data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Adva is a regular tetrahedron with side length \\( s \\), and there are three spheres associated with it. The first sphere passes through the vertices of the tetrahedron, the second intersects the midpoints of the edges, and the third is inscribed such that it touches the faces... | math | {
"style": "rule",
"ground_truth": "9:3:1"
} | {
"index": 4300,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Aerith timed herself solving a contest and noted the time both as days:hours:minutes:seconds and in seconds. For example, if she spent 1,000,000 seconds, she recorded it as 11:13:46:40 and 1,000,000 seconds. Bob subtracts these numbers, ignoring punctuation. In this case, he co... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 4301,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "After Euclid High School's last basketball game, it was determined that $\\frac{1}{4}$ of the team's points were scored by Alexa and $\\frac{2}{7}$ were scored by Brittany. Chelsea scored $15$ points. None of the other $7$ team members scored more than $2$ points. What was the ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 4302,
"solution": "Let $x$ be the total number of points scored by the team. According to the problem, the points can be broken down as follows:\n- Alexa scored $\\frac{1}{4}x$ points.\n- Brittany scored $\\frac{2}{7}x$ points.\n- Chelsea scored $15$ points.\n- The other $7$ team members scored $y$ point... |
DeepScaleR | [
{
"role": "user",
"content": "After Natasha ate a third of the peaches from the jar, the level of the compote lowered by one quarter. By how much (relative to the new level) will the level of the compote lower if all the remaining peaches are eaten?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{9}"
} | {
"index": 4303,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "After Sally takes 20 shots, she has made $55\\%$ of her shots. After she takes 5 more shots, she raises her percentage to $56\\%$. How many of the last 5 shots did she make?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 4304,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "After a fair die with faces numbered 1 to 6 is rolled, the number on the top face is $x$. What is the most likely outcome?"
}
] | math | {
"style": "rule",
"ground_truth": "x > 2"
} | {
"index": 4305,
"solution": "With a fair die that has faces numbered from 1 to 6, the probability of rolling each of 1 to 6 is $\\frac{1}{6}$. We calculate the probability for each of the five choices. There are 4 values of $x$ that satisfy $x>2$, so the probability is $\\frac{4}{6}=\\frac{2}{3}$. There are 2 valu... |
DeepScaleR | [
{
"role": "user",
"content": "After a gymnastics meet, each gymnast shook hands once with every gymnast on every team (except herself). Afterwards, a coach came down and only shook hands with each gymnast from her own team. There were a total of 281 handshakes. What is the fewest number of handshakes the co... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 4306,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "After a track and field event, each athlete shook hands once with every athlete from every other team, but not with their own team members. Afterwards, two coaches arrived, each only shaking hands with each athlete from their respective teams. If there were a total of 300 hands... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 4307,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "After eating lunch together, Jay and Paul start walking in opposite directions. Jay walks 0.75 miles every 15 minutes and Paul power walks 2.5 miles every 30 minutes. In miles, how far apart are they after 1.5 hours?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 4308,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "After expanding and combining like terms in the expression \\((x+y+z)^{2028} + (x-y-z)^{2028}\\), how many monomials of the form \\(x^{a} y^{b} z^{c}\\) have a non-zero coefficient?"
}
] | math | {
"style": "rule",
"ground_truth": "1030225"
} | {
"index": 4309,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "After finding the average of $35$ scores, a student carelessly included the average with the $35$ scores and found the average of these $36$ numbers. The ratio of the second average to the true average was"
}
] | math | {
"style": "rule",
"ground_truth": "1:1"
} | {
"index": 4310,
"solution": "1. **Calculate the True Average of the Original Scores**:\n Let the $35$ scores be denoted as $x_1, x_2, \\dots, x_{35}$. The true average of these scores is given by:\n \\[\n \\text{True Average} = \\frac{x_1 + x_2 + \\cdots + x_{35}}{35}\n \\]\n Let's denote this average as... |
DeepScaleR | [
{
"role": "user",
"content": "After folding the long rope in half 6 times, calculate the number of segments the rope will be cut into."
