data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "From the set \\( M = \\{1, 2, \\cdots, 2008\\} \\) of the first 2008 positive integers, a \\( k \\)-element subset \\( A \\) is chosen such that the sum of any two numbers in \\( A \\) cannot be divisible by the difference of those two numbers. What is the maximum value of \\( ... | math | {
"style": "rule",
"ground_truth": "670"
} | {
"index": 12400,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the set \\( \\{1, 2, 3, \\ldots, 999, 1000\\} \\), select \\( k \\) numbers. If among the selected numbers, there are always three numbers that can form the side lengths of a triangle, what is the smallest value of \\( k \\)? Explain why."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 12401,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the set \\(\\left\\{-3, -\\frac{5}{4}, -\\frac{1}{2}, 0, \\frac{1}{3}, 1, \\frac{4}{5}, 2\\right\\}\\), two numbers are drawn without replacement. Find the concept of the two numbers being the slopes of perpendicular lines."
}
] | math | {
"style": "rule",
"ground_truth": "3/28"
} | {
"index": 12402,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the set of integers $\\{1,2,3,\\dots,2009\\}$, choose $k$ pairs $\\{a_i,b_i\\}$ with $a_i<b_i$ so that no two pairs have a common element. Suppose that all the sums $a_i+b_i$ are distinct and less than or equal to $2009$. Find the maximum possible value of $k$."
}
] | math | {
"style": "rule",
"ground_truth": "803"
} | {
"index": 12403,
"solution": "Suppose that we have a valid solution with $k$ pairs. As all $a_i$ and $b_i$ are distinct, their sum is at least $1+2+3+\\cdots+2k=k(2k+1)$. On the other hand, as the sum of each pair is distinct and at most equal to $2009$, the sum of all $a_i$ and $b_i$ is at most $2009 + (2009-1) +... |
DeepScaleR | [
{
"role": "user",
"content": "From the set of integers $\\{1,2,3,\\dots,3009\\}$, choose $k$ pairs $\\{a_i,b_i\\}$ such that $a_i < b_i$ and no two pairs have a common element. Assume all the sums $a_i+b_i$ are distinct and less than or equal to 3009. Determine the maximum possible value of $k$."
}
] | math | {
"style": "rule",
"ground_truth": "1203"
} | {
"index": 12404,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the set of integers $\\{1,2,3,\\dots,3009\\}$, choose $k$ pairs $\\{a_i,b_i\\}$ with $a_i<b_i$ so that no two pairs share a common element. Each sum $a_i+b_i$ must be distinct and less than or equal to $3009$. Determine the maximum possible value of $k$."
}
] | math | {
"style": "rule",
"ground_truth": "1504"
} | {
"index": 12405,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the set of numbers \\( 1, 2, 3, 4, \\cdots, 1982 \\), remove some numbers so that in the remaining numbers, no number is equal to the product of any two other numbers. What is the minimum number of numbers that need to be removed to achieve this? How can this be done?"
}... | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 12406,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the set of three-digit numbers that do not contain the digits $0,1,2,3,4,5$, several numbers were written down in such a way that no two numbers could be obtained from each other by swapping two adjacent digits. What is the maximum number of such numbers that could have be... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 12407,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the set {1, 2, 3, ..., 14}, select \\(a_{1}, a_{2}, a_{3}\\) in increasing order such that \\(a_{2} - a_{1} \\geq 3\\) and \\(a_{3} - a_{2} \\geq 3\\). Find the total number of different ways to select these numbers."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 12408,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the six digits 0, 1, 2, 3, 4, 5, select two odd numbers and two even numbers to form a four-digit number without repeating digits. The total number of such four-digit numbers is ______."
