data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Two particles move along the sides of a right $\\triangle ABC$ with $\\angle B = 90^\\circ$ in the direction \\[A\\Rightarrow B\\Rightarrow C\\Rightarrow A,\\] starting simultaneously. One starts at $A$ moving at speed $v$, the other starts at $C$ moving at speed $2v$. The midp... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 36500,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two passenger trains, A and B, are 150 meters and 200 meters long, respectively. They are traveling in opposite directions on parallel tracks. A passenger on train A measures that train B passes by his window in 10 seconds. How long does a passenger on train B see train A pass ... | math | {
"style": "rule",
"ground_truth": "7.5"
} | {
"index": 36501,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two pedestrians departed simultaneously from point A in the same direction. The first pedestrian met a tourist heading towards point A 20 minutes after leaving point A, and the second pedestrian met the tourist 5 minutes after the first pedestrian. The tourist arrived at point ... | math | {
"style": "rule",
"ground_truth": "15/8"
} | {
"index": 36502,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two people agreed to meet at a specific location between 12 PM and 1 PM. The condition is that the first person to arrive will wait for the second person for 15 minutes and then leave. What is the probability that these two people will meet if each of them chooses their moment ... | math | {
"style": "rule",
"ground_truth": "7/16"
} | {
"index": 36503,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two people are flipping a coin: one flipped it 10 times, and the other 11 times.\n\nWhat is the probability that the second person gets more heads than the first person?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 36504,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two people are flipping a coin: one flipped it 10 times, and the other flipped it 11 times. Find the probability that the second person gets more heads than the first person."
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"index": 36505,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two people are flipping a coin: one flipped it 10 times, and the other flipped it 11 times. Find the probability that the second person got heads more times than the first person."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 36506,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two people are playing \"Easter egg battle.\" In front of them is a large basket of eggs. They randomly pick one egg each and hit them against each other. One of the eggs breaks, the defeated player takes a new egg, and the winner keeps their egg for the next round (the outcome... | math | {
"style": "rule",
"ground_truth": "11/12"
} | {
"index": 36507,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two people are playing a game. One person thinks of a ten-digit number, and the other can ask questions about which digits are in specific sets of positions in the number's sequence. The first person answers the questions without indicating which digits are in which exact posit... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 36508,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two people are selected from each of the two groups. Group A has 5 boys and 3 girls, and Group B has 6 boys and 2 girls. Calculate the number of ways to select 4 people such that exactly 1 girl is included."
}
] | math | {
"style": "rule",
"ground_truth": "345"
} | {
"index": 36509,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two people are tossing a coin: one tossed it 10 times, and the other tossed it 11 times.\n\nWhat is the probability that the second person's coin landed on heads more times than the first person's coin?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 36510,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two people, A and B, alternately pick distinct numbers from the set \\(\\{0, 1, \\cdots, 81\\}\\). A picks first, and each person picks one number per turn. After all 82 numbers are picked, let \\(S_A\\) and \\(S_B\\) be the sums of the numbers chosen by A and B, respectively. ... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"index": 36511,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two people, A and B, are collaborating to type a document. Initially, A types 100 characters per minute, and B types 200 characters per minute. When they reach half of the total amount, A's speed triples while B takes a 5-minute break and then continues at the original speed. B... | math | {
"style": "rule",
"ground_truth": "18000"
} | {
"index": 36512,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two people, A and B, are working together to type a document. Initially, A types 100 characters per minute, and B types 200 characters per minute. When they have completed half of the document, A's typing speed triples, while B takes a 5-minute break and then continues typing a... | math | {
"style": "rule",
"ground_truth": "18000"
} | {