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"index": 4311,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "After learning about functions, the mathematics team of a high school first grade conducted a mathematical modeling activity. Through a survey of the sales of a certain product in a supermarket near the school, it was found that the relationship between the daily sales price P(... | math | {
"style": "rule",
"ground_truth": "121"
} | {
"index": 4312,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "After lunch, there are dark spots with a total area of $S$ on a transparent square tablecloth. It turns out that if the tablecloth is folded in half along any of the two lines connecting the midpoints of its opposite sides or along one of its two diagonals, the total visible ar... | math | {
"style": "rule",
"ground_truth": "2/3"
} | {
"index": 4313,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "After reading the following solution, answer the question: Xiaofang found in the simplification of $\\sqrt{7+4\\sqrt{3}}$ that first, $\\sqrt{7+4\\sqrt{3}}$ can be simplified to $\\sqrt{7+2\\sqrt{12}}$. Since $4+3=7$ and $4\\times 3=12$, that is, ${(\\sqrt{4})^2}+{(\\sqrt{3})^2... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{10}}{2}-\\frac{\\sqrt{6}}{2}"
} | {
"index": 4314,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "After school, Maya and Naomi headed to the beach, $6$ miles away. Maya decided to bike while Naomi took a bus. The graph below shows their journeys, indicating the time and distance traveled. What was the difference, in miles per hour, between Naomi's and Maya's average speeds?... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 4315,
"solution": "1. **Convert Naomi's travel time to hours**: Naomi travels $6$ miles in $10$ minutes. To convert minutes to hours, we divide by $60$:\n \\[\n 10 \\text{ minutes} = \\frac{10}{60} \\text{ hours} = \\frac{1}{6} \\text{ hours}\n \\]\n\n2. **Calculate Naomi's average speed**: Using t... |
DeepScaleR | [
{
"role": "user",
"content": "After shifting the graph of the function $y=\\sin^2x-\\cos^2x$ to the right by $m$ units, the resulting graph is symmetric to the graph of $y=k\\sin x\\cos x$ ($k>0$) with respect to the point $\\left( \\frac{\\pi}{3}, 0 \\right)$. Find the minimum positive value of $k+m$."
}... | math | {
"style": "rule",
"ground_truth": "2+ \\frac{5\\pi}{12}"
} | {
"index": 4316,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "After the Guts round ends, HMMT organizers will collect all answers submitted to all 66 questions (including this one) during the individual rounds and the guts round. Estimate $N$, the smallest positive integer that no one will have submitted at any point during the tournament... | math | {
"style": "rule",
"ground_truth": "139"
} | {
"index": 4317,
"solution": "The correct answer was 139. Remark: Until the end of the Guts round, no team had submitted 71 as the answer to any question. One team, however, submitted 71 as their answer to this question, increasing the answer up to 139."
} |
DeepScaleR | [
{
"role": "user",
"content": "After the appearance of purple sand flowerpots in the Ming and Qing dynasties, their development momentum was soaring, gradually becoming the collection target of collectors. With the development of pot-making technology, purple sand flowerpots have been integrated into the dai... | math | {
"style": "rule",
"ground_truth": "39"
} | {
"index": 4318,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "After the implementation of the \"double reduction\" policy, schools have attached importance to extended services and increased the intensity of sports activities in these services. A sports equipment store seized the opportunity and planned to purchase 300 sets of table tenni... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 4319,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "After the year 2002, which is a palindrome, identify the next year where the sum of the product of its digits is greater than 15. Find the sum of the product of the digits of that year."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 4320,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "After traveling 50 miles by taxi, Ann is charged a fare of $\\$120$. Assuming the taxi fare is directly proportional to distance traveled, how much would Ann be charged (in dollars) if she had traveled 70 miles?"
}
] | math | {
"style": "rule",
"ground_truth": "168"
} | {
"index": 4321,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "After viewing the John Harvard statue, a group of tourists decides to estimate the distances of nearby locations on a map by drawing a circle, centered at the statue, of radius $\\sqrt{n}$ inches for each integer $2020 \\leq n \\leq 10000$, so that they draw 7981 circles altoge... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"index": 4322,
"solution": "Consider a coordinate system on any line $\\ell$ where 0 is placed at the foot from $(0,0)$ to $\\ell$. Then, by the Pythagorean theorem, a point $(x, y)$ on $\\ell$ is assigned a coordinate $u$ for which $x^{2}+y^{2}=u^{2}+a$ for some fixed $a$ (dependent only on $\\ell$ ). Consider t... |
DeepScaleR | [
{
"role": "user",
"content": "After walking so much that his feet get really tired, the beaver staggers so that, at each step, his coordinates change by either $(+1,+1)$ or $(+1,-1)$. Now he walks from $(0,0)$ to $(8,0)$ without ever going below the $x$-axis. How many such paths are there?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 4323,
"solution": "$C(4)=14$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Agricultural researchers randomly selected $50$ plants of a certain crop in an experimental field for research. The frequency distribution of the mass (in grams) of a single plant falling into each group is shown in the table below:\n\n| Mass of a Single Plant (in grams) | Freq... | math | {
"style": "rule",
"ground_truth": "0.8413"
} | {
"index": 4324,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Ahn chooses a two-digit integer, subtracts it from 200, and doubles the result. What is the largest number Ahn can get?"
