data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Alice and Bob play a game on a circle with 8 marked points. Alice places an apple beneath one of the points, then picks five of the other seven points and reveals that none of them are hiding the apple. Bob then drops a bomb on any of the points, and destroys the apple if he dr... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 4400,
"solution": "Let the points be $0, \\ldots, 7(\\bmod 8)$, and view Alice's reveal as revealing the three possible locations of the apple. If Alice always picks $0,2,4$ and puts the apple randomly at 0 or 4 , by symmetry Bob cannot achieve more than $\\frac{1}{2}$. Here's a proof that $\\frac{1}{2}$... |
DeepScaleR | [
{
"role": "user",
"content": "Alice and Bob play a game with a baseball. On each turn, if Alice has the ball, there is a 1/2 chance that she will toss it to Bob and a 1/2 chance that she will keep the ball. If Bob has the ball, there is a 2/5 chance that he will toss it to Alice, and if he doesn't toss it t... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{20}"
} | {
"index": 4401,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice and Bob play a similar game with a basketball. On each turn, if Alice has the ball, there is a 2/3 chance that she will toss it to Bob and a 1/3 chance that she will keep the ball. If Bob has the ball, there is a 1/4 chance that he will toss it to Alice, and a 3/4 chance ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{18}"
} | {
"index": 4402,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice and Bob play on a $20 \\times 20$ grid. Initially, all the cells are empty. Alice starts and the two players take turns placing stones on unoccupied cells. On her turn, Alice places a red stone on an empty cell that is not at a distance of $\\sqrt{5}$ from any other cell ... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 4403,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice and Bob play the following \"point guessing game.\" First, Alice marks an equilateral triangle $A B C$ and a point $D$ on segment $B C$ satisfying $B D=3$ and $C D=5$. Then, Alice chooses a point $P$ on line $A D$ and challenges Bob to mark a point $Q \\neq P$ on line $A ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}, 1, \\frac{3 \\sqrt{3}}{5}"
} | {
"index": 4404,
"solution": "First, if $P=A$ then clearly Bob cannot choose a $Q$. So we can have $B P: P C=1$. Otherwise, we need $A P$ to be tangent to the Apollonius Circle. The key claim is that $A B=A C=A P$. To see why, simply note that since $B$ and $C$ are inverses with respect to the Apollonius Circle, we... |
DeepScaleR | [
{
"role": "user",
"content": "Alice and Bob stand atop two different towers in the Arctic. Both towers are a positive integer number of meters tall and are a positive (not necessarily integer) distance away from each other. One night, the sea between them has frozen completely into reflective ice. Alice shi... | math | {
"style": "rule",
"ground_truth": "7,15"
} | {
"index": 4405,
"solution": "Let Alice's tower be of a height $a$, and Bob's tower a height $b$. Reflect the diagram over the ice to obtain an isosceles trapezoid. Then we get that by Ptolemy's Theorem, $4 a b=26^{2}-16^{2}=4 \\cdot 105$, thus $a b=105$. Hence $a \\in\\{1,3,5,7,15,21,35,105\\}$. But $\\max (a, b) ... |
DeepScaleR | [
{
"role": "user",
"content": "Alice and Bob take turns removing balls from a bag containing 10 black balls and 10 white balls, with Alice going first. Alice always removes a black ball if there is one, while Bob removes one of the remaining balls uniformly at random. Once all balls have been removed, the ex... | math | {
"style": "rule",
"ground_truth": "4519"
} | {
"index": 4406,
"solution": "Suppose $a$ is the number of black balls and $b$ is the number of white balls, and let $E_{a, b}$ denote the expected number of black balls Bob has once all the balls are removed with Alice going first. Then we want to find $E_{10,10}$. It is evident that if $E_{0, b}=0$. Also, since B... |
DeepScaleR | [
{
"role": "user",
"content": "Alice and Carol each have a rectangular sheet of paper. Alice has a sheet of paper measuring 10 inches by 12 inches and rolls it into a tube by taping the two 10-inch sides together. Carol rolls her sheet, which measures 8 inches by 15 inches, by taping the two 15-inch sides to... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 4407,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice and the Cheshire Cat play a game. At each step, Alice either (1) gives the cat a penny, which causes the cat to change the number of (magic) beans that Alice has from $n$ to $5n$ or (2) gives the cat a nickel, which causes the cat to give Alice another bean. Alice wins (a... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 4408,
"solution": "Consider the number of beans Alice has in base 5. Note that $2008=31013_{5}, 42=132_{5}$, and $100=400_{5}$. Now, suppose Alice has $d_{k} \\cdots d_{2} d_{1}$ beans when she wins; the conditions for winning mean that these digits must satisfy $d_{2} d_{1}=32, d_{k} \\cdots d_{3} \\geq... |
DeepScaleR | [
{
"role": "user",
"content": "Alice and the White Rabbit left the Rabbit's house together at noon to go to the Duchess's reception. Halfway through, the Rabbit remembered that he forgot his gloves and fan, and ran back home at twice the speed he had been walking with Alice. Grabbing the gloves and fan, he t... | math | {
"style": "rule",
"ground_truth": "12:40"
} | {
"index": 4409,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice buys three burgers and two sodas for $\\$3.20$, and Bill buys two burgers and a soda for $\\$2.00$. How many cents does a burger cost?"