}
] | math | {
"style": "rule",
"ground_truth": "180"
} | {
"index": 12409,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the sixteen natural numbers $1, 2, 3, 4, \\cdots, 15, 16$, if $n$ numbers are randomly selected, there will always be such two numbers: one is three times the other. What is the smallest $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 12410,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the vertex $ A$ of the equilateral triangle $ ABC$ a line is drown that intercepts the segment $ [BC]$ in the point $ E$ . The point $ M \\in (AE$ is such that $ M$ external to $ ABC$ , $ \\angle AMB \\equal{} 20 ^\\circ$ and $ \\angle AMC \\equal{} 30 ^ \\c... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 12411,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From the vertex $B$ of an isosceles triangle $ABC$, a height $BD$ is dropped to its base $AC$. Each of the legs $AB$ and $BC$ of triangle $ABC$ is equal to 8. In triangle $BCD$, a median $DE$ is drawn. A circle is inscribed in triangle $BDE$, touching side $BE$ at point $K$ and... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 12412,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From three red balls numbered $1$, $2$, $3$ and two white balls numbered $2$, $3$, find the probability that two balls drawn at random have different numbers and colors."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{2}{5}"
} | {
"index": 12413,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From time $t=0$ to time $t=1$ a population increased by $i\\%$, and from time $t=1$ to time $t=2$ the population increased by $j\\%$. Therefore, from time $t=0$ to time $t=2$ the population increased by"
}
] | math | {
"style": "rule",
"ground_truth": "$\\left(i+j+\\frac{ij}{100}\\right)\\%$"
} | {
"index": 12414,
"solution": "1. **Identify the scale factors for each time period:**\n - From $t=0$ to $t=1$, the population increases by $i\\%$. The scale factor for this increase is $1 + \\frac{i}{100}$.\n - From $t=1$ to $t=2$, the population increases by $j\\%$. The scale factor for this increase is $1 + ... |
DeepScaleR | [
{
"role": "user",
"content": "From vertex $A$ of an equilateral triangle $ABC$ , a ray $Ax$ intersects $BC$ at point $D$ . Let $E$ be a point on $Ax$ such that $BA =BE$ . Calculate $\\angle AEC$ ."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 12415,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Fully factor the following expression: $2x^2-8$"
}
] | math | {
"style": "rule",
"ground_truth": "(2) (x+2) (x-2)"
} | {
"index": 12416,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Fully simplify the following expression: $[(2+3+4+5)\\div2] + [(2\\cdot5+8)\\div3]$."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 12417,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Function $f(x, y): \\mathbb N \\times \\mathbb N \\to \\mathbb Q$ satisfies the conditions:\n\n(i) $f(1, 1) =1$ ,\n\n(ii) $f(p + 1, q) + f(p, q + 1) = f(p, q)$ for all $p, q \\in \\mathbb N$ , and\n\n(iii) $qf(p + 1, q) = pf(p, q + 1)$ for all $p, q \\in \\mathbb N$ ... | math | {
"style": "rule",
"ground_truth": "\\frac{30! \\cdot 1989!}{2020!}"
} | {
"index": 12418,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Functions $f$ and $g$ are quadratic, $g(x) = - f(100 - x)$, and the graph of $g$ contains the vertex of the graph of $f$. The four $x$-intercepts on the two graphs have $x$-coordinates $x_1$, $x_2$, $x_3$, and $x_4$, in increasing order, and $x_3 - x_2 = 150$. Then $x_4 - x_1 =... | math | {
"style": "rule",
"ground_truth": "752"
} | {
"index": 12419,
"solution": "1. **Understanding the relationship between $f$ and $g$:**\n Given that $g(x) = -f(100 - x)$, we can infer that the graph of $g$ is a $180^\\circ$ rotation of the graph of $f$ around the point $(50, 0)$. This is because replacing $x$ with $100 - x$ reflects the graph across the line... |
DeepScaleR | [
{
"role": "user",
"content": "G.H. Hardy once went to visit Srinivasa Ramanujan in the hospital, and he started the conversation with: \"I came here in taxi-cab number 1729. That number seems dull to me, which I hope isn't a bad omen.\" \"Nonsense,\" said Ramanujan. \"The number isn't dull at all. It's quit... | math | {
"style": "rule",
"ground_truth": "251"
} | {
"index": 12420,
"solution": "Let this smallest positive integer be represented as $a^{3} + b^{3} + c^{3} = d^{3} + e^{3} + f^{3}$. By inspection, a solution is not possible with the first 4 cubes. We prove that it is impossible to write the same number as two different sums of the first 5 cubes. Because we necess... |
DeepScaleR | [
{
"role": "user",
"content": "Gabor wanted to design a maze. He took a piece of grid paper and marked out a large square on it. From then on, and in the following steps, he always followed the lines of the grid, moving from grid point to grid point. Then he drew some lines within the square, totaling 400 un... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 12421,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Gabriela found an encyclopedia with $2023$ pages, numbered from $1$ to $2023$ . She noticed that the pages formed only by even digits have a blue mark, and that every three pages since page two have a red mark. How many pages of the encyclopedia have both colors?"
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"index": 12422,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Gage skated 1 hr 15 min each day for 5 days and 1 hr 30 min each day for 3 days. How many minutes would he have to skate the ninth day in order to average 85 minutes of skating each day for the entire time?"
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 12423,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Gamma and Delta both participated in a two-day science quiz. Each attempted questions totaling 600 points after the second day. On the first day, Gamma scored 210 points out of 350 points attempted, and on the second day scored 150 points out of 250 points attempted. Delta, who... | math | {
"style": "rule",
"ground_truth": "\\frac{359}{600}"
} | {
"index": 12424,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Gari is seated in a jeep, and at the moment, has one 10-peso coin, two 5-peso coins, and six 1-peso coins in his pocket. If he picks four coins at random from his pocket, what is the probability that these will be enough to pay for his jeepney fare of 8 pesos?"