}
] | math | {
"style": "rule",
"ground_truth": "380"
} | {
"index": 4325,
"solution": "1. **Identify the operation**: Ahn performs the operation $2(200 - n)$, where $n$ is a two-digit integer.\n\n2. **Determine the range of $n$**: Since $n$ is a two-digit integer, $10 \\leq n \\leq 99$.\n\n3. **Maximize the expression**: To maximize $2(200 - n)$, we need to minimize $n$ ... |
DeepScaleR | [
{
"role": "user",
"content": "Aileen plays badminton where she and her opponent stand on opposite sides of a net and attempt to bat a birdie back and forth over the net. A player wins a point if their opponent fails to bat the birdie over the net. When Aileen is the server (the \ffirst player to try to hit ... | math | {
"style": "rule",
"ground_truth": "73"
} | {
"index": 4326,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Ainsley and Buddy play a game where they repeatedly roll a standard fair six-sided die. Ainsley wins if two multiples of 3 in a row are rolled before a non-multiple of 3 followed by a multiple of 3, and Buddy wins otherwise. If the probability that Ainsley wins is $\\frac{a}{b}... | math | {
"style": "rule",
"ground_truth": "109"
} | {
"index": 4327,
"solution": "We let $X$ be the event of a multiple of 3 being rolled and $Y$ be the event of a nonmultiple of 3 being rolled. In order for Ainsley to win, she needs event $X$ to happen consecutively; meanwhile, Buddy just needs $Y$ then $X$ to occur. Thus, if $Y$ occurs in the first two rolls, Budd... |
DeepScaleR | [
{
"role": "user",
"content": "Ajay is standing at point $A$ near Pontianak, Indonesia, $0^\\circ$ latitude and $110^\\circ \\text{ E}$ longitude. Billy is standing at point $B$ near Big Baldy Mountain, Idaho, USA, $45^\\circ \\text{ N}$ latitude and $115^\\circ \\text{ W}$ longitude. Assume that Earth is a ... | math | {
"style": "rule",
"ground_truth": "120^\\circ"
} | {
"index": 4328,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Al and Barb start their new jobs on the same day. Al's schedule is 3 work-days followed by 1 rest-day. Barb's schedule is 7 work-days followed by 3 rest-days. On how many of their first 1000 days do both have rest-days on the same day?"
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 4329,
"solution": "1. **Identify the Cycles**: Al's schedule is a cycle of 4 days (3 work-days followed by 1 rest-day), and Barb's schedule is a cycle of 10 days (7 work-days followed by 3 rest-days).\n\n2. **Calculate the Least Common Multiple (LCM)**: To find when their schedules align completely, we c... |
DeepScaleR | [
{
"role": "user",
"content": "Al and Bert must arrive at a town 22.5 km away. They have one bicycle between them and must arrive at the same time. Bert sets out riding at 8 km/h, leaves the bicycle, and then walks at 5 km/h. Al walks at 4 km/h, reaches the bicycle, and rides at 10 km/h. For how many minutes... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"index": 4330,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs $1 more than a pink pill, and Al's pills cost a total of $546 for the two weeks. How much does one green pill cost?"
}
] | math | {
"style": "rule",
"ground_truth": "$19"
} | {
"index": 4331,
"solution": "1. **Identify the total number of days and total cost**: Al takes the pills for two weeks, which is 14 days. The total cost for these two weeks is $\\$546$.\n\n2. **Calculate daily expenditure**: \n \\[\n \\text{Daily cost} = \\frac{\\text{Total cost}}{\\text{Number of days}} = \\f... |
DeepScaleR | [
{
"role": "user",
"content": "Al walks down to the bottom of an escalator that is moving up and he counts 150 steps. His friend, Bob, walks up to the top of the escalator and counts 75 steps. If Al's speed of walking (in steps per unit time) is three times Bob's walking speed, how many steps are visible on ... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 4332,
"solution": "Let $e$ and $b$ be the speeds of the escalator and Bob, respectively.\n When Al was on his way down, he took $150$ steps with a speed of $3b-e$ per step. When Bob was on his way up, he took $75$ steps with a speed of $b+e$ per step. Since Al and Bob were walking the same distance, we h... |
DeepScaleR | [
{
"role": "user",
"content": "Al's age is $16$ more than the sum of Bob's age and Carl's age, and the square of Al's age is $1632$ more than the square of the sum of Bob's age and Carl's age. What is the sum of the ages of Al, Bob, and Carl?"