"index": 36513,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two people, A and B, form a team called \"Star Team\" to participate in a guessing idiom activity. In each round, A and B each guess an idiom. It is known that the probability of A guessing correctly in each round is $\\frac{2}{3}$, and the probability of B guessing correctly i... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{3}"
} | {
"index": 36514,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two people, A and B, participate in a general knowledge competition, with a total of 4 different questions, including 2 multiple-choice questions and 2 true/false questions. A and B each draw one question (without repetition). \n$(1)$ What is the probability that A draws a mul... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{6}"
} | {
"index": 36515,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two people, A and B, play a \"guess the number\" game using a fair six-sided die (the faces of the die are numbered $1, 2, \\cdots, 6$). Each person independently thinks of a number on the die, denoted as $a$ and $b$. If $|a - b| \\leqslant 1$, they are said to be \"in sync.\" ... | math | {
"style": "rule",
"ground_truth": "4/9"
} | {
"index": 36516,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two people, A and B, play a guessing game. First, A thinks of a number denoted as $a$, then B guesses the number A thought of, denoting B's guess as $b$. Both $a$ and $b$ belong to the set $\\{0,1,2,…,9\\}$. If $|a-b|=1$, then A and B are said to have a \"telepathic connection\... | math | {
"style": "rule",
"ground_truth": "\\dfrac {9}{50}"
} | {
"index": 36517,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two people, A and B, start from the same point on a 300-meter circular track and run in opposite directions. A runs at 2 meters per second, and B runs at 4 meters per second. When they first meet, A turns around and runs back. When A and B meet again, B turns around and runs ba... | math | {
"style": "rule",
"ground_truth": "250"
} | {
"index": 36518,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two people, A and B, take turns to draw candies from a bag. A starts by taking 1 candy, then B takes 2 candies, A takes 4 candies next, then B takes 8 candies, and so on. This continues. When the number of candies remaining in the bag is less than the number they are supposed t... | math | {
"style": "rule",
"ground_truth": "260"
} | {
"index": 36519,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two people, A and B, visit the \"2011 Xi'an World Horticultural Expo\" together. They agree to independently choose 4 attractions from numbered attractions 1 to 6 to visit, spending 1 hour at each attraction. Calculate the probability that they will be at the same attraction du... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{6}"
} | {
"index": 36520,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two people, Person A and Person B, start at the same time from point $A$ to point $B$: Person A is faster than Person B. After reaching point $B$, Person A doubles their speed and immediately returns to point $A$. They meet Person B at a point 240 meters from point $B$. After m... | math | {
"style": "rule",
"ground_truth": "420"
} | {
"index": 36521,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two places, A and B, are 12 km apart. Cars A and B depart from place A towards place B one after the other at a constant speed. Car A takes 15 minutes to travel from A to B, while Car B takes 10 minutes. If Car B departs 2 minutes later than Car A:\n1. Write the functions repre... | math | {
"style": "rule",
"ground_truth": "7.2"
} | {
"index": 36522,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two players in turn play a game. First Player has cards with numbers $2, 4, \\ldots, 2000$ while Second Player has cards with numbers $1, 3, \\ldots, 2001$ . In each his turn, a player chooses one of his cards and puts it on a table; the opponent sees it and puts his card ne... | math | {
"style": "rule",
"ground_truth": "999"
} | {
"index": 36523,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two players play a game where they are each given 10 indistinguishable units that must be distributed across three locations. (Units cannot be split.) At each location, a player wins at that location if the number of units they placed there is at least 2 more than the units of ... | math | {
"style": "rule",
"ground_truth": "1011"
} | {
"index": 36524,
"solution": "By stars and bars, the total number of distributions is $\\binom{12}{2}^{2}=66^{2}$. If no locations are won, either both distributions are identical or the difference between the two is $(1,0,-1)$, in some order. The first case has 66 possibilities. If the difference is $(1,0,-1)$, w... |
DeepScaleR | [
{
"role": "user",
"content": "Two players play alternately on a $ 5 \\times 5$ board. The first player always enters a $ 1$ into an empty square and the second player always enters a $ 0$ into an empty square. When the board is full, the sum of the numbers in each of the nine $ 3 \\times 3$ squares is calcu... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 36525,
"solution": "\nTo determine the largest score the first player can achieve, we must analyze how the scores are calculated and devise a strategy for maximizing the score in any \\(3 \\times 3\\) square.\n\nThe board is a \\(5 \\times 5\\) grid, so we have several overlapping \\(3 \\times 3\\) squar... |