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 4410,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice celebrated her birthday on Friday, March 15 in the year 2012. Determine the next year when her birthday will next fall on a Monday.\nA) 2021\nB) 2022\nC) 2023\nD) 2024\nE) 2025"
}
] | math | {
"style": "rule",
"ground_truth": "2025"
} | {
"index": 4411,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice chose five positive integers and found that their product was even. What is the maximum number of odd integers she could have chosen?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 4412,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice conducted a survey among a group of students regarding their understanding of snakes. She found that $92.3\\%$ of the students surveyed believed that snakes are venomous. Of the students who believed this, $38.4\\%$ erroneously thought that all snakes are venomous. Knowin... | math | {
"style": "rule",
"ground_truth": "88"
} | {
"index": 4413,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice cycled 240 miles in 4 hours, 30 minutes. Then, she cycled another 300 miles in 5 hours, 15 minutes. What was Alice's average speed in miles per hour for her entire journey?"
}
] | math | {
"style": "rule",
"ground_truth": "55.38"
} | {
"index": 4414,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice draws three cards from a standard 52-card deck with replacement. Ace through 10 are worth 1 to 10 points respectively, and the face cards King, Queen, and Jack are each worth 10 points. The probability that the sum of the point values of the cards drawn is a multiple of 1... | math | {
"style": "rule",
"ground_truth": "26597"
} | {
"index": 4415,
"solution": "The probability that all three cards drawn are face cards is $\\left(\\frac{3}{13}\\right)^{3}=\\frac{27}{2197}$. In that case, the sum is 30 and therefore a multiple of 10 . Otherwise, one of the cards is not a face card, so its point value $p$ is drawn uniformly from values from 1 to... |
DeepScaleR | [
{
"role": "user",
"content": "Alice drew a regular $2021$-gon in the plane. Bob then labeled each vertex of the $2021$-gon with a real number, in such a way that the labels of consecutive vertices differ by at most $1$. Then, for every pair of non-consecutive vertices whose labels differ by at most $1$, Ali... | math | {
"style": "rule",
"ground_truth": "2018"
} | {
"index": 4416,
"solution": "\nTo solve this problem, we need to find the least possible number of diagonals, \\( d \\), that Alice can draw given Bob's labeling constraints on the vertices of a regular 2021-gon.\n\n### Step 1: Understanding the Problem\n\nAlice has a regular 2021-gon, and Bob labels each vertex w... |
DeepScaleR | [
{
"role": "user",
"content": "Alice has $24$ apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least two apples?"
}
] | math | {
"style": "rule",
"ground_truth": "190"
} | {
"index": 4417,
"solution": "To solve the problem, we need to find the number of ways Alice, Becky, and Chris can each receive at least two apples from a total of 24 apples. Let's denote the number of apples received by Alice, Becky, and Chris as $a$, $b$, and $c$ respectively. The condition given is that each per... |
DeepScaleR | [
{
"role": "user",
"content": "Alice has 10 green marbles and 5 purple marbles in a bag. She removes a marble at random, records the color, puts it back, and then repeats this process until she has withdrawn 8 marbles. What is the probability that exactly four of the marbles that she removes are green? Expre... | math | {
"style": "rule",
"ground_truth": "0.171"
} | {
"index": 4418,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice has 4 sisters and 6 brothers. Given that Alice's sister Angela has S sisters and B brothers, calculate the product of S and B."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 4419,
"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."
}
] | math | {
"style": "rule",
"ground_truth": "0.4"
} | {
"index": 4420,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice is bored in class, so she thinks of a positive integer. Every second after that, she subtracts from her current number its smallest prime divisor, possibly itself. After 2022 seconds, she realizes that her number is prime. Find the sum of all possible values of her initia... | math | {
"style": "rule",
"ground_truth": "8093"
} | {
"index": 4421,
"solution": "Let $a_{k}$ denote Alice's number after $k$ seconds, and let $p_{k}$ be the smallest prime divisor of $a_{k}$. We are given that $a_{2022}$ is prime, and want to find $a_{0}$. If $a_{0}$ is even, then $a_{n+1}=a_{n}-2$, since every $a_{n}$ is even. Then we need $a_{2022}=2$, so $a_{0}=... |
DeepScaleR | [
{
"role": "user",
"content": "Alice is jogging north at a speed of 6 miles per hour, and Tom is starting 3 miles directly south of Alice, jogging north at a speed of 9 miles per hour. Moreover, assume Tom changes his path to head north directly after 10 minutes of eastward travel. How many minutes after thi... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 4422,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice is making a batch of cookies and needs $2\\frac{1}{2}$ cups of sugar. Unfortunately, her measuring cup holds only $\\frac{1}{4}$ cup of sugar. How many times must she fill that cup to get the correct amount of sugar?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 4423,
"solution": "1. **Convert the mixed number to an improper fraction**: Alice needs $2\\frac{1}{2}$ cups of sugar. This can be converted to an improper fraction:\n \\[\n 2\\frac{1}{2} = 2 + \\frac{1}{2} = \\frac{4}{2} + \\frac{1}{2} = \\frac{5}{2}\n \\]\n\n2. **Set up the division problem**: Al... |
DeepScaleR | [
{
"role": "user",
"content": "Alice is once again very bored in class. On a whim, she chooses three primes $p, q, r$ independently and uniformly at random from the set of primes of at most 30. She then calculates the roots of $p x^{2}+q x+r$. What is the probability that at least one of her roots is an inte... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{200}"
} | {
"index": 4424,
"solution": "Since all of the coefficients are positive, any root $x$ must be negative. Moreover, by the rational root theorem, in order for $x$ to be an integer we must have either $x=-1$ or $x=-r$. So we must have either $p r^{2}-q r+r=0 \\Longleftrightarrow p r=q-1$ or $p-q+r=0$. Neither of thes... |
DeepScaleR | [
{
"role": "user",
"content": "Alice is sitting in a teacup ride with infinitely many layers of spinning disks. The largest disk has radius 5. Each succeeding disk has its center attached to a point on the circumference of the previous disk and has a radius equal to $2 / 3$ of the previous disk. Each disk sp... | math | {
"style": "rule",
"ground_truth": "18 \\pi"
} | {
"index": 4425,
"solution": "Suppose the center of the largest teacup is at the origin in the complex plane, and let $z=\\frac{2}{3} e^{\\pi i t / 6}$. The center of the second disk is at $5 e^{\\pi i t / 6}$ at time $t$; that is, \\frac{15}{2} z$. Then the center of the third disk relative to the center of the se... |
DeepScaleR | [
{
"role": "user",
"content": "Alice is thinking of a positive real number $x$, and Bob is thinking of a positive real number $y$. Given that $x^{\\sqrt{y}}=27$ and $(\\sqrt{x})^{y}=9$, compute $x y$."