}
] | math | {
"style": "rule",
"ground_truth": "102"
} | {
"index": 4333,
"solution": "1. **Define Variables:**\n Let $a$ be Al's age, $b$ be Bob's age, and $c$ be Carl's age. \n\n2. **Set Up Equations:**\n From the problem, we have two equations:\n \\[ a = 16 + b + c \\]\n \\[ a^2 = 1632 + (b+c)^2 \\]\n\n3. **Substitute and Expand:**\n Substitute $a = 16 + b +... |
DeepScaleR | [
{
"role": "user",
"content": "Al, Bert, Carl, and Dan are the winners of a school contest for a pile of books, which they are to divide in a ratio of $4:3:2:1$, respectively. Due to some confusion, they come at different times to claim their prizes, and each assumes he is the first to arrive. If each takes ... | math | {
"style": "rule",
"ground_truth": "\\frac{1701}{2500}"
} | {
"index": 4334,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Al, Betty, and Clare split $\\$1000$ among them to be invested in different ways. Each begins with a different amount. At the end of one year they have a total of $\\$1500$. Betty and Clare have both doubled their money, whereas Al has managed to lose $\\$100$. What was Al's or... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"index": 4335,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Al, Bill, and Cal will each randomly be assigned a whole number from $1$ to $10$, inclusive, with no two of them getting the same number. What is the probability that Al's number will be a whole number multiple of Bill's and Bill's number will be a whole number multiple of Cal'... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{80}"
} | {
"index": 4336,
"solution": "To solve this problem, we need to count the number of valid assignments of numbers to Al, Bill, and Cal such that Al's number is a multiple of Bill's number, and Bill's number is a multiple of Cal's number. We then divide this count by the total number of ways to assign the numbers.\n\... |
DeepScaleR | [
{
"role": "user",
"content": "Al-Karhi's rule for approximating the square root. If \\(a^{2}\\) is the largest square contained in the given number \\(N\\), and \\(r\\) is the remainder, then\n\n$$\n\\sqrt{N}=\\sqrt{a^{2}+r}=a+\\frac{r}{2a+1}, \\text{ if } r<2a+1\n$$\n\nExplain how Al-Karhi might have deriv... | math | {
"style": "rule",
"ground_truth": "20.366"
} | {
"index": 4337,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alain and Louise are driving on a circular track with radius 25 km. Alain leaves the starting line first, going clockwise at 80 km/h. Fifteen minutes later, Louise leaves the same starting line, going counterclockwise at 100 km/h. For how many hours will Louise have been drivin... | math | {
"style": "rule",
"ground_truth": "\\\\frac{10\\\\pi-1}{9}"
} | {
"index": 4338,
"solution": "Since the track is circular with radius 25 km, then its circumference is $2 \\pi(25)=50 \\pi$ km. In the 15 minutes that Alain drives at 80 km/h, he drives a distance of $\\frac{1}{4}(80)=20$ km (because 15 minutes is one-quarter of an hour). When Louise starts driving, she drives in t... |
DeepScaleR | [
{
"role": "user",
"content": "Alan, Beth, Carla, and Dave weigh themselves in pairs. Together, Alan and Beth weigh 280 pounds, Beth and Carla weigh 230 pounds, Carla and Dave weigh 250 pounds, and Alan and Dave weigh 300 pounds. How many pounds do Alan and Carla weigh together?"
}
] | math | {
"style": "rule",
"ground_truth": "250"
} | {
"index": 4339,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alan, Jason, and Shervin are playing a game with MafsCounts questions. They each start with $2$ tokens. In each round, they are given the same MafsCounts question. The first person to solve the MafsCounts question wins the round and steals one token from each of the other pla... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 4340,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Albert now decides to extend his list to the 2000th digit. He writes down positive integers in increasing order with a first digit of 1, such as $1, 10, 11, 12, \\ldots$. Determine the three-digit number formed by the 1998th, 1999th, and 2000th digits."
}
] | math | {
"style": "rule",
"ground_truth": "141"
} | {
"index": 4341,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Albert starts to make a list, in increasing order, of the positive integers that have a first digit of 1. He writes $1, 10, 11, 12, \\ldots$ but by the 1,000th digit he (finally) realizes that the list would contain an infinite number of elements. Find the three-digit number fo... | math | {
"style": "rule",
"ground_truth": "116"
} | {
"index": 4342,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Albert's choice of burgers, sides, and drinks are independent events. How many different meals can Albert get if there are 5 choices of burgers, 3 choices of sides, and 12 choices of drinks?"
}
] | math | {
"style": "rule",
"ground_truth": "180"
} | {
"index": 4343,
"solution": "The total number of different meals that Albert can get is: $$5 \\times 3 \\times 12=180$$"
} |
DeepScaleR | [
{
"role": "user",
"content": "Alberto, Bernardo, and Carlos are collectively listening to three different songs. Each is simultaneously listening to exactly two songs, and each song is being listened to by exactly two people. In how many ways can this occur?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 4344,
"solution": "We have $\\binom{3}{2}=3$ choices for the songs that Alberto is listening to. Then, Bernardo and Carlos must both be listening to the third song. Thus, there are 2 choices for the song that Bernardo shares with Alberto. From here, we see that the songs that everyone is listening to are... |
DeepScaleR | [
{
"role": "user",
"content": "Alec must purchase 14 identical shirts and only has $\\$130$. There is a flat $\\$2$ entrance fee for shopping at the warehouse store where he plans to buy the shirts. The price of each shirt is the same whole-dollar amount. Assuming a $5\\%$ sales tax is added to the price ... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 4345,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alec wishes to construct a string of 6 letters using the letters A, C, G, and N, such that: - The first three letters are pairwise distinct, and so are the last three letters; - The first, second, fourth, and fifth letters are pairwise distinct. In how many ways can he construc... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"index": 4346,
"solution": "There are $4!=24$ ways to decide the first, second, fourth, and fifth letters because these letters can be selected sequentially without replacement from the four possible letters. Once these four letters are selected, there are 2 ways to select the third letter because two distinct le... |
DeepScaleR | [
{
"role": "user",
"content": "Alex and Bob have 30 matches. Alex picks up somewhere between one and six matches (inclusive), then Bob picks up somewhere between one and six matches, and so on. The player who picks up the last match wins. How many matches should Alex pick up at the beginning to guarantee tha... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 4347,
"solution": "2."