}
] | math | {
"style": "rule",
"ground_truth": "37/42"
} | {
"index": 12425,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Gary plays the following game with a fair $n$-sided die whose faces are labeled with the positive integers between 1 and $n$, inclusive: if $n=1$, he stops; otherwise he rolls the die, and starts over with a $k$-sided die, where $k$ is the number his $n$-sided die lands on. (In... | math | {
"style": "rule",
"ground_truth": "\\frac{197}{60}"
} | {
"index": 12426,
"solution": "If we let $a_{n}$ be the expected number of rolls starting with an $n$-sided die, we see immediately that $a_{1}=0$, and $a_{n}=1+\\frac{1}{n} \\sum_{i=1}^{n} a_{i}$ for $n>1$. Thus $a_{2}=2$, and for $n \\geq 3$, $a_{n}=1+\\frac{1}{n} a_{n}+\\frac{n-1}{n}\\left(a_{n-1}-1\\right)$, or... |
DeepScaleR | [
{
"role": "user",
"content": "Gary purchased a large beverage, but only drank $m/n$ of it, where $m$ and $n$ are relatively prime positive integers. If he had purchased half as much and drunk twice as much, he would have wasted only $2/9$ as much beverage. Find $m+n$."
}
] | math | {
"style": "rule",
"ground_truth": "37"
} | {
"index": 12427,
"solution": "Let $x$ be the fraction consumed, then $(1-x)$ is the fraction wasted. We have $\\frac{1}{2} - 2x =\\frac{2}{9} (1-x)$, or $9 - 36x = 4 - 4x$, or $32x = 5$ or $x = 5/32$. Therefore, $m + n = 5 + 32 = \\boxed{037}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Gauss is a famous German mathematician, known as the \"Prince of Mathematics\". There are 110 achievements named after \"Gauss\". Let $x\\in \\mathbb{R}$, $[x]$ denotes the greatest integer less than or equal to $x$, and $\\{x\\}=x-[x]$ represents the non-negative fractional pa... | math | {
"style": "rule",
"ground_truth": "3024+\\sqrt{3}"
} | {
"index": 12428,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Gauss is a famous German mathematician, known as the \"Prince of Mathematics\". There are 110 achievements named after \"Gauss\". Let $x\\in\\mathbb{R}$, use $[x]$ to represent the largest integer not exceeding $x$, and use $\\{x\\}=x-[x]$ to represent the non-negative fraction... | math | {
"style": "rule",
"ground_truth": "3024+ \\sqrt {3}"
} | {
"index": 12429,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Gauss is a famous German mathematician, known as the \"Prince of Mathematics\". There are 110 achievements named after his name \"Gauss\". For $x\\in R$, let $[x]$ represent the largest integer not greater than $x$, and let $\\{x\\}=x-[x]$ represent the non-negative fractional ... | math | {
"style": "rule",
"ground_truth": "3027+ \\sqrt{3}"
} | {
"index": 12430,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Gavin has a collection of 50 songs that are each 3 minutes in length and 50 songs that are each 5 minutes in length. What is the maximum number of songs from his collection that he can play in 3 hours?"
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 12431,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Gavrila found that the front tires of the car last for 21,000 km, and the rear tires last for 28,000 km. Therefore, he decided to swap them at some point so that the car would travel the maximum possible distance. Find this maximum distance (in km)."