DeepScaleR | [
{
"role": "user",
"content": "Two players take turns placing Xs and Os in the cells of a $9 \\times 9$ square (the first player places Xs, and their opponent places Os). At the end of the game, the number of rows and columns where there are more Xs than Os are counted as points for the first player. The num... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 36526,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two players, A and B, play a game called \"draw the joker card\". In the beginning, Player A has $n$ different cards. Player B has $n+1$ cards, $n$ of which are the same with the $n$ cards in Player A's hand, and the rest one is a Joker (different from all other $n$ cards). The... | math | {
"style": "rule",
"ground_truth": "n=32"
} | {
"index": 36527,
"solution": "We denote $a_{n}$ to be the probability that A wins the game when A has $n$ cards in the beginning. So we have $$ \\begin{equation*} a_{1}=\\frac{1}{2}+\\frac{1}{2} \\cdot \\frac{1}{2} a_{1} \\tag{1} \\end{equation*} $$ Therefore, $a_{1}=\\frac{2}{3}$. In addition, we have $$ \\begin{... |
DeepScaleR | [
{
"role": "user",
"content": "Two players, A and B, take turns shooting baskets. The probability of A making a basket on each shot is $\\frac{1}{2}$, while the probability of B making a basket is $\\frac{1}{3}$. The rules are as follows: A goes first, and if A makes a basket, A continues to shoot; otherwise... | math | {
"style": "rule",
"ground_truth": "\\frac{247}{432}"
} | {
"index": 36528,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two players, \\(A\\) and \\(B\\), play rock-paper-scissors continuously until player \\(A\\) wins 2 consecutive games. Suppose each player is equally likely to use each hand sign in every game. What is the expected number of games they will play?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 36529,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two points $A, B$ are randomly chosen on a circle with radius $100.$ For a positive integer $x$ , denote $P(x)$ as the probability that the length of $AB$ is less than $x$ . Find the minimum possible integer value of $x$ such that $\\text{P}(x) > \\frac{2}{3}$ ."
... | math | {
"style": "rule",
"ground_truth": "174"
} | {
"index": 36530,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two points $A(x_1, y_1)$ and $B(x_2, y_2)$ are chosen on the graph of $f(x) = \\ln x$, with $0 < x_1 < x_2$. The points $C$ and $D$ trisect $\\overline{AB}$, with $AC < CB$. Through $C$ a horizontal line is drawn to cut the curve at $E(x_3, y_3)$. Find $x_3$ if $x_1 = 1$ and $x... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 36531,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two points are chosen inside the square $\\{(x, y) \\mid 0 \\leq x, y \\leq 1\\}$ uniformly at random, and a unit square is drawn centered at each point with edges parallel to the coordinate axes. The expected area of the union of the two squares can be expressed as $\\frac{a}{... | math | {
"style": "rule",
"ground_truth": "1409"
} | {
"index": 36532,
"solution": "Let $\\Delta x$ and $\\Delta y$ be the positive differences between the $x$ coordinates and $y$ coordinates of the centers of the squares, respectively. Then, the length of the intersection of the squares along the $x$ dimension is $1-\\Delta x$, and likewise the length along the $y$ ... |
DeepScaleR | [
{
"role": "user",
"content": "Two points are drawn on each side of a square with an area of 81 square units, dividing the side into 3 congruent parts. Quarter-circle arcs connect the points on adjacent sides to create the figure shown. What is the length of the boundary of the bolded figure? Express your... | math | {
"style": "rule",
"ground_truth": "30.8"
} | {
"index": 36533,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two points are taken on the lateral edge of a pyramid, dividing the edge into three equal parts. Planes parallel to the base are drawn through these points. Find the volume of the section of the pyramid between these planes, given that the volume of the entire pyramid is 1."
... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{27}"
} | {
"index": 36534,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two poles, one 30 feet high and the other 90 feet high, are 150 feet apart. Halfway between the poles, there is an additional smaller pole of 10 feet high. The lines go from the top of each main pole to the foot of the opposite main pole, intersecting somewhere above the smalle... | math | {
"style": "rule",
"ground_truth": "22.5"
} | {
"index": 36535,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two positive integers $m$ and $n$ are both less than $500$ and $\\text{lcm}(m,n) = (m-n)^2$ . What is the maximum possible value of $m+n$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "840"
} | {
"index": 36536,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two positive integers $m$ and $n$ are chosen such that $m$ is the smallest positive integer with only two positive divisors and $n$ is the largest integer less than $100$ with exactly three positive divisors. What is $m+n$?"