}
] | math | {
"style": "rule",
"ground_truth": "16 \\sqrt[4]{3}"
} | {
"index": 4426,
"solution": "Note that $$27^{\\sqrt{y}}=\\left(x^{\\sqrt{y}}\\right)^{\\sqrt{y}}=x^{y}=(\\sqrt{x})^{2 y}=81$$ so $\\sqrt{y}=4 / 3$ or $y=16 / 9$. It follows that $x^{4 / 3}=27$ or $x=9 \\sqrt[4]{3}$. The final answer is $9 \\sqrt[4]{3} \\cdot 16 / 9=16 \\sqrt[4]{3}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice knows that $3$ red cards and $3$ black cards will be revealed to her one at a time in random order. Before each card is revealed, Alice must guess its color. If Alice plays optimally, the expected number of cards she will guess correctly is $\\frac{m}{n},$ where $m$ and $... | math | {
"style": "rule",
"ground_truth": "051"
} | {
"index": 4427,
"solution": "Denote by $N_{i,j}$ the optimal expected number of cards that Alice guesses correctly, where the number of cards are $i$ and $j \\ge i.$\n If $i = 0$ then Alice guesses correctly all cards, so $N_{0,j} = j.$\n If $j = i$ then Alice guesses next card with probability $\\frac {1}{2} \\im... |
DeepScaleR | [
{
"role": "user",
"content": "Alice needs to replace a light bulb located $10$ centimeters below the ceiling in her kitchen. The ceiling is $2.4$ meters above the floor. Alice is $1.5$ meters tall and can reach $46$ centimeters above the top of her head. Standing on a stool, she can just reach the light bul... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"index": 4428,
"solution": "1. **Convert all measurements to centimeters**: \n - The ceiling height is $2.4$ meters, which is $2.4 \\times 100 = 240$ centimeters.\n - Alice's height is $1.5$ meters, which is $1.5 \\times 100 = 150$ centimeters.\n - Alice can reach $46$ centimeters above her head.\n\n2. **Ca... |
DeepScaleR | [
{
"role": "user",
"content": "Alice picks a number uniformly at random from the first $5$ even positive integers, and Palice picks a number uniformly at random from the first $5$ odd positive integers. If Alice picks a larger number than Palice with probability $\\frac{m}{n}$ for relatively prime posi... | math | {
"style": "rule",
"ground_truth": "39"
} | {
"index": 4429,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice places a coin, heads up, on a table then turns off the light and leaves the room. Bill enters the room with 2 coins and flips them onto the table and leaves. Carl enters the room, in the dark, and removes a coin at random. Alice reenters the room, turns on the light and n... | math | {
"style": "rule",
"ground_truth": "3/5"
} | {
"index": 4430,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice places six ounces of coffee into a twelve-ounce cup and two ounces of coffee plus four ounces of cream into a second twelve-ounce cup. She then pours half the contents from the first cup into the second and, after stirring thoroughly, pours half the liquid in the second c... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{15}"
} | {
"index": 4431,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice plays a game where she rolls a fair eight-sided die each morning. If Alice rolls a number divisible by 3, she wears red; otherwise, she wears blue. If she rolls a perfect square (1, 4), then she rolls again. In a leap year, what is the expected number of times Alice will ... | math | {
"style": "rule",
"ground_truth": "488"
} | {
"index": 4432,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions?"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 4433,
"solution": "We need to consider different seating arrangements for Alice, as her position affects the seating of the others due to her restrictions with Bob and Carla.\n\n1. **Alice sits in the center chair (3rd position):**\n - The 2nd and 4th chairs must be occupied by Derek and Eric in either... |
DeepScaleR | [
{
"role": "user",
"content": "Alice sells an item at $10 less than the list price and receives $10\\%$ of her selling price as her commission. \nBob sells the same item at $20 less than the list price and receives $20\\%$ of his selling price as his commission. \nIf they both get the same commission, then t... | math | {
"style": "rule",
"ground_truth": "$30"
} | {
"index": 4434,
"solution": "Let $x$ be the list price of the item. \n\n1. **Alice's Selling Price and Commission:**\n - Alice sells the item at $x - 10$ dollars.\n - Her commission is $10\\%$ of her selling price, which is $0.10(x - 10)$.\n\n2. **Bob's Selling Price and Commission:**\n - Bob sells the item ... |
DeepScaleR | [
{
"role": "user",
"content": "Alice starts to make a list, in increasing order, of the positive integers that have a first digit of 2. She writes $2, 20, 21, 22, \\ldots$ but by the 1000th digit she (finally) realizes that the list would contain an infinite number of elements. Find the three-digit number fo... | math | {
"style": "rule",
"ground_truth": "216"
} | {
"index": 4435,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice starts with the number 0. She can apply 100 operations on her number. In each operation, she can either add 1 to her number, or square her number. After applying all operations, her score is the minimum distance from her number to any perfect square. What is the maximum s... | math | {
"style": "rule",
"ground_truth": "94"
} | {
"index": 4436,
"solution": "Note that after applying the squaring operation, Alice's number will be a perfect square, so she can maximize her score by having a large number of adding operations at the end. However, her scores needs to be large enough that the many additions do not bring her close to a larger squa... |
DeepScaleR | [
{
"role": "user",
"content": "Alice thinks of four positive integers $a \\leq b \\leq c \\leq d$ satisfying $\\{a b+c d, a c+b d, a d+b c\\}=\\{40,70,100\\}$. What are all the possible tuples $(a, b, c, d)$ that Alice could be thinking of?"