}
] | math | {
"style": "rule",
"ground_truth": "24000"
} | {
"index": 12432,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Gavrila is in an elevator cabin which is moving downward with a deceleration of 5 m/s². Find the force with which Gavrila presses on the floor. Gavrila's mass is 70 kg, and the acceleration due to gravity is 10 m/s². Give the answer in newtons, rounding to the nearest whole num... | math | {
"style": "rule",
"ground_truth": "350"
} | {
"index": 12433,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Geoff and Trevor each roll a fair eight-sided die (the sides are labeled 1 through 8). What is the probability that the product of the numbers they roll is even or a prime number?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{8}"
} | {
"index": 12434,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Geoff and Trevor each roll a fair six-sided die. What is the probability that the product of the numbers they roll is even?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac34"
} | {
"index": 12435,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Geoff walks on the number line for 40 minutes, starting at the point 0. On the $n$th minute, he flips a fair coin. If it comes up heads he walks $\\frac{1}{n}$ in the positive direction and if it comes up tails he walks $\\frac{1}{n}$ in the negative direction. Let $p$ be the p... | math | {
"style": "rule",
"ground_truth": "8101"
} | {
"index": 12436,
"solution": "To estimate it by hand, we'll do casework on the most likely ways that Geoff will go past +2, and double the answer. If Geoff starts with one of the three sequences below, he will be past 2 or very close to 2: $$(+,+,+,+),(+,+,+,-,+,+),(+,+,-,+,+,+)$$ The probability of one of these h... |
DeepScaleR | [
{
"role": "user",
"content": "George and Henry started a race from opposite ends of the pool. After a minute and a half, they passed each other in the center of the pool. If they lost no time in turning and maintained their respective speeds, how many minutes after starting did they pass each other the seco... | math | {
"style": "rule",
"ground_truth": "4\\frac{1}{2}"
} | {
"index": 12437,
"solution": "1. **Understanding the Problem:**\n George and Henry start from opposite ends of the pool and meet in the center after $1.5$ minutes. This means each of them has traveled half the length of the pool in $1.5$ minutes.\n\n2. **Analyzing the Speeds:**\n Since they meet in the center,... |
DeepScaleR | [
{
"role": "user",
"content": "George flips an unfair coin 7 times. The coin has a $\\frac{1}{4}$ probability of coming up heads and a $\\frac{3}{4}$ probability of coming up tails. What is the probability that he flips exactly 2 tails?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{189}{16384}"
} | {
"index": 12438,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "George has 10 different colors available to paint his room and he must choose exactly 2 of them. However, George prefers to include the color blue as one of his choices if possible. In how many ways can he choose the colors?"
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 12439,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "George has a quadratic of the form $x^2+bx+\\frac13$, where $b$ is a specific negative number. Using his knowledge of how to complete the square, George is able to rewrite this quadratic in the form $(x+m)^2+\\frac{1}{12}$. What is $b$?"
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 12440,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "George has an unfair six-sided die. The probability that it rolls a 6 is $\\frac{1}{2}$, and the probability that it rolls any other number is $\\frac{1}{10}$. What is the expected value of the number shown when this die is rolled? Express your answer as a decimal."
}
] | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"index": 12441,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "George is about to get a certain amount of change less than one dollar from the cash register. If he gets the most quarters possible and the rest in pennies, he would need to receive 3 pennies to meet the amount. If he gets the most dimes possible and the rest in pennies, he wo... | math | {
"style": "rule",
"ground_truth": "106"
} | {
"index": 12442,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "George is planning a dinner party for three other couples, his wife, and himself. He plans to seat the four couples around a circular table for 8, and wants each husband to be seated opposite his wife. How many seating arrangements can he make, if rotations and reflections of e... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 12443,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "George is trying to find the Fermat point $P$ of $\\triangle ABC$, where $A$ is at the origin, $B$ is at $(10,2)$, and $C$ is at $(5,4)$. He guesses that the point is at $P = (3,1)$. Compute the sum of the distances from $P$ to the vertices of $\\triangle ABC$. If he obtains $x... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 12444,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "George now has an unfair eight-sided die. The probabilities of rolling each number from 1 to 5 are each $\\frac{1}{15}$, the probability of rolling a 6 or a 7 is $\\frac{1}{6}$ each, and the probability of rolling an 8 is $\\frac{1}{5}$. What is the expected value of the number... | math | {
"style": "rule",
"ground_truth": "4.7667"
} | {