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"index": 36537,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two positive integers $p,q \\in \\mathbf{Z}^{+}$ are given. There is a blackboard with $n$ positive integers written on it. A operation is to choose two same number $a,a$ written on the blackboard, and replace them with $a+p,a+q$. Determine the smallest $n$ so that such operati... | math | {
"style": "rule",
"ground_truth": "\\frac{p+q}{\\gcd(p,q)}"
} | {
"index": 36538,
"solution": "\nGiven two positive integers \\( p \\) and \\( q \\), we are to determine the smallest number \\( n \\) such that the operation of choosing two identical numbers \\( a, a \\) on the blackboard and replacing them with \\( a+p \\) and \\( a+q \\) can go on infinitely.\n\nTo solve this,... |
DeepScaleR | [
{
"role": "user",
"content": "Two positive integers \\( x \\) and \\( y \\) are such that:\n\n\\[ \\frac{2010}{2011} < \\frac{x}{y} < \\frac{2011}{2012} \\]\n\nFind the smallest possible value for the sum \\( x + y \\)."
}
] | math | {
"style": "rule",
"ground_truth": "8044"
} | {
"index": 36539,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two positive integers \\( x \\) and \\( y \\) have \\( xy=24 \\) and \\( x-y=5 \\). What is the value of \\( x+y \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 36540,
"solution": "The positive integer divisors of 24 are \\( 1,2,3,4,6,8,12,24 \\). The pairs of divisors that give a product of 24 are \\( 24 \\times 1,12 \\times 2,8 \\times 3 \\), and \\( 6 \\times 4 \\). We want to find two positive integers \\( x \\) and \\( y \\) whose product is 24 and whose di... |
DeepScaleR | [
{
"role": "user",
"content": "Two positive integers \\(m\\) and \\(n\\) are chosen such that \\(m\\) is the smallest positive integer with only two positive divisors, and \\(n\\) is the largest integer less than 200 that has exactly four positive divisors. What is \\(m+n\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "192"
} | {
"index": 36541,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two positive integers are written on the blackboard, one being 2002 and the other being a number less than 2002. If the arithmetic mean of the two numbers is an integer $m$, then the following operation can be performed: one of the numbers is erased and replaced by $m$. What is... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 36542,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two positive integers differ by $60$. The sum of their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers?"
}
] | math | {
"style": "rule",
"ground_truth": "156"
} | {
"index": 36543,
"solution": "Call the two integers $b$ and $b+60$, so we have $\\sqrt{b}+\\sqrt{b+60}=\\sqrt{c}$. Square both sides to get $2b+60+2\\sqrt{b^2+60b}=c$. Thus, $b^2+60b$ must be a square, so we have $b^2+60b=n^2$, and $(b+n+30)(b-n+30)=900$. The sum of these two factors is $2b+60$, so they must both ... |
DeepScaleR | [
{
"role": "user",
"content": "Two positive integers differ by 6 and their product is 135. What is the larger integer?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 36544,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two positive integers differ by 8 and their product is 168. What is the larger integer?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 36545,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two positive integers that only have 1 as a common factor are called coprime numbers. For example, 2 and 7 are coprime, as are 3 and 4. In any permutation of 2, 3, 4, 5, 6, 7, where each pair of adjacent numbers are coprime, there are a total of \\_\\_\\_\\_\\_\\_\\_\\_ differe... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 36546,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two positive rational numbers $x$ and $y$, when written in lowest terms, have the property that the sum of their numerators is 9 and the sum of their denominators is 10 . What is the largest possible value of $x+y$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "73 / 9"
} | {
"index": 36547,
"solution": "For fixed denominators $a<b$ (with sum 10), we maximize the sum of the fractions by giving the smaller denominator as large a numerator as possible: $8 / a+1 / b$. Then, if $a \\geq 2$, this quantity is at most $8 / 2+1 / 1=5$, which is clearly smaller than the sum we get by setting $... |
DeepScaleR | [
{
"role": "user",
"content": "Two positive real numbers have geometric mean $\\sqrt{3}$ and harmonic mean $\\frac{3}{2}.$ Enter the two numbers, separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "1,3"
} | {
"index": 36548,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two prime numbers that differ by exactly 2 are called twin primes. For example, 3 and 5 are twin primes, as are 29 and 31. In number theory research, twin primes are one of the most popular topics. If a pair of twin primes both do not exceed 200, what is the maximum sum of thes... | math | {
"style": "rule",
"ground_truth": "396"
} | {