}
] | math | {
"style": "rule",
"ground_truth": "(1,4,6,16)"
} | {
"index": 4437,
"solution": "Since $a b \\cdot c d=a c \\cdot b d=a d \\cdot b c$, the largest sum among $a b+c d, a c+b d, a d+b c$ will be the one with the largest difference between the two quantities, so $a b+c d=100, a c+b d=70, a d+b c=40$. Consider the sum of each pair of equations, which gives $(a+b)(c+d)=... |
DeepScaleR | [
{
"role": "user",
"content": "Alice wants to compare the percentage increase in area when her pizza size increases first from an 8-inch pizza to a 10-inch pizza, and then from the 10-inch pizza to a 14-inch pizza. Calculate the percent increase in area for both size changes."
}
] | math | {
"style": "rule",
"ground_truth": "96\\%"
} | {
"index": 4438,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice wants to write down a list of prime numbers less than 100, using each of the digits 1, 2, 3, 4, and 5 once and no other digits. Which prime number must be in her list?"
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"index": 4439,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice writes 1001 letters on a blackboard, each one chosen independently and uniformly at random from the set $S=\\{a, b, c\\}$. A move consists of erasing two distinct letters from the board and replacing them with the third letter in $S$. What is the probability that Alice ca... | math | {
"style": "rule",
"ground_truth": "\\frac{3-3^{-999}}{4}"
} | {
"index": 4440,
"solution": "Let $n_{a}, n_{b}$, and $n_{c}$ be the number of $a$ 's, $b$ 's, and $c$ 's on the board, respectively. The key observation is that each move always changes the parity of all three of $n_{a}, n_{b}$, and $n_{c}$. Since the final configuration must have $n_{a}, n_{b}$, and $n_{c}$ equal... |
DeepScaleR | [
{
"role": "user",
"content": "Alice's favorite number has the following properties:\n\n\n- It has 8 distinct digits.\n- The digits are decreasing when read from left to right.\n- It is divisible by 180.\n\nWhat is Alice's favorite number?\n\n*Author: Anderson Wang*"
}
] | math | {
"style": "rule",
"ground_truth": "97654320"
} | {
"index": 4441,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice's favorite number is between $90$ and $150$. It is a multiple of $13$, but not a multiple of $4$. The sum of its digits should be a multiple of $4$. What is Alice's favorite number?"
}
] | math | {
"style": "rule",
"ground_truth": "143"
} | {
"index": 4442,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice's password consists of a two-digit number, followed by a symbol from the set {$!, @, #, $, %}, followed by another two-digit number. Calculate the probability that Alice's password consists of an even two-digit number followed by one of {$, %, @}, and another even two-dig... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{20}"
} | {
"index": 4443,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice, Bob, and Carol each start with $3. Every 20 seconds, a bell rings, at which time each of the players who has an odd amount of money must give $1 to one of the other two players randomly. What is the probability that after the bell has rung 100 times, each player has exac... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{13}"
} | {
"index": 4444,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice, Bob, and Carol play a game in which each of them chooses a real number between 0 and 1. The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{24}"
} | {
"index": 4445,
"solution": "Let $a$, $b$, and $c$ be the numbers that Alice, Bob, and Carol choose, respectively. Alice chooses $a$ uniformly from $[0,1]$, Bob chooses $b$ uniformly from $[\\frac{1}{2}, \\frac{2}{3}]$, and Carol aims to choose $c$ optimally.\n\nCarol wins if her number $c$ is between the numbers ... |
DeepScaleR | [
{
"role": "user",
"content": "Alice, Bob, and Carol repeatedly take turns tossing a die. Alice begins; Bob always follows Alice; Carol always follows Bob; and Alice always follows Carol. Find the probability that Carol will be the first one to toss a six. (The probability of obtaining a six on any toss is $... | math | {
"style": "rule",
"ground_truth": "\\frac{36}{91}"
} | {
"index": 4446,
"solution": "1. **Identify the Event Sequence**:\n - Alice, Bob, and Carol take turns tossing a die in the order: Alice, Bob, Carol, Alice, Bob, Carol, and so on.\n - We need to find the probability that Carol is the first to toss a six.\n\n2. **Calculate the Probability of Carol Winning in the... |
DeepScaleR | [
{
"role": "user",
"content": "Alice, Bob, and Charlie are playing a game with 6 cards numbered 1 through 6. Each player is dealt 2 cards uniformly at random. On each player's turn, they play one of their cards, and the winner is the person who plays the median of the three cards played. Charlie goes last, s... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{15}"
} | {
"index": 4447,
"solution": "If Alice has a card that is adjacent to one of Bob's, then Alice and Bob will play those cards as one of them is guaranteed to win. If Alice and Bob do not have any adjacent cards, since Charlie goes last, Charlie can always choose a card that will win. Let $A$ denote a card that is he... |
DeepScaleR | [
{
"role": "user",
"content": "Alice, Bob, and Charlie each flip a fair coin repeatedly until they each flip heads. In a separate event, three more people, Dave, Eve, and Frank, each flip a biased coin (with a probability of $\\frac{1}{3}$ of getting heads) until they first flip heads. Determine the probabil... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{702}"