} |
DeepScaleR | [
{
"role": "user",
"content": "Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are $6$ cm in diameter and $12$ cm high. Felicia buys cat food in cylindrical cans that are $12$ cm in diameter and $6$ cm high. What is the ratio of the volume of one of Alex's cans to the vol... | math | {
"style": "rule",
"ground_truth": "1:2"
} | {
"index": 4348,
"solution": "To find the ratio of the volumes of Alex's and Felicia's cat food cans, we first need to calculate the volume of each can. The volume $V$ of a cylinder is given by the formula:\n\\[ V = \\pi r^2 h \\]\nwhere $r$ is the radius and $h$ is the height of the cylinder.\n\n1. **Calculate the... |
DeepScaleR | [
{
"role": "user",
"content": "Alex and Katy play a game on an \\(8 \\times 8\\) grid made of 64 unit cells. They take turns to play, with Alex going first. On Alex's turn, he writes 'A' in an empty cell. On Katy's turn, she writes 'K' in two empty cells that share an edge. The game ends when one player cann... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 4349,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alex bakes a total of $24$ pies, and each pie is apple, blueberry, or cherry. The ratio of apple to blueberry to cherry pies is $1:4:3$. How many cherry pies did Alex bake?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 4350,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alex chose positive integers $a, b, c, d, e, f$ and completely multiplied out the polynomial product $(1-x)^{a}(1+x)^{b}\\left(1-x+x^{2}\\right)^{c}\\left(1+x^{2}\\right)^{d}\\left(1+x+x^{2}\\right)^{e}\\left(1+x+x^{2}+x^{3}+x^{4}\\right)^{f}$. After she simplified her result, ... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 4351,
"solution": "Define $f(x)=(1-x)^{a}(1+x)^{b}\\left(1-x+x^{2}\\right)^{c}\\left(1+x^{2}\\right)^{d}\\left(1+x+x^{2}\\right)^{e}\\left(1+x+x^{2}+x^{3}+x^{4}\\right)^{f}$. We note several algebraic identities, each of which can be checked by expanding and simplifying: $1-x^{5}=(1-x)\\left(1+x+x^{2}+x^... |
DeepScaleR | [
{
"role": "user",
"content": "Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues ... | math | {
"style": "rule",
"ground_truth": "103"
} | {
"index": 4352,
"solution": "1. **Define the variables and equations:**\n Let $x$ be the number of times Alex visits the first booth, and $y$ be the number of times he visits the second booth. Each visit to the first booth changes the token counts as follows: Alex loses 2 red tokens and gains 1 blue token and 1 ... |
DeepScaleR | [
{
"role": "user",
"content": "Alex has 10 different kinds of lunch meat and 9 different kinds of cheese. If he wants to make a sandwich with one kind of meat and two kinds of cheese, how many different sandwiches could he make? (It does not matter in which order he chooses the two types of cheese.)"
}
] | math | {
"style": "rule",
"ground_truth": "360"
} | {
"index": 4353,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alex has 12 friends and 63 coins. What is the minimum number of additional coins he needs so that he can give each friend at least one coin and no two friends receive the same number of coins?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 4354,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alex has 5 pieces of paper, each with a different math problem. In how many ways can he give these problems to his 10 friends (he can give each friend more than one problem)?"
}
] | math | {
"style": "rule",
"ground_truth": "100,\\!000"
} | {
"index": 4355,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alex is at the candy store buying jellybeans. He wants to buy at least 100 jellybeans. He wants to buy the least amount such that he would have exactly $11$ leftover after dividing the jellybeans evenly among $13$ people. How many jellybeans should Alex buy?"