"index": 36549,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two quadratic equations with unequal leading coefficients,\n$$\n(a-1) x^{2} - \\left(a^{2}+2\\right) x + \\left(a^{2}+2a\\right) = 0\n$$\nand\n$$\n(b-1) x^{2} - \\left(b^{2}+2\\right) x + \\left(b^{2}+2b\\right) = 0\n$$\n(where \\(a\\) and \\(b\\) are positive integers), have a... | math | {
"style": "rule",
"ground_truth": "256"
} | {
"index": 36550,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How many different convex cyclic quadrilaterals are there with integer sides and perimeter equal to 32?\n$\\textbf{(A)}\\ 560 \\qquad \\textbf{(B)}\\ 564 \\qquad \\... | math | {
"style": "rule",
"ground_truth": "568"
} | {
"index": 36551,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two railway tracks intersect at a right angle. Two trains are simultaneously speeding towards the intersection point from different tracks: one from a station located 40 km from the intersection point, and the other from a station located 50 km away. The first train travels at ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 36552,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two random points are chosen on a segment and the segment is divided at each of these two points. Of the three segments obtained, find the probability that the largest segment is more than three times longer than the smallest segment."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{27}{35}"
} | {
"index": 36553,
"solution": "We interpret the problem with geometric probability. Let the three segments have lengths $x, y, 1-x-y$ and assume WLOG that $x \\geq y \\geq 1-x-y$. The every possible $(x, y)$ can be found in the triangle determined by the points $\\left(\\frac{1}{3}, \\frac{1}{3}\\right),\\left(\\fr... |
DeepScaleR | [
{
"role": "user",
"content": "Two rays with common endpoint $O$ form a $30^\\circ$ angle. Point $A$ lies on one ray, point $B$ on the other ray, and $AB=1$. What is the maximum possible length of $\\overline{OB}$?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 36554,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two real numbers $a$ and $b$ are chosen independently and uniformly at random from the interval $(0, 75)$. Let $O$ and $P$ be two points on the plane with $OP = 200$. Let $Q$ and $R$ be on the same side of line $OP$ such that the degree measures of $\\angle POQ$ and $\\angle PO... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"index": 36555,
"solution": "Noting that $\\angle OQP$ and $\\angle ORP$ are right angles, we realize that we can draw a semicircle with diameter $\\overline{OP}$ and points $Q$ and $R$ on the semicircle. Since the radius of the semicircle is $100$, if $\\overline{QR} \\leq 100$, then $\\overarc{QR}$ must be less... |
DeepScaleR | [
{
"role": "user",
"content": "Two real numbers $a$ and $b$ satisfy $a+b=8$ and $a^3+b^3=172$. Compute the value of $ab$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{85}{6}"
} | {
"index": 36556,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two real numbers $x$ and $y$ are such that $8 y^{4}+4 x^{2} y^{2}+4 x y^{2}+2 x^{3}+2 y^{2}+2 x=x^{2}+1$. Find all possible values of $x+2 y^{2}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 36557,
"solution": "Writing $a=x+2 y^{2}$, the given quickly becomes $4 y^{2} a+2 x^{2} a+a+x=x^{2}+1$. We can rewrite $4 y^{2} a$ for further reduction to $a(2 a-2 x)+2 x^{2} a+a+x=x^{2}+1$, or $$\\begin{equation*} 2 a^{2}+\\left(2 x^{2}-2 x+1\\right) a+\\left(-x^{2}+x-1\\right)=0 \\tag{*} \\end{equatio... |
DeepScaleR | [
{
"role": "user",
"content": "Two real numbers $x$ and $y$ satisfy $x-y=4$ and $x^3-y^3=28$. Compute $xy$."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 36558,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two real numbers are selected independently at random from the interval $[-15, 15]$. The product of those numbers is considered only if both numbers are outside the interval $[-5, 5]$. What is the probability that the product of those numbers, when considered, is greater than z... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{9}"
} | {
"index": 36559,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two real numbers are selected independently at random from the interval $[-20, 10]$. What is the probability that the product of those numbers is greater than zero?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{9}"
} | {
"index": 36560,
"solution": "1. **Define the intervals for positive and negative numbers**:\n - The interval $[-20, 10]$ can be split into two parts: $[-20, 0)$ for negative numbers and $(0, 10]$ for positive numbers.\n - The length of the interval $[-20, 0)$ is $20$ units.\n - The length of the interval $(... |
DeepScaleR | [
{
"role": "user",
"content": "Two reals $a$ and $b$ are such that $a+b=7$ and $a^3+b^3=91$. Compute $ab$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 36561,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two reals \\( x \\) and \\( y \\) are such that \\( x-y=4 \\) and \\( x^{3}-y^{3}=28 \\). Compute \\( x y \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 36562,
"solution": "We have \\( 28=x^{3}-y^{3}=(x-y)\\left(x^{2}+x y+y^{2}\\right)=(x-y)\\left((x-y)^{2}+3 x y\\right)=4 \\cdot(16+3 x y) \\), from which \\( x y=-3 \\)."