} | {
"index": 4448,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice, Bob, and Charlie each pick a 2-digit number at random. What is the probability that all of their numbers' tens' digits are different from each others' tens' digits and all of their numbers' ones digits are different from each others' ones' digits?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{112}{225}"
} | {
"index": 4449,
"solution": "$\\frac{9}{10} \\frac{8}{10} \\frac{8}{9} \\frac{7}{9}=\\frac{112}{225}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice, Bob, and Charlie roll a 4, 5, and 6-sided die, respectively. What is the probability that a number comes up exactly twice out of the three rolls?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{30}"
} | {
"index": 4450,
"solution": "There are $4 \\cdot 5 \\cdot 6=120$ different ways that the dice can come up. The common number can be any of $1,2,3,4$, or 5: there are $3+4+5=12$ ways for it to be each of $1,2,3$, or 4, because we pick one of the three people's rolls to disagree, and there are 3,4, and 5 ways that r... |
DeepScaleR | [
{
"role": "user",
"content": "Alice, Bob, and Conway are playing rock-paper-scissors. Each player plays against each of the other $2$ players and each pair plays until a winner is decided (i.e. in the event of a tie, they play again). What is the probability that each player wins exactly once?"
}
] | math | {
"style": "rule",
"ground_truth": "1/4"
} | {
"index": 4451,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alicia earns 20 dollars per hour, of which $1.45\\%$ is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes?"
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 4452,
"solution": "1. **Convert Alicia's hourly wage to cents**: Alicia earns $20$ dollars per hour. Since there are $100$ cents in a dollar, her hourly wage in cents is:\n \\[\n 20 \\text{ dollars} \\times 100 \\text{ cents/dollar} = 2000 \\text{ cents}\n \\]\n\n2. **Calculate the tax deduction in... |
DeepScaleR | [
{
"role": "user",
"content": "Alicia had two containers. The first was $\\frac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\\frac{3}{4}$ full of water. What is the ratio of the volume of ... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{10}"
} | {
"index": 4453,
"solution": "\n1. **Define Variables:**\n Let the volume of the first container be $A$ and the volume of the second container be $B$.\n\n2. **Set Up the Equation:**\n Since Alicia poured $\\frac{5}{6}$ of the first container into the second container, and this amount filled $\\frac{3}{4}$ of th... |
DeepScaleR | [
{
"role": "user",
"content": "Alicia has $n$ candies, where $n$ is a positive integer with three digits. If she buys $5$ more, she will have a multiple of $8$. If she loses $8$, she will have a multiple of $5$. What is the smallest possible value of $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "123"
} | {
"index": 4454,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alicia starts a sequence with $m=3$. What is the fifth term of her sequence following the algorithm: Step 1: Alicia writes down the number $m$ as the first term. Step 2: If $m$ is even, Alicia sets $n=\frac{1}{2} m$. If $m$ is odd, Alicia sets $n=m+1$. Step 3: Alicia writes dow... | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 4455,
"solution": "We follow Alicia's algorithm carefully: Step 1: Alicia writes down $m=3$ as the first term. Step 2: Since $m=3$ is odd, Alicia sets $n=m+1=4$. Step 3: Alicia writes down $m+n+1=8$ as the second term. Step 4: Alicia sets $m=8$. Step 2: Since $m=8$ is even, Alicia sets $n=\frac{1}{2} m=4... |
DeepScaleR | [
{
"role": "user",
"content": "Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the votes were distributed among the three candidates. If Brenda received $36$ votes, then how many votes were cast all together?"
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 4456,
"solution": "1. **Understanding the problem**: We are given that Brenda received $36$ votes, which represents $\\frac{3}{10}$ of the total votes in the election.\n\n2. **Calculating the total number of votes**:\n - Since $36$ votes is $\\frac{3}{10}$ of the total votes, we can find the total numb... |
DeepScaleR | [
{
"role": "user",
"content": "Alina and Masha wanted to create an interesting version of the school tour of the Olympiad. Masha proposed several problems and rejected every second problem of Alina (exactly half). Alina also proposed several problems and only accepted every third problem of Masha (exactly on... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 4457,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alison is eating 2401 grains of rice for lunch. She eats the rice in a very peculiar manner: every step, if she has only one grain of rice remaining, she eats it. Otherwise, she finds the smallest positive integer $d>1$ for which she can group the rice into equal groups of size... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 4458,
"solution": "Note that $2401=7^{4}$. Also, note that the operation is equivalent to replacing $n$ grains of rice with $n \\cdot \\frac{p-1}{p}$ grains of rice, where $p$ is the smallest prime factor of $n$. Now, suppose that at some moment Alison has $7^{k}$ grains of rice. After each of the next f... |
DeepScaleR | [
{
"role": "user",
"content": "All $20$ diagonals are drawn in a regular octagon. At how many distinct points in the interior of the octagon (not on the boundary) do two or more diagonals intersect?"