"index": 12445,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "George purchases a sack of apples, a bunch of bananas, a cantaloupe, and a carton of dates for $ \\$ 20$. If a carton of dates costs twice as much as a sack of apples and the price of a cantaloupe is equal to the price of a sack of apples minus a bunch of bananas, how much woul... | math | {
"style": "rule",
"ground_truth": "\\$ 5"
} | {
"index": 12446,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "George walks $1$ mile to school. He leaves home at the same time each day, walks at a steady speed of $3$ miles per hour, and arrives just as school begins. Today he was distracted by the pleasant weather and walked the first $\\frac{1}{2}$ mile at a speed of only $2$ miles per... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 12447,
"solution": "1. **Calculate the normal time to get to school**: George walks 1 mile to school at a speed of 3 miles per hour. The time taken to walk this distance is calculated by the formula:\n \\[\n \\text{Time} = \\frac{\\text{Distance}}{\\text{Speed}} = \\frac{1 \\text{ mile}}{3 \\text{ mp... |
DeepScaleR | [
{
"role": "user",
"content": "George, Jeff, Brian, and Travis decide to play a game of hot potato. They begin by arranging themselves clockwise in a circle in that order. George and Jeff both start with a hot potato. On his turn, a player gives a hot potato (if he has one) to a randomly chosen player among ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{27}"
} | {
"index": 12448,
"solution": "Notice that Travis can only have the hot potato at the end if he has two potatoes before his turn. A little bit of casework shows that this can only happen when Case 1: George gives Travis his potato, while Jeff gives Brian his potato, which in then goes to Travis. The probability of ... |
DeepScaleR | [
{
"role": "user",
"content": "Gerard cuts a large rectangle into four smaller rectangles. The perimeters of three of these smaller rectangles are 16, 18, and 24. What is the perimeter of the fourth small rectangle?\nA) 8\nB) 10\nC) 12\nD) 14\nE) 16"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 12449,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Gilda has a bag of marbles. She gives $20\\%$ of them to her friend Pedro. Then Gilda gives $10\\%$ of what is left to another friend, Ebony. Finally, Gilda gives $25\\%$ of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gi... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"index": 12450,
"solution": "1. **Initial Amount of Marbles**: Let's denote the initial number of marbles Gilda has as $M$.\n\n2. **Marbles Given to Pedro**: Gilda gives $20\\%$ of her marbles to Pedro. Therefore, the number of marbles she gives to Pedro is $0.20M$. The number of marbles left with Gilda after giv... |
DeepScaleR | [
{
"role": "user",
"content": "Gina's running app tracked her average rate in minutes per kilometre. After starting the app, Gina stood still for 15 seconds and then ran at a constant rate of 7 minutes per kilometre for the rest of the run. How many kilometres did Gina run between when her app showed her ave... | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"index": 12451,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Giraldo wrote five distinct natural numbers on the vertices of a pentagon. And next he wrote on each side of the pentagon the least common multiple of the numbers written of the two vertices who were on that side and noticed that the five numbers written on the sides were equal... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 12452,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Giselle will combine blue paint, green paint, and white paint in the ratio $3:2:4$, respectively. If she uses $12$ quarts of white paint, how many quarts of green paint should she use?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 12453,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Give an example of a function that, when \\( x \\) is equal to a known number, takes the form \\( \\frac{0}{0} \\), but as \\( x \\) approaches this number, tends to a certain limit."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{7}"
} | {
"index": 12454,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Give an example of a number $x$ for which the equation $\\sin 2017 x - \\operatorname{tg} 2016 x = \\cos 2015 x$ holds. Justify your answer."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"index": 12455,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Give an example of a quadratic function that has zeroes at $x=2$ and $x=4$, and that takes the value $6$ when $x=3$.\n\nEnter your answer in the expanded form \"ax^2 + bx + c\", where a,b,c are replaced by appropriate numbers."
}
] | math | {
"style": "rule",
"ground_truth": "-6x^2+36x-48"
} | {
"index": 12456,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Give an example of an expression consisting of ones, parentheses, the symbols \"+\", and \"×\" such that:\n\n- Its value is equal to 11\n- If in this expression all \"+\" signs are replaced with \"×\" signs, and all \"×\" signs are replaced with \"+\" signs, it will still resul... | math | {
"style": "rule",
"ground_truth": "1+1+1+1+1+1+1+1+1+1+1"
} | {
"index": 12457,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Give the set of all positive integers $n$ such that $\\varphi(n)=2002^{2}-1$."
}
] | math | {
"style": "rule",
"ground_truth": "\\varnothing"
} | {
"index": 12458,
