
{
"role": "user",
"content": "Let the number of elements in the set \\( S \\) be denoted by \\( |S| \\), and the number of subsets of the set \\( S \\) be denoted by \\( n(S) \\). Given three non-empty finite sets \\( A, B, C \\) that satisfy the following conditions:\n$$\n\\begin{array}{l}\n|A| = |B| = 201... | math | {
"style": "rule",
"ground_truth": "2018"
} | {
"split": "train",
"index": 18230,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rectangles \\( A B C D, D E F G, C E I H \\) have equal areas and integer sides. Find \\( D G \\) if \\( B C = 19 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "380"
} | {
"split": "train",
"index": 18231,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a parabola $y=x^2+bx+c$ intersects the y-axis at point Q(0, -3), and the sum of the squares of the x-coordinates of the two intersection points with the x-axis is 15, find the equation of the function and its axis of symmetry. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 18232,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The slopes of lines $l_1$ and $l_2$ are the two roots of the equation $6x^2+x-1=0$, respectively. The angle between lines $l_1$ and $l_2$ is __________. 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": 18233,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Pete thinks of a number. He doubles it, adds 10, multiplies by 4, and ends up with 120. What was his original number? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 18234,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many zeroes does the product of \\( 25^{5} \\), \\( 150^{4} \\), and \\( 2008^{3} \\) end with? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 18235,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many two-digit numbers are there in which the tens digit is greater than the ones digit? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 18236,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The marked price of a book was 30% less than the suggested retail price. Alice purchased the book for half the marked price at a Fiftieth Anniversary sale. What percent of the suggested retail price did Alice pay? Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "35\\%"
} | {
"split": "train",
"index": 18237,
"solution": "1. **Assume the suggested retail price (SRP) of the book**: Let the suggested retail price of the book be $P$. For simplicity, we can assume $P = 100$ USD.\n\n2. **Calculate the marked price (MP)**: The problem states that the marked price is 30% less than the sugg... |
deepscaler | [
{
"role": "user",
"content": "Given an angle $a$ and a point $P(-4,3)$ on its terminal side, find the value of $\\dfrac{\\cos \\left( \\dfrac{\\pi}{2}+a\\right)\\sin \\left(-\\pi-a\\right)}{\\cos \\left( \\dfrac{11\\pi}{2}-a\\right)\\sin \\left( \\dfrac{9\\pi}{2}+a\\right)}$. Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "- \\dfrac{3}{4}"
} | {
"split": "train",
"index": 18238,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle of radius $10$ inches has its center at the vertex $C$ of an equilateral triangle $ABC$ and passes through the other two vertices. The side $AC$ extended through $C$ intersects the circle at $D$. Calculate the measure of angle $ADB$. Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"split": "train",
"index": 18239,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that each side of a large square is divided into four equal parts, a smaller square is inscribed in such a way that its corners are at the division points one-fourth and three-fourths along each side of the large square, calculate the ratio of the area of this inscribed s... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 18240,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Recall that the conjugate of the complex number $w = a + bi$, where $a$ and $b$ are real numbers and $i = \\sqrt{-1}$, is the complex number $\\overline{w} = a - bi$. For any complex number $z$, let $f(z) = 4i\\overline{z}$. The polynomial $P(z) = z^4 + 4z^3 + 3z^2 + 2z + 1$ ha... | math | {
"style": "rule",
"ground_truth": "208"
} | {
"split": "train",
"index": 18241,
"solution": "1. **Understanding the function and polynomial transformation**:\n Given a complex number $w = a + bi$, its conjugate is $\\overline{w} = a - bi$. The function $f(z) = 4i\\overline{z}$ transforms $z$ into $4i\\overline{z}$. We are given a polynomial $P(z) = z^4 +... |
deepscaler | [
{
"role": "user",
"content": "In a $k \\times k$ chessboard, a set $S$ of 25 cells that are in a $5 \\times 5$ square is chosen uniformly at random. The probability that there are more black squares than white squares in $S$ is $48 \\%$. Find $k$. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 18242,
"solution": "We know that there must be fewer black squares than white squares, and $k$ must be odd. Additionally, we know that there are $k-4$ ways to pick the left column of the $5 \\times 5$ square so that the right column can fit within the $k \\times k$ grid, and $k-4$ way... |
deepscaler | [
{
"role": "user",
"content": "For each even positive integer $x$, let $g(x)$ denote the greatest power of 2 that divides $x.$ For example, $g(20)=4$ and $g(16)=16.$ For each positive integer $n,$ let $S_n=\\sum_{k=1}^{2^{n-1}}g(2k).$ Find the greatest integer $n$ less than 1000 such that $S_n$ is a perfect ... | math | {
"style": "rule",
"ground_truth": "899"
} | {
"split": "train",
"index": 18243,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle \\(ABCD\\), \\(AB = 20 \\, \\text{cm}\\) and \\(BC = 10 \\, \\text{cm}\\). Points \\(M\\) and \\(N\\) are taken on \\(AC\\) and \\(AB\\), respectively, such that the value of \\(BM + MN\\) is minimized. Find this minimum value. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 18244,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Expand the product ${(x+2)(x+5)}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x^2 + 7x + 10"
} | {
"split": "train",
"index": 18245,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$44 \\times 22$ is equal to Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "88 \\times 11"
} | {
"split": "train",
"index": 18246,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an institute, there are truth-tellers, who always tell the truth, and liars, who always lie. One day, each of the employees made two statements:\n\n1) There are fewer than ten people in the institute who work more than I do.\n2) In the institute, at least one hundred people ... | math | {
"style": "rule",
"ground_truth": "110"
} | {
"split": "train",
"index": 18247,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The maximum number that can be formed by the digits 0, 3, 4, 5, 6, 7, 8, 9, given that there are only thirty 2's and twenty-five 1's. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "199"
} | {
"split": "train",
"index": 18248,
"solution": ""
} |
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