} |
DeepScaleR | [
{
"role": "user",
"content": "Two rectangles have integer dimensions, and both have a perimeter of 144 cm. What is the greatest possible difference between the areas of two such rectangles?"
}
] | math | {
"style": "rule",
"ground_truth": "1225"
} | {
"index": 36563,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two rectangles, each measuring 7 cm in length and 3 cm in width, overlap to form the shape shown on the right. What is the perimeter of this shape in centimeters?"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 36564,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two rectangles, one $8 \\times 10$ and the other $12 \\times 9$, are overlaid as shown in the picture. The area of the black part is 37. What is the area of the gray part? If necessary, round the answer to 0.01 or write the answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"index": 36565,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two rectangles, one measuring $2 \\times 4$ and another measuring $3 \\times 5$, along with a circle of diameter 3, are to be contained within a square. The sides of the square are parallel to the sides of the rectangles and the circle must not overlap any rectangle at any poin... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"index": 36566,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two rectangles, one measuring \\(8 \\times 10\\) and the other \\(12 \\times 9\\), are overlaid as shown in the picture. The area of the black part is 37. What is the area of the gray part? If necessary, round the answer to 0.01 or write the answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"index": 36567,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two regular polygons have the same perimeter. If the first has 38 sides and a side length twice as long as the second, how many sides does the second have?"
}
] | math | {
"style": "rule",
"ground_truth": "76"
} | {
"index": 36568,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two regular tetrahedra are inscribed in a cube in such a way that four vertices of the cube serve as the vertices of one tetrahedron, and the remaining four vertices of the cube serve as the vertices of the other. What fraction of the volume of the cube is occupied by the volum... | math | {
"style": "rule",
"ground_truth": "1/6"
} | {
"index": 36569,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two regular tetrahedrons $A$ and $B$ are made with the 8 vertices of a unit cube. (this way is unique)\r\n\r\nWhat's the volume of $A\\cup B$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"index": 36570,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two riders simultaneously departed from points \\( A \\) and \\( C \\) towards point \\( B \\). Despite the fact that \\( C \\) was 20 km farther from \\( B \\) than \\( A \\) was from \\( B \\), both riders arrived at \\( B \\) at the same time. Find the distance from \\( C \\... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 36571,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two right circular cones with vertices facing down as shown in the figure below contain the same amount of liquid. The radii of the tops of the liquid surfaces are $3$ cm and $6$ cm. Into each cone is dropped a spherical marble of radius $1$ cm, which sinks to the bottom and is... | math | {
"style": "rule",
"ground_truth": "4:1"
} | {
"index": 36572,
"solution": "\n#### Initial Scenario\nLet the heights of the narrow cone and the wide cone be \\( h_1 \\) and \\( h_2 \\), respectively. The volumes of the cones before the marbles are dropped are given by:\n- Narrow Cone: \\( V_1 = \\frac{1}{3} \\pi r_1^2 h_1 = \\frac{1}{3} \\pi (3)^2 h_1 = 3\\pi... |
DeepScaleR | [
{
"role": "user",
"content": "Two right triangles $\\triangle ABC$ and $\\triangle ABD$ share a side $AB$ in such way that $AB=8$, $BC=12$, and $BD=10$. Let $E$ be a point on $BC$ such that $BE=9$. Determine the area of $\\triangle ABE$."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 36573,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two right triangles \\( \\triangle AXY \\) and \\( \\triangle BXY \\) have a common hypotenuse \\( XY \\) and side lengths (in units) \\( AX=5 \\), \\( AY=10 \\), and \\( BY=2 \\). Sides \\( AY \\) and \\( BX \\) intersect at \\( P \\). Determine the area (in square units) of \... | math | {
"style": "rule",
"ground_truth": "25/3"
} | {