"solution": "The empty set, $\\varnothing$. If $m$ is relatively prime to $n$ and $m<n$, then $n-m$ must likewise be relatively prime to $n$, and these are distinct for $n>2$ since $n / 2, n$ are not relatively prime. Therefore, for all $n>2, \\varphi(n)$ must be even. $2002^{2}-1$ is odd, and $\\... |
DeepScaleR | [
{
"role": "user",
"content": "Given $10$ points in the space such that each $4$ points are not lie on a plane. Connect some points with some segments such that there are no triangles or quadrangles. Find the maximum number of the segments."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 12459,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $2014$ points in the plane, no three of which are collinear, what is the minimum number of line segments that can be drawn connecting pairs of points in such a way that adding a single additional line segment of the same sort will always produce a triangle of three conn... | math | {
"style": "rule",
"ground_truth": "2013"
} | {
"index": 12460,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $ \\dfrac {3\\pi}{4} < \\alpha < \\pi$, $\\tan \\alpha+ \\dfrac {1}{\\tan \\alpha}=- \\dfrac {10}{3}$. \n$(1)$ Find the value of $\\tan \\alpha$; \n$(2)$ Find the value of $ \\dfrac {5\\sin ^{2} \\dfrac {\\alpha}{2}+8\\sin \\dfrac {\\alpha}{2}\\cos \\dfrac {\\alpha}{2}+... | math | {
"style": "rule",
"ground_truth": "- \\dfrac {5}{4}"
} | {
"index": 12461,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $ \\frac {\\pi}{2} < \\alpha < \\pi$ and $0 < \\beta < \\frac {\\pi}{2}$, with $\\tan \\alpha= -\\frac {3}{4}$ and $\\cos (\\beta-\\alpha)= \\frac {5}{13}$, find the value of $\\sin \\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {63}{65}"
} | {
"index": 12462,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $ \\frac {\\pi}{2}\\leqslant \\beta\\leqslant \\alpha\\leqslant \\frac {3\\pi}{4} $, $ \\cos (\\alpha-\\beta) = \\frac {12}{13} $, $ \\sin (\\alpha+\\beta) = -\\frac {3}{5} $, find the values of $ \\sin 2\\alpha $ and $ \\cos 2\\beta $."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{63}{65}"
} | {
"index": 12463,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $ \\sin \\alpha = \\frac{1}{3} $, and $ 0 < \\alpha < \\pi $, then $ \\tan \\alpha = $_____, and $ \\sin \\frac{\\alpha}{2} + \\cos \\frac{\\alpha}{2} = $_____."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2 \\sqrt{3}}{3}"
} | {
"index": 12464,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $ \\tan \\left( \\alpha + \\frac{\\pi}{4} \\right) = -\\frac{1}{2} $ and $\\frac{\\pi}{2} < \\alpha < \\pi$, calculate the value of $ \\frac{\\sin 2\\alpha - 2\\cos^2 \\alpha}{\\sin \\left(\\alpha - \\frac{\\pi}{4} \\right)} $."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{2 \\sqrt{5}}{5}"
} | {
"index": 12465,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $$(1+x)(2-x)^{6}=a_{0}+a_{1}(x-1)+a_{2}(x-1)^{2}+\\ldots+a_{7}(x-1)^{7}$$, find the value of $a_{3}$."
}
] | math | {
"style": "rule",
"ground_truth": "-25"
} | {
"index": 12466,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $$(5x- \\frac {1}{ \\sqrt {x}})^{n}$$, the sum of the binomial coefficients in its expansion is 64. Find the constant term in the expansion."
}
] | math | {
"style": "rule",
"ground_truth": "375"
} | {
"index": 12467,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $$\\alpha \\in \\left( \\frac{5}{4}\\pi, \\frac{3}{2}\\pi \\right)$$ and it satisfies $$\\tan\\alpha + \\frac{1}{\\tan\\alpha} = 8$$, then $\\sin\\alpha\\cos\\alpha = \\_\\_\\_\\_\\_\\_$; $\\sin\\alpha - \\cos\\alpha = \\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{3}}{2}"
} | {
"index": 12468,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $$\\alpha, \\beta \\in (0, \\frac{\\pi}{2})$$, and $$\\alpha + \\beta \\neq \\frac{\\pi}{2}, \\sin\\beta = \\sin\\alpha\\cos(\\alpha + \\beta)$$.\n(1) Express $\\tan\\beta$ in terms of $\\tan\\alpha$;\n(2) Find the maximum value of $\\tan\\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{4}"
} | {
"index": 12469,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $$\\frac {1}{3} \\leq a \\leq 1$$, if the function $f(x) = ax^2 - 2x + 1$ has a domain of $[1, 3]$. \n(1) Find the minimum value of $f(x)$ in its domain (expressed in terms of $a$); \n(2) Let the maximum value of $f(x)$ in its domain be $M(a)$, and the minimum value be ... | math | {
"style": "rule",
"ground_truth": "\\frac {1}{2}"
} | {
"index": 12470,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $$\\frac {1}{3}$$≤a≤1, if the function f(x)=ax<sup>2</sup>-2x+1 has its maximum value M(a) and minimum value N(a) in the interval [1,3], let g(a)=M(a)-N(a).\n(1) Find the expression for g(a);\n(2) Describe the intervals where g(a) is increasing and decreasing (no proof re... | math | {
"style": "rule",
"ground_truth": "\\frac {1}{2}"
} | {
"index": 12471,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $$\\frac {\\pi}{2} < \\alpha < \\pi$$, $$0 < \\beta < \\frac {\\pi}{2}$$, $$\\tan\\alpha = -\\frac {3}{4}$$, and $$\\cos(\\beta-\\alpha) = \\frac {5}{13}$$, find the value of $\\sin\\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {63}{65}"
} | {
"index": 12472,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $$\\frac {\\sin\\alpha-2\\cos\\alpha}{\\sin\\alpha +2\\cos\\alpha}=3,$$ calculate: \n(1) $$\\frac {\\sin\\alpha+2\\cos\\alpha}{5\\cos\\alpha -\\sin\\alpha};$$ \n(2) $({\\sin\\alpha+\\cos\\alpha})^{2}.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{17}"
} | {
"index": 12473,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $$\\frac{1}{C_5^m} - \\frac{1}{C_6^m} = \\frac{7}{10C_7^m}$$, find $C_{21}^m$."