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 4459,
"solution": "To solve this problem, we need to count the number of distinct intersection points formed by the diagonals of a regular octagon. We will use combinatorial geometry to determine this count.\n\n#### Step 1: Counting the Total Number of Diagonals\nIn an $n$-sided polygon, the number of di... |
DeepScaleR | [
{
"role": "user",
"content": "All dwarves are either liars or knights. Liars always lie, while knights always tell the truth. Each cell of a $4 \\times 4$ board contains one dwarf. It is known that among them there are both liars and knights. Each dwarf stated: \"Among my neighbors (by edge), there are an e... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 4460,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "All letters in the word $VUQAR$ are different and chosen from the set $\\{1,2,3,4,5\\}$. Find all solutions to the equation \\[\\frac{(V+U+Q+A+R)^2}{V-U-Q+A+R}=V^{{{U^Q}^A}^R}.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "(5, 2, 1, 3, 4) \\text{ and } (5, 2, 1, 4, 3)"
} | {
"index": 4461,
"solution": "\nLet's consider the given problem: we need to find all solutions for the letters \\( V, U, Q, A, R \\) in the equation:\n\n\\[\n\\frac{(V+U+Q+A+R)^2}{V-U-Q+A+R} = V^{{{U^Q}^A}^R},\n\\]\n\nwhere each letter is from the set \\(\\{1,2,3,4,5\\}\\) and all letters are different.\n\n### Ste... |
DeepScaleR | [
{
"role": "user",
"content": "All lines with equation $ax+by=c$ such that $a,b,c$ form an arithmetic progression pass through a common point. What are the coordinates of that point?"
}
] | math | {
"style": "rule",
"ground_truth": "(-1,2)"
} | {
"index": 4462,
"solution": "1. **Identify the Arithmetic Progression (AP) Relationship**: Given that $a$, $b$, and $c$ form an arithmetic progression, we can express $b$ and $c$ in terms of $a$ and a common difference $d$. Thus, $b = a + d$ and $c = a + 2d$.\n\n2. **Substitute in Line Equation**: Substitute $b$ a... |
DeepScaleR | [
{
"role": "user",
"content": "All natural numbers from 1 to 1000 inclusive are divided into two groups: even and odd. In which group is the sum of all the digits used to write the numbers greater and by how much?"
}
] | math | {
"style": "rule",
"ground_truth": "499"
} | {
"index": 4463,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "All natural numbers from 1 to 2017 inclusive were written in a row. How many times was the digit 7 written?"
}
] | math | {
"style": "rule",
"ground_truth": "602"
} | {
"index": 4464,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "All of David's telephone numbers have the form $555-abc-defg$, where $a$, $b$, $c$, $d$, $e$, $f$, and $g$ are distinct digits and in increasing order, and none is either $0$ or $1$. How many different telephone numbers can David have?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 4465,
"solution": "1. **Identify the digits available for use**: The digits that can be used in the telephone number are $2, 3, 4, 5, 6, 7, 8, 9$. This is because the digits $a, b, c, d, e, f, g$ must be distinct, in increasing order, and cannot include $0$ or $1$.\n\n2. **Count the available digits**: T... |
DeepScaleR | [
{
"role": "user",
"content": "All of Marcy's marbles are blue, red, green, or yellow. One third of her marbles are blue, one fourth of them are red, and six of them are green. What is the smallest number of yellow marbles that Macy could have?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 4466,
"solution": "Let the total number of marbles Marcy has be denoted by $n$. According to the problem, the distribution of marbles is as follows:\n- Blue marbles: $\\frac{1}{3}n$\n- Red marbles: $\\frac{1}{4}n$\n- Green marbles: $6$\n\nWe need to find the number of yellow marbles, which can be express... |
DeepScaleR | [
{
"role": "user",
"content": "All of the roots of $x^3+ax^2+bx+c$ are positive integers greater than $2$ , and the coeffi\u000ecients satisfy $a+b+c+1=-2009$ . Find $a$"
}
] | math | {
"style": "rule",
"ground_truth": "-58"
} | {
"index": 4467,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "All of the triangles in the diagram below are similar to isosceles triangle $ABC$, in which $AB=AC$. Each of the $7$ smallest triangles has area $1,$ and $\\triangle ABC$ has area $40$. What is the area of trapezoid $DBCE$?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 4468,
"solution": "1. **Identify the relationship between the triangles**: All of the triangles in the diagram are similar to isosceles triangle $ABC$, where $AB = AC$. Each of the 7 smallest triangles has an area of 1, and $\\triangle ABC$ has an area of 40.\n\n2. **Express the area of trapezoid $DBCE$*... |
DeepScaleR | [
{
"role": "user",
"content": "All positive integers whose binary representations (excluding leading zeroes) have at least as many 1's as 0's are put in increasing order. Compute the number of digits in the binary representation of the 200th number."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 4469,
"solution": "We do a rough estimation. There are 255 positive integers with at most 8 digits and a majority of them, but not more than 200, satisfy the property. Meanwhile, there are 511 positive integers with at most 9 digits, and a majority of them satisfy this property. Thus, the answer must be ... |
DeepScaleR | [
{
"role": "user",
"content": "All positive integers whose digits add up to 11 are listed in increasing order: $29, 38, 47, ...$. What is the eleventh number in that list?"