}
] | math | {
"style": "rule",
"ground_truth": "102"
} | {
"index": 4356,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alex is thinking of a number that is divisible by all of the positive integers 1 through 200 inclusive except for two consecutive numbers. What is the smaller of these numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "128"
} | {
"index": 4357,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alex is trying to open a lock whose code is a sequence that is three letters long, with each of the letters being one of $\\text A$ , $\\text B$ or $\\text C$ , possibly repeated. The lock has three buttons, labeled $\\text A$ , $\\text B$ and $\\text C$ . When the most... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 4358,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alex needs to catch a train. The train arrives randomly some time between 1:00 and 2:00, waits for 10 minutes, and then leaves. If Alex also arrives randomly between 1:00 and 2:00, what is the probability that the train will be there when Alex arrives?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{72}"
} | {
"index": 4359,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alex picks his favorite point $(x, y)$ in the first quadrant on the unit circle $x^{2}+y^{2}=1$, such that a ray from the origin through $(x, y)$ is $\\theta$ radians counterclockwise from the positive $x$-axis. He then computes $\\cos ^{-1}\\left(\\frac{4 x+3 y}{5}\\right)$ an... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 4360,
"solution": "$x=\\cos (\\theta), y=\\sin (\\theta)$. By the trig identity you never thought you'd need, $\\frac{4 x+3 y}{5}=\\cos (\\theta-\\phi)$, where $\\phi$ has sine $3 / 5$ and cosine $4 / 5$. Now $\\theta-\\phi=\\theta$ is impossible, since $\\phi \\neq 0$, so we must have $\\theta-\\phi=-\\... |
DeepScaleR | [
{
"role": "user",
"content": "Alex sent 150 text messages and talked for 28 hours, given a cell phone plan that costs $25 each month, $0.10 per text message, $0.15 per minute used over 25 hours, and $0.05 per minute within the first 25 hours. Calculate the total amount Alex had to pay in February."
}
] | math | {
"style": "rule",
"ground_truth": "142.00"
} | {
"index": 4361,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alex the Kat has written $61$ problems for a math contest, and there are a total of $187$ problems submitted. How many more problems does he need to write (and submit) before he has written half of the total problems?"
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"index": 4362,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alex thought of a two-digit number (from 10 to 99). Grisha tries to guess it by naming two-digit numbers. It is considered that he guessed the number if he correctly guessed one digit, and the other digit is off by no more than one (for example, if the number thought of is 65, ... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 4363,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alex wrote all natural divisors of a natural number \\( n \\) on the board in ascending order. Dima erased several of the first and several of the last numbers of the resulting sequence so that 151 numbers remained. What is the maximum number of these 151 divisors that could be... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 4364,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alex, Bonnie, and Chris each have $3$ blocks, colored red, blue, and green; and there are $3$ empty boxes. Each person independently places one of their blocks into each box. Each block placement by Bonnie and Chris is picked such that there is a 50% chance that the color match... | math | {
"style": "rule",
"ground_truth": "\\frac{37}{64}"
} | {
"index": 4365,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alex, Mel, and Chelsea play a game that has $6$ rounds. In each round there is a single winner, and the outcomes of the rounds are independent. For each round the probability that Alex wins is $\\frac{1}{2}$, and Mel is twice as likely to win as Chelsea. What is the probabil... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{36}"
} | {
"index": 4366,
"solution": "1. **Assign probabilities to each player:**\n Let $m$ be the probability that Mel wins a round, and $c$ be the probability that Chelsea wins a round. We know that:\n - Alex's probability of winning is $\\frac{1}{2}$.\n - Mel is twice as likely to win as Chelsea, so $m = 2c$.\n ... |
DeepScaleR | [
{
"role": "user",
"content": "Alexa wrote the first $16$ numbers of a sequence:\n\\[1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, …\\]\nThen she continued following the same pattern, until she had $2015$ numbers in total. \nWhat was the last number she wrote?"
}
] | math | {
"style": "rule",
"ground_truth": "1344"
} | {
"index": 4367,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alexey plans to buy one of two car brands: \"A\" costing 900,000 rubles or \"B\" costing 600,000 rubles. On average, Alexey drives 15,000 km per year. The cost of gasoline is 40 rubles per liter. The cars use the same type of gasoline. The car is planned to be used for 5 years,... | math | {
"style": "rule",
"ground_truth": "160000"
} | {
"index": 4368,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alexia designed a logo 2 inches wide and 1.5 inches tall to be used on her school's notebooks. The school wants the logo on the notebook to be 8 inches wide. How tall, in inches, will it be if it is enlarged proportionately?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 4369,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alexio has 100 cards numbered 1-100, inclusive, and places them in a box. Alexio then chooses a card from the box at random. What is the probability that the number on the card he chooses is a multiple of 2, 3 or 5? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{37}{50}"
} | {
"index": 4370,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alexio has 120 cards numbered 1-120, inclusive, and places them in a box. Alexio then chooses a card from the box at random. What is the probability that the number on the card he chooses is a multiple of 3, 4, or 7? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{69}{120}"
} | {
"index": 4371,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alexio has 120 cards numbered from 1 to 120, inclusive, and places them in a box. He then randomly picks a card. What is the probability that the number on the card is a multiple of 2, 4, or 6? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 4372,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alexio has 200 cards numbered 1-200, inclusive, and places them in a box. Alexio then chooses a card from the box at random. What is the probability that the number on the card he chooses is a multiple of 4, 5, or 7? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{97}{200}"