}
] | math | {
"style": "rule",
"ground_truth": "210"
} | {
"index": 12474,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $$\\frac{\\cos\\alpha + \\sin\\alpha}{\\cos\\alpha - \\sin\\alpha} = 2$$, find the value of $$\\frac{1 + \\sin4\\alpha - \\cos4\\alpha}{1 + \\sin4\\alpha + \\cos4\\alpha}$$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 12475,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $$\\frac{\\pi}{2} < \\alpha < \\pi$$ and $$0 < \\beta < \\frac{\\pi}{2}$$, with $\\tan\\alpha = -\\frac{3}{4}$ and $\\cos(\\beta - \\alpha) = \\frac{5}{13}$, find the value of $\\sin\\beta$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{63}{65}"
} | {
"index": 12476,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $$\\overrightarrow {a} = (x-1, y)$$, $$\\overrightarrow {b} = (x+1, y)$$, and $$|\\overrightarrow {a}| + |\\overrightarrow {b}| = 4$$ \n(1) Find the equation of the trajectory C of point M(x, y). \n(2) Let P be a moving point on curve C, and F<sub>1</sub>(-1, 0), F<sub>... | math | {
"style": "rule",
"ground_truth": "\\frac {2 \\sqrt {21}}{7}"
} | {
"index": 12477,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $$\\overrightarrow {m} = (\\sin \\omega x + \\cos \\omega x, \\sqrt {3} \\cos \\omega x)$$, $$\\overrightarrow {n} = (\\cos \\omega x - \\sin \\omega x, 2\\sin \\omega x)$$ ($\\omega > 0$), and the function $f(x) = \\overrightarrow {m} \\cdot \\overrightarrow {n}$, if the... | math | {
"style": "rule",
"ground_truth": "\\sqrt {3}"
} | {
"index": 12478,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $$\\overrightarrow {m} = (\\sin x, \\frac {3}{4})$$, $$\\overrightarrow {n} = (\\cos x, -1)$$, in triangle ABC, $\\sin A + \\cos A = \\sqrt {2}$. \n(1) When $$\\overrightarrow {m} \\parallel \\overrightarrow {n}$$, find the value of $\\sin^2 x + \\sin 2x$; \n(2) Let the... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"index": 12479,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $$\\sin\\left(\\alpha+ \\frac {\\pi}{3}\\right)=- \\frac {4}{5}$$, and $$- \\frac {\\pi}{2}<\\alpha<0$$, find the value of $\\cos\\alpha$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {3-4 \\sqrt {3}}{10}"
} | {
"index": 12480,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $$\\sqrt {2 \\frac {2}{3}}=2 \\sqrt { \\frac {2}{3}}$$, $$\\sqrt {3 \\frac {3}{8}}=3 \\sqrt { \\frac {3}{8}}$$, $$\\sqrt {4 \\frac {4}{15}}=4 \\sqrt { \\frac {4}{15}}$$, ..., if $$\\sqrt {6 \\frac {a}{t}}=6 \\sqrt { \\frac {a}{t}}$$ (where $a$, $t$∈$R^*$), then $a=$ \\_\\... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 12481,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $$a_{n}= \\frac {n(n+1)}{2}$$, remove all the numbers in the sequence $\\{a_n\\}$ that can be divided by 2, and arrange the remaining numbers in ascending order to form the sequence $\\{b_n\\}$. Find the value of $b_{21}$."
}
] | math | {
"style": "rule",
"ground_truth": "861"
} | {
"index": 12482,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $$f(x)= \\begin{cases} f(x+1), x < 4 \\\\ ( \\frac {1}{2})^{x}, x\\geq4\\end{cases}$$, find $f(\\log_{2}3)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {1}{24}"
} | {
"index": 12483,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $$x \\in \\left(- \\frac{\\pi}{2}, \\frac{\\pi}{2}\\right)$$ and $$\\sin x + \\cos x = \\frac{1}{5}$$, calculate the value of $\\tan 2x$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{24}{7}"
} | {
"index": 12484,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $$|\\vec{a}|=3, |\\vec{b}|=2$$. If $$\\vec{a} \\cdot \\vec{b} = -3$$, then the angle between $$\\vec{a}$$ and $$\\vec{b}$$ is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}\\pi"
} | {
"index": 12485,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $$∫_{ 0 }^{ 2 }(\\cos \\frac {π}{4}x+ \\sqrt {4-x^{2}})dx$$, evaluate the definite integral."