"index": 36574,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two right triangles share a side as follows: Triangle ABC and triangle ABD have AB as their common side. AB = 8 units, AC = 12 units, and BD = 8 units. There is a rectangle BCEF where point E is on line segment BD and point F is directly above E such that CF is parallel to AB. ... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 36575,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two right triangles share a side as follows: [asy]\npair pA, pB, pC, pD, pE;\npA = (0, 0);\npB = pA + 6 * dir(0);\npC = pA + 10 * dir(90);\npD = pB + 6 * dir(90);\npE = (6 * pA + 10 * pD) / 16;\ndraw(pA--pB--pC--pA);\ndraw(pA--pB--pD--pA);\nlabel(\"$A$\", pA, SW);\nlabel(\"$B$\... | math | {
"style": "rule",
"ground_truth": "\\frac{75}{4}"
} | {
"index": 36576,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two right triangles share a side such that the common side AB has a length of 8 units, and both triangles ABC and ABD have respective heights from A of 8 units each. Calculate the area of triangle ABE where E is the midpoint of side CD and CD is parallel to AB. Assume that side... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 36577,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two right triangles, $ABC$ and $ACD$, are joined as shown. Squares are drawn on four of the sides. The areas of three of the squares are 25, 49, and 64 square units. What is the number of square units in the area of the fourth square?\n\nNote that the diagram is not provided, b... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 36578,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two right triangles, $ABC$ and $ACD$, are joined as shown. Squares are drawn on four of the sides. The areas of three of the squares are 9, 16 and 36 square units. What is the number of square units in the area of the fourth square?\n\nNote that the diagram is not drawn to scal... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"index": 36579,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two right triangles, $ABC$ and $ACD$, are joined at side $AC$. Squares are drawn on four of the sides. The areas of three of the squares are 25, 49, and 64 square units. Determine the number of square units in the area of the fourth square."
}
] | math | {
"style": "rule",
"ground_truth": "138"
} | {
"index": 36580,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two runners start simultaneously and run on a circular track in opposite directions at constant speeds. One runner completes a lap in 5 minutes, and the other runner completes a lap in 8 minutes. Find the number of distinct meeting points of the runners on the track if they run... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 36581,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two runners started simultaneously in the same direction from the same point on a circular track. The first runner, moving ahead, caught up with the second runner at the moment when the second runner had only run half a lap. From that moment, the second runner doubled their spe... | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"index": 36582,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two sectors of a circle of radius $10$ overlap as shown, with centers at points $A$ and $B$. Each sector subtends an angle of $45^\\circ$. Determine the area of the overlapping region.\n\n[asy]\ndraw((0,0)--(7.07,-7.07)--(14.14,0)--(7.07,7.07)--cycle,black+linewidth(1));\nfilld... | math | {
"style": "rule",
"ground_truth": "25\\pi - 50\\sqrt{2}"
} | {
"index": 36583,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two sectors of a circle of radius $12$ are placed side by side, as shown. Determine the $\\textit{area}$ of figure $ABCD.$ [asy]\ndraw((0,0)--(12,0)..(10.3923,6)..(6,10.3923)--(-6,10.3923)..(-4.3923,4.3923)..(0,0),black+linewidth(1));\ndraw((0,0)--(6,10.3923),black+linewidth(1)... | math | {
"style": "rule",
"ground_truth": "48\\pi"
} | {
"index": 36584,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two sectors of a circle of radius $12$ overlap as shown, with $P$ and $R$ as the centers of the respective circles. Determine the area of the shaded region.\n\n[asy]\ndraw((0,0)--(10.3923,-6)--(20.7846,0)--(10.3923,6)--cycle,black+linewidth(1));\nfilldraw((10.3923,6)..(12,0)..(... | math | {
"style": "rule",
"ground_truth": "48\\pi-72\\sqrt{3}"
} | {
"index": 36585,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two sectors of a circle of radius $15$ overlap in the same manner as the original problem, with $P$ and $R$ as the centers of the respective circles. The angle at the centers for both sectors is now $45^\\circ$. Determine the area of the shaded region."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{225\\pi - 450\\sqrt{2}}{4}"
} | {
"index": 36586,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two semicircles of diameters 2 and 4 are placed such that the smaller is inside the larger one and their diameters are along the same line. A square of side 1 is inscribed in the larger semicircle such that one of its sides is along the diameter of the semicircles. Calculate th... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}\\pi - 1"