} | {
"index": 4373,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alexio now has 100 cards numbered from 1 to 100. He again randomly selects one card from the box. What is the probability that the number on the chosen card is a multiple of 3, 5, or 7? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{20}"
} | {
"index": 4374,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alexio now has 150 cards numbered from 1 to 150, inclusive, and places them in a box. He chooses a card at random. What is the probability that the number on the card he picks is a multiple of 4, 5 or 6? Express your answer as a reduced fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{15}"
} | {
"index": 4375,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alexio now has 150 cards numbered from 1 to 150, inclusive, and places them in a box. He then chooses a card from the box at random. What is the probability that the number on the card he chooses is a multiple of 2, 3, or 7? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{107}{150}"
} | {
"index": 4376,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alexis imagines a $2008\\times 2008$ grid of integers arranged sequentially in the following way:\n\n\\[\\begin{array}{r@{\\hspace{20pt}}r@{\\hspace{20pt}}r@{\\hspace{20pt}}r@{\\hspace{20pt}}r}1,&2,&3,&\\ldots,&20082009,&2010,&2011,&\\ldots,&40264017,&4018,&4019,&\\ldots,&602... | math | {
"style": "rule",
"ground_truth": "1004"
} | {
"index": 4377,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alfred and Bonnie play a game in which they take turns tossing a fair coin. The winner of a game is the first person to obtain a head. Alfred and Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred g... | math | {
"style": "rule",
"ground_truth": "93"
} | {
"index": 4378,
"solution": "The probability that the $n$th flip in each game occurs and is a head is $\\frac{1}{2^n}$. The first person wins if the coin lands heads on an odd numbered flip. So, the probability of the first person winning the game is $\\frac{1}{2}+\\frac{1}{8}+\\frac{1}{32}+\\cdots = \\frac{\\frac... |
DeepScaleR | [
{
"role": "user",
"content": "Ali Baba and the 40 thieves are dividing their loot. The division is considered fair if any 30 participants receive at least half of the loot in total. What is the maximum share that Ali Baba can receive in a fair division?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 4379,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Ali Baba and the 40 thieves decided to divide a treasure of 1987 gold coins in the following manner: the first thief divides the entire treasure into two parts, then the second thief divides one of the parts into two parts, and so on. After the 40th division, the first thief pi... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"index": 4380,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Ali Baba and the thief are dividing a hoard consisting of 100 gold coins, distributed in 10 piles of 10 coins each. Ali Baba chooses 4 piles, places a cup near each of them, and puts a certain number of coins (at least one, but not the entire pile) into each cup. The thief must... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 4381,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Ali chooses one of the stones from a group of $2005$ stones, marks this stone in a way that Betül cannot see the mark, and shuffles the stones. At each move, Betül divides stones into three non-empty groups. Ali removes the group with more stones from the two groups that do n... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 4382,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Ali wants to move from point $A$ to point $B$. He cannot walk inside the black areas but he is free to move in any direction inside the white areas (not only the grid lines but the whole plane). Help Ali to find the shortest path between $A$ and $B$. Only draw the path and writ... | math | {
"style": "rule",
"ground_truth": "7 + 5\\sqrt{2}"
} | {
"index": 4383,
"solution": "\nThe task is to find the shortest path for Ali to move from point \\( A \\) to point \\( B \\), only navigating through the white areas in the given plane. Based on the diagram provided, we will employ geometric considerations to determine the path and length.\n\n### Geometric Analysi... |
DeepScaleR | [
{
"role": "user",
"content": "Alice Czarina is bored and is playing a game with a pile of rocks. The pile initially contains 2015 rocks. At each round, if the pile has $N$ rocks, she removes $k$ of them, where $1 \\leq k \\leq N$, with each possible $k$ having equal probability. Alice Czarina continues unti... | math | {
"style": "rule",
"ground_truth": "-501"
} | {
"index": 4384,
"solution": "We claim that $p=\\frac{1}{5} \\frac{6}{10} \\frac{11}{15} \\frac{16}{20} \\cdots \\frac{2006}{2010} \\frac{2011}{2015}$. Let $p_{n}$ be the probability that, starting with $n$ rocks, the number of rocks left after each round is a multiple of 5. Indeed, using recursions we have $$p_{5 ... |
DeepScaleR | [
{
"role": "user",
"content": "Alice and Ali each select a positive integer less than 250. Alice's number is a multiple of 25, and Ali's number is a multiple of 30. What is the probability that they selected the same number? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{80}"
} | {
"index": 4385,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice and Bob are playing a game where Alice declares, \"My number is 36.\" Bob has to choose a number such that all the prime factors of Alice's number are also prime factors of his, but with the condition that the exponent of at least one prime factor in Bob's number is stric... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 4386,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice and Bob are playing a game with dice. They each roll a die six times and take the sums of the outcomes of their own rolls. The player with the higher sum wins. If both players have the same sum, then nobody wins. Alice's first three rolls are 6, 5, and 6, while Bob's firs... | math | {
"style": "rule",
"ground_truth": "3895"
} | {
"index": 4387,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice and Bob are playing in the forest. They have six sticks of length $1,2,3,4,5,6$ inches. Somehow, they have managed to arrange these sticks, such that they form the sides of an equiangular hexagon. Compute the sum of all possible values of the area of this hexagon."