}
] | math | {
"style": "rule",
"ground_truth": "137"
} | {
"index": 4470,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "All positive integers whose digits add up to 12 are listed in increasing order. What is the eleventh number in that list?"
}
] | math | {
"style": "rule",
"ground_truth": "156"
} | {
"index": 4471,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "All positive integers whose digits add up to 12 are listed in increasing order: $39, 48, 57, ...$. What is the tenth number in that list?"
}
] | math | {
"style": "rule",
"ground_truth": "147"
} | {
"index": 4472,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "All positive integers whose digits add up to 12 are listed in increasing order: $39, 48, 57, ...$. What is the twelfth number in that list?"
}
] | math | {
"style": "rule",
"ground_truth": "165"
} | {
"index": 4473,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "All positive integers whose digits add up to 14 are listed in increasing order. What is the eleventh number in that list?"
}
] | math | {
"style": "rule",
"ground_truth": "194"
} | {
"index": 4474,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "All positive integers whose digits add up to 14 are listed in increasing order: $59, 68, 77, ...$. What is the fifteenth number in that list?"
}
] | math | {
"style": "rule",
"ground_truth": "266"
} | {
"index": 4475,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "All positive odd numbers are arranged in the following table (the number of numbers in the next row is twice the number of numbers in the previous row) \nFirst row 1 \nSecond row 3 5 \nThird row 7 9 11 13 \n… \nThen, the third number in the sixth row is ．"... | math | {
"style": "rule",
"ground_truth": "67"
} | {
"index": 4476,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "All sides of the convex pentagon $ABCDE$ are of equal length, and $\\angle A = \\angle B = 90^\\circ$. What is the degree measure of $\\angle E$?"
}
] | math | {
"style": "rule",
"ground_truth": "150^\\circ"
} | {
"index": 4477,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "All students at Adams High School and at Baker High School take a certain exam. The average scores for boys, for girls, and for boys and girls combined, at Adams HS and Baker HS are shown in the table, as is the average for boys at the two schools combined. What is the average ... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"index": 4478,
"solution": "1. **Define Variables:**\n Let $A$ and $a$ be the number of boys and girls at Adams High School, respectively. Let $B$ and $b$ be the number of boys and girls at Baker High School, respectively.\n\n2. **Set Up Equations Based on Given Averages:**\n - For Adams High School, the aver... |
DeepScaleR | [
{
"role": "user",
"content": "All subscripts in this problem are to be considered modulo 6 , that means for example that $\\omega_{7}$ is the same as $\\omega_{1}$. Let $\\omega_{1}, \\ldots \\omega_{6}$ be circles of radius $r$, whose centers lie on a regular hexagon of side length 1 . Let $P_{i}$ be the i... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 4479,
"solution": "Consider two consecutive circles $\\omega_{i}$ and $\\omega_{i+1}$. Let $Q_{i}, Q_{i}^{\\prime}$ be two points on $\\omega_{i}$ and $Q_{i+1}, Q_{i+1}^{\\prime}$ on $\\omega_{i+1}$ such that $Q_{i}, P_{i}$ and $Q_{i+1}$ are colinear and also $Q_{i}^{\\prime}, P_{i}$ and $Q_{i+1}^{\\prim... |
DeepScaleR | [
{
"role": "user",
"content": "All the complex roots of $(z + 1)^4 = 16z^4,$ when plotted in the complex plane, lie on a circle. Find the radius of this circle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 4480,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "All the complex roots of $(z + 1)^5 = 32z^5,$ when plotted in the complex plane, lie on a circle. Find the radius of this circle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 4481,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "All the complex roots of $(z + 2)^6 = 64z^6$, when plotted in the complex plane, lie on a circle. Find the radius of this circle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{\\sqrt{3}}"
} | {
"index": 4482,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "All the complex roots of $(z - 2)^6 = 64z^6$ when plotted in the complex plane, lie on a circle. Find the radius of this circle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"index": 4483,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "All the edges of the regular tetrahedron \\(P-ABC\\) have length \\(1\\). Let \\( L, M, N \\) be the midpoints of the edges \\( PA, PB, PC \\) respectively. Find the area of the cross-section of the circumsphere of the tetrahedron when cut by the plane \\( LMN \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"index": 4484,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "All the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ are written in a $3\\times3$ array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to $18$. What is the numbe... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 4485,
"solution": "1. **Understanding the Problem**: We need to arrange the numbers $1$ through $9$ in a $3 \\times 3$ grid such that consecutive numbers are adjacent (share an edge), and the sum of the numbers in the four corners is $18$.\n\n2. **Finding Corner Numbers**: Since consecutive numbers must ... |
DeepScaleR | [
{
"role": "user",
"content": "All the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 are written in a 3x3 array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to 18. What is the number in the cen... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 4486,
"solution": "1. **Understanding the Problem**: We are given a $3 \\times 3$ grid where each cell contains a unique number from $1$ to $9$. The condition is that consecutive numbers must be adjacent (share an edge). Additionally, the sum of the numbers in the four corner cells is $18$. We need to fi... |
DeepScaleR | [
{
"role": "user",
"content": "All the numbers 2, 3, 4, 5, 6, 7 are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. W... | math | {
"style": "rule",
"ground_truth": "729"
} | {
"index": 4487,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "All the positive integers greater than 1 are arranged in five columns (A, B, C, D, E) as shown. Continuing the pattern, in what column will the integer 800 be written?\n\n[asy]\nlabel(\"A\",(0,0),N);\nlabel(\"B\",(10,0),N);\nlabel(\"C\",(20,0),N);\nlabel(\"D\",(30,0),N);\nlabel... | math | {
"style": "rule",
"ground_truth": "\\text{B}"
} | {
"index": 4488,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "All the roots of the polynomial $z^6-10z^5+Az^4+Bz^3+Cz^2+Dz+16$ are positive integers, possibly repeated. What is the value of $B$?"