} | {
"index": 36587,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two semicircles, each with radius \\(\\sqrt{2}\\), are tangent to each other. If \\( AB \\parallel CD \\), determine the length of segment \\( AD \\)."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{2}"
} | {
"index": 36588,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two siblings, Andy and Alexa, bake 24 cookies. Alexa eats some number of cookies that is a positive multiple of the number of cookies eaten by her brother. If the siblings finish all 24 cookies, then what is the maximum number of cookies that the brother, Andy, could have eaten... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 36589,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two sides of a regular $n$-gon are extended to meet at a $28^{\\circ}$ angle. What is the smallest possible value for $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 36590,
"solution": "We note that if we inscribe the $n$-gon in a circle, then according to the inscribed angle theorem, the angle between two sides is $\\frac{1}{2}$ times some $x-y$, where $x$ and $y$ are integer multiples of the arc measure of one side of the $n$-gon. Thus, the angle is equal to $\\fra... |
DeepScaleR | [
{
"role": "user",
"content": "Two sides of a right triangle have the lengths 4 and 5. What is the product of the possible lengths of the third side? Express the product as a decimal rounded to the nearest tenth."
}
] | math | {
"style": "rule",
"ground_truth": "19.2"
} | {
"index": 36591,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two sides of a right triangle have the lengths 8 and 15. What is the product of the possible lengths of the third side? Express the product as a decimal rounded to the nearest tenth."
}
] | math | {
"style": "rule",
"ground_truth": "215.7"
} | {
"index": 36592,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two sides of a triangle are $8 \\mathrm{dm}$ and $5 \\mathrm{dm}$; the angle opposite to the first side is twice as large as the angle opposite to the second side. What is the length of the third side of the triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "7.8"
} | {
"index": 36593,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two sides of a triangle measure 4 cm and 9 cm. What is the greatest integer number of centimeters that could be the third side of this triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 36594,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two sides of an isosceles triangle are 10 inches and 20 inches. If the shortest side of a similar triangle is 50 inches, what is the perimeter of the larger triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "250\\text{ inches}"
} | {
"index": 36595,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two similar right triangles have areas of 6 square inches and 150 square inches. The length of the hypotenuse of the smaller triangle is 5 inches. What is the sum of the lengths of the legs of the larger triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 36596,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two skaters, Allie and Billie, are at points $A$ and $B$, respectively, on a flat, frozen lake. The distance between $A$ and $B$ is $100$ meters. Allie leaves $A$ and skates at a speed of $8$ meters per second on a straight line that makes a $60^\\circ$ angle with $AB$. At the ... | math | {
"style": "rule",
"ground_truth": "160"
} | {
"index": 36597,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Two skaters, Allie and Billie, are at points $A$ and $B$, respectively, on a flat, frozen lake. The distance between $A$ and $B$ is $100$ meters. Allie leaves $A$ and skates at a speed of $8$ meters per second on a straight line that makes a $60^\\circ$ angle with $AB$. At the ... | math | {
"style": "rule",
"ground_truth": "160"
} | {
"index": 36598,
"solution": "Label the point of intersection as $C$. Since $d = rt$, $AC = 8t$ and $BC = 7t$. According to the law of cosines,\n[asy] pointpen=black; pathpen=black+linewidth(0.7); pair A=(0,0),B=(10,0),C=16*expi(pi/3); D(B--A); D(A--C); D(B--C,dashed); MP(\"A\",A,SW);MP(\"B\",B,SE);MP(\"C\",C,N);M... |
DeepScaleR | [
{
"role": "user",
"content": "Two skiers started from the same point one after another with an interval of 9 minutes. The second skier caught up with the first one 9 km from the starting point. After reaching the “27 km” mark, the second skier turned back and met the first skier at a distance of 2 km from t... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 36599,
"solution": ""
} |
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