}
] | math | {
"style": "rule",
"ground_truth": "33 \\sqrt{3}"
} | {
"index": 4388,
"solution": "Let the side lengths, in counterclockwise order, be $a, b, c, d, e, f$. Place the hexagon on the coordinate plane with edge $a$ parallel to the $x$-axis and the intersection between edge $a$ and edge $f$ at the origin (oriented so that edge $b$ lies in the first quadrant). If you trave... |
DeepScaleR | [
{
"role": "user",
"content": "Alice and Bob are playing the Smallest Positive Integer Game. Alice says, \"My number is 24.\" Bob says, \"What kind of silly smallest number is that? Every prime factor of your number is also a prime factor of my number.\"\n\nWhat is the smallest possible number that Bob could... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 4389,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice and Bob are stuck in quarantine, so they decide to play a game. Bob will write down a polynomial $f(x)$ with the following properties:\n\n(a) for any integer $n$ , $f(n)$ is an integer;\n(b) the degree of $f(x)$ is less than $187$ .\n\nAlice knows that $f(x)$ sa... | math | {
"style": "rule",
"ground_truth": "187"
} | {
"index": 4390,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice and Bob each arrive at a gathering at a random time between 12:00 noon and 1:00 PM. If Alice arrives after Bob, what is the probability that Bob arrived before 12:45 PM?"
}
] | math | {
"style": "rule",
"ground_truth": "0.5625"
} | {
"index": 4391,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice and Bob each arrive at a meeting at a random time between 8:00 and 9:00 AM. If Alice arrives after Bob, what is the probability that Bob arrived before 8:45 AM?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{16}"
} | {
"index": 4392,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice and Bob each arrive at a party at a random time between 1:00 and 2:00. If Alice arrives after Bob, what is the probability that Bob arrived before 1:30?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 4393,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice and Bob each draw one number from 50 slips of paper numbered from $1$ to $50$ placed in a hat. Alice says, \"I can't tell who has the larger number.\" Bob then says, \"I am certain who has the larger number.\" Understanding Bob's certainty, Alice asks Bob if his number is... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"index": 4394,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice and Bob have an $8 \\times 8$ chessboard in front of them. Initially, all the squares are white. Each turn, Alice selects a white square and colors it black. Bob then chooses to color one of the neighboring squares (sharing an edge) black or does nothing. Alice can stop t... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 4395,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice and Bob live on the same road. At time $t$ , they both decide to walk to each other's houses at constant speed. However, they were busy thinking about math so that they didn't realize passing each other. Alice arrived at Bob's house at $3:19\\text{pm}$ , and Bob arrived... | math | {
"style": "rule",
"ground_truth": "179"
} | {
"index": 4396,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice and Bob play a game around a circle divided into 15 equally spaced points, numbered 1 through 15. Alice moves 7 points clockwise per turn, and Bob moves 4 points counterclockwise per turn. Determine how many turns will be required for Alice and Bob to land on the same poi... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 4397,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Alice and Bob play a game involving a circle whose circumference is divided by 12 equally-spaced points. The points are numbered clockwise, from 1 to 12. Both start on point 12. Alice moves clockwise and Bob, counterclockwise.\nIn a turn of the game, Alice moves 5 points clockw... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 4398,
"solution": "1. **Identify the movement of Alice and Bob**: \n - Alice moves 5 points clockwise.\n - Bob moves 9 points counterclockwise, which is equivalent to moving 3 points clockwise (since 9 counterclockwise on a 12-point circle is the same as 3 clockwise: $12 - 9 = 3$).\n\n2. **Calculate ... |
DeepScaleR | [
{
"role": "user",
"content": "Alice and Bob play a game on a board consisting of one row of 2022 consecutive squares. They take turns placing tiles that cover two adjacent squares, with Alice going first. By rule, a tile must not cover a square that is already covered by another tile. The game ends when no ... | math | {
"style": "rule",
"ground_truth": "290"
} | {
"index": 4399,
"solution": "We show that the number in question equals 290. More generally, let $a(n)$ (resp.\\ $b(n)$) be the optimal final score for Alice (resp.\\ Bob) moving first in a position with $n$ consecutive squares. We show that \\begin{align*} a(n) &= \\left\\lfloor \\frac{n}{7} \\right\\rfloor + a\\... |
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