}
] | math | {
"style": "rule",
"ground_truth": "-88"
} | {
"index": 4489,
"solution": "1. **Identify the Roots**: Given that all roots of the polynomial $z^6 - 10z^5 + Az^4 + Bz^3 + Cz^2 + Dz + 16$ are positive integers, and their sum (as coefficients of $z^5$ with a sign change) is 10, we consider possible sets of roots that sum to 10. The roots are $2, 2, 2, 2, 1, 1$.\... |
DeepScaleR | [
{
"role": "user",
"content": "All the sequences consisting of five letters from the set $\\{T, U, R, N, I, P\\}$ (with repetitions allowed) are arranged in alphabetical order in a dictionary. Two sequences are called \"anagrams\" of each other if one can be obtained by rearranging the letters of the other. ... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 4490,
"solution": "Convert each letter to a digit in base $6: I \\mapsto 0, N \\mapsto 1, P \\mapsto 2, R \\mapsto 3, T \\mapsto 4, U \\mapsto 5$. Then the dictionary simply consists of all base-6 integers from $00000_{6}$ to $555555_{6}$ in numerical order. If one number can be obtained from another by ... |
DeepScaleR | [
{
"role": "user",
"content": "All the students in an algebra class took a $100$-point test. Five students scored $100$, each student scored at least $60$, and the mean score was $76$. What is the smallest possible number of students in the class?"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 4491,
"solution": "1. **Define Variables and Known Values:**\n Let $n$ be the number of students in the class. We know that five students scored $100$ points each, every student scored at least $60$ points, and the mean score of the class was $76$ points.\n\n2. **Set Up the Equation for Total Score:**\... |
DeepScaleR | [
{
"role": "user",
"content": "All three vertices of $\\triangle ABC$ lie on the parabola defined by $y=x^2$, with $A$ at the origin and $\\overline{BC}$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length of $BC$?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 4492,
"solution": "1. **Identify the Coordinates of Points**: \n - Since $A$ is at the origin, $A = (0,0)$.\n - The line $\\overline{BC}$ is parallel to the $x$-axis, and since all vertices lie on the parabola $y = x^2$, the coordinates of $B$ and $C$ must be symmetric about the $y$-axis. Let's denot... |
DeepScaleR | [
{
"role": "user",
"content": "All triangles have the same value, and all circles have the same value. What is the sum of three circles? \\begin{align*}\n\\Delta + \\bigcirc + \\Delta + \\bigcirc + \\Delta&= 21\\\\\n\\bigcirc + \\Delta+\\bigcirc+\\Delta+\\bigcirc &= 19\\\\\n\\bigcirc + \\bigcirc + \\bigcirc ... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 4493,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "All two-digit numbers divisible by 5, where the number of tens is greater than the number of units, were written on the board. There were \\( A \\) such numbers. Then, all two-digit numbers divisible by 5, where the number of tens is less than the number of units, were written ... | math | {
"style": "rule",
"ground_truth": "413"
} | {
"index": 4494,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "All vertices of a regular 2016-gon are initially white. What is the least number of them that can be painted black so that:\n(a) There is no right triangle\n(b) There is no acute triangle\nhaving all vertices in the vertices of the 2016-gon that are still white?"
}
] | math | {
"style": "rule",
"ground_truth": "1008"
} | {
"index": 4495,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "All vertices of a regular tetrahedron \\( A B C D \\) are located on one side of the plane \\( \\alpha \\). It turns out that the projections of the vertices of the tetrahedron onto the plane \\( \\alpha \\) are the vertices of a certain square. Find the value of \\(A B^{2}\\),... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 4496,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Allen and Ben are painting a fence. The ratio of the amount of work Allen does to the amount of work Ben does is $3:5$. If the fence requires a total of $240$ square feet to be painted, how many square feet does Ben paint?"
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 4497,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Allen and Bethany each arrive at a party at a random time between 1:00 and 2:00. Each stays for 15 minutes, then leaves. What is the probability that Allen and Bethany see each other at the party?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{16}"
} | {
"index": 4498,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Allen and Brian are playing a game in which they roll a 6-sided die until one of them wins. Allen wins if two consecutive rolls are equal and at most 3. Brian wins if two consecutive rolls add up to 7 and the latter is at most 3. What is the probability that Allen wins?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{12}"
} | {
"index": 4499,
"solution": "Note that at any point in the game after the first roll, the probability that Allen wins depends only on the most recent roll, and not on any rolls before that one. So we may define $p$ as the probability that Allen wins at any point in the game, given that the last roll was a 1,2, or ... |
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