}
] | math | {
"style": "rule",
"ground_truth": "\\pi+\\frac{4}{\\pi}"
} | {
"index": 12486,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $( \\sqrt {x}+ \\dfrac {2}{x^{2}})^{n}$, the ratio of the coefficient of the fifth term to the coefficient of the third term in its expansion is $56:3$. \n(Ⅰ) Find the constant term in the expansion; \n(Ⅱ) When $x=4$, find the term with the maximum binomial coefficient ... | math | {
"style": "rule",
"ground_truth": "\\dfrac {63}{256}"
} | {
"index": 12487,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $(1+x) + (1+x)^2 + \\ldots + (1+x)^5 = a_0 + a_1(1-x) + a_2(1-x)^2 + \\ldots + a_5(1-x)^5$, calculate the value of $a_1 + a_2 + a_3 + a_4 + a_5$."
}
] | math | {
"style": "rule",
"ground_truth": "-57"
} | {
"index": 12488,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $(1-2x)^7 = a + a_1x + a_2x^2 + \\ldots + a_7x^7$, calculate the value of $a_2 + a_3 + a_4 + a_5 + a_6 + a_7$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 12489,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $(1-2x)^{2017} = a_0 + a_1(x-1) + a_2(x-1)^2 + \\ldots + a_{2016}(x-1)^{2016} + a_{2017}(x-1)^{2017}$ ($x \\in \\mathbb{R}$), find the value of $a_1 - 2a_2 + 3a_3 - 4a_4 + \\ldots - 2016a_{2016} + 2017a_{2017}$."
}
] | math | {
"style": "rule",
"ground_truth": "-4034"
} | {
"index": 12490,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $(2x-1)^{2015} = a_{0} + a_{1}x + a_{2}x^{2} + \\ldots + a_{2015}x^{2015}$ ($x \\in \\mathbb{R}$), evaluate the expression $\\frac {1}{2}+ \\frac {a_{2}}{2^{2}a_{1}}+ \\frac {a_{3}}{2^{3}a_{1}}+\\ldots+ \\frac {a_{2015}}{2^{2015}a_{1}}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac {1}{4030}"
} | {
"index": 12491,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $(2x-1)^{5}=a\\_{0}+a\\_{1}x+a\\_{2}x^{2}+…+a\\_{5}x^{5}$, find the values of:\n\n(1) $a\\_{0}+a\\_{1}+a\\_{2}+a\\_{3}+a\\_{4}$,\n\n(2) $|a\\_{0}|+|a\\_{1}|+|a\\_{2}|+|a\\_{3}|+|a\\_{4}|+|a\\_{5}|$,\n\n(3) $a\\_{1}+a\\_{3}+a\\_{5}$,\n\n(4) $(a\\_{0}+a\\_{2}+a\\_{4})^{2}-(... | math | {
"style": "rule",
"ground_truth": "-243"
} | {
"index": 12492,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $(3-2x)^{5}=a_{0}+a_{1}x+a_{2}x^{2}+…+a_{5}x^{5}$, find the value of $a_{0}+a_{1}+2a_{2}+…+5a_{5}$."
}
] | math | {
"style": "rule",
"ground_truth": "233"
} | {
"index": 12493,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $(625^{\\log_2 250})^{\\frac{1}{4}}$, find the value of the expression."
}
] | math | {
"style": "rule",
"ground_truth": "250"
} | {
"index": 12494,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $(a+i)i=b+ai$, solve for $|a+bi|$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 12495,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $(b_1, b_2, ... b_7)$ be a list of the first 7 even positive integers such that for each $2 \\le i \\le 7$, either $b_i + 2$ or $b_i - 2$ or both appear somewhere before $b_i$ in the list, determine the number of such lists."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 12496,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $(b_1, b_2, ..., b_{12})$ is a list of the first 12 positive integers, where for each $2 \\leq i \\leq 12$, either $b_i + 1$, $b_i - 1$, or both appear somewhere in the list before $b_i$, and all even integers precede any of their immediate consecutive odd integers, find ... | math | {
"style": "rule",
"ground_truth": "2048"
} | {
"index": 12497,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $(x+1)^4(x+4)^8 = a + a_1(x+3) + a_2(x+3)^2 + \\ldots + a_{12}(x+3)^{12}$, find the value of $a_2 + a_4 + \\ldots + a_{12}$."
}
] | math | {
"style": "rule",
"ground_truth": "112"
} | {
"index": 12498,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given $(x^2+1)(2x+1)^9 = a_0 + a_1(x+2) + a_2(x+2)^2 + \\ldots + a_{11}(x+2)^{11}$, find the value of $a_0 + a_1 + a_2 + \\ldots + a_{11}$."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 12499,
"solution": ""
} |
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