data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "A circular spinner for a game has a radius of 10 cm. The probability of winning on one spin of this spinner is $\\frac{2}{5}$. What is the area, in sq cm, of the WIN sector? Express your answer in terms of $\\pi$.\n\n[asy]import graph;\n\ndraw(Circle((0,0),25),black);\n\ndraw((... | math | {
"style": "rule",
"ground_truth": "40\\pi"
} | {
"index": 1300,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circular spinner for a game has a radius of 5 cm. The probability of winning on one spin of this spinner is $\\frac{2}{5}$. What is the area, in sq cm, of the WIN sector? Express your answer in terms of $\\pi$.\n\n[asy]import graph;\ndraw(Circle((0,0),25),black);\ndraw((0,0)-... | math | {
"style": "rule",
"ground_truth": "10\\pi"
} | {
"index": 1301,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circular spinner used in a game has a radius of 15 cm. The probability of winning on one spin of this spinner is $\\frac{1}{3}$ for the WIN sector and $\\frac{1}{4}$ for the BONUS sector. What is the area, in square centimeters, of both the WIN sector and the BONUS sector? Ex... | math | {
"style": "rule",
"ground_truth": "56.25\\pi"
} | {
"index": 1302,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A circular table has 60 chairs around it. There are $N$ people seated at this table in such a way that the next person seated must sit next to someone. What is the smallest possible value for $N$?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 1303,
"solution": "To find the smallest possible value for $N$, we need to ensure that any new person seated must sit next to someone already seated. We will analyze the seating arrangement to determine the minimum $N$ that satisfies this condition.\n\n1. **Understanding the seating arrangement**: The ta... |
DeepScaleR | [
{
"role": "user",
"content": "A city adopts a lottery system for \"price-limited housing,\" where winning families can randomly draw a house number from the available housing in a designated community. It is known that two friendly families, Family A and Family B, have both won the lottery and decided to go... | math | {
"style": "rule",
"ground_truth": "\\dfrac{3}{5}"
} | {
"index": 1304,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A city does not have electric lighting yet, so candles are used in houses at night. In João's house, one candle is used per night without burning it completely, and with four of these candle stubs, João makes a new candle. How many nights can João light up his house with 43 can... | math | {
"style": "rule",
"ground_truth": "57"
} | {
"index": 1305,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A city uses a lottery system for assigning car permits, with 300,000 people participating in the lottery and 30,000 permits available each month.\n1. If those who win the lottery each month exit the lottery, and those who do not win continue in the following month's lottery, wi... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 1306,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A class collects 50 dollars to buy flowers for a classmate who is in the hospital. Roses cost 3 dollars each, and carnations cost 2 dollars each. No other flowers are to be used. How many different bouquets could be purchased for exactly 50 dollars?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 1307,
"solution": "1. **Formulate the Problem:**\n Let $r$ represent the number of roses, and $c$ represent the number of carnations. The cost of roses is $3$ dollars each and carnations $2$ dollars each. The total amount spent is $50$ dollars. Therefore, we can set up the equation:\n \\[\n 3r + 2c... |
DeepScaleR | [
{
"role": "user",
"content": "A class has $25$ students. The teacher wants to stock $N$ candies, hold the Olympics and give away all $N$ candies for success in it (those who solve equally tasks should get equally, those who solve less get less, including, possibly, zero candies). At what smallest $N... | math | {
"style": "rule",
"ground_truth": "600"
} | {
"index": 1308,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A class has $50$ students. The math scores $\\xi$ of an exam follow a normal distribution $N(100, 10^{2})$. Given that $P(90 \\leqslant \\xi \\leqslant 100)=0.3$, estimate the number of students with scores of $110$ or higher."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 1309,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A class has 50 students, and their scores in a math test $\\xi$ follow a normal distribution $N(100, 10^2)$. It is known that $P(90 \\leq \\xi \\leq 100) = 0.3$. Estimate the number of students scoring above 110."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 1310,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A class has 54 students, and there are 4 tickets for the Shanghai World Expo to be distributed among the students using a systematic sampling method. If it is known that students with numbers 3, 29, and 42 have already been selected, then the student number of the fourth select... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 1311,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A class has a group of 7 people, and now 3 of them are chosen to swap seats with each other, while the remaining 4 people's seats remain unchanged. Calculate the number of different rearrangement plans."
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 1312,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A class has a group of 7 students, and now select 3 of them to swap seats with each other, while the remaining 4 students keep their seats unchanged. Calculate the number of different ways to adjust their seats."
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 1313,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A class has a total of 54 students. Now, using the systematic sampling method based on the students' ID numbers, a sample of 4 students is drawn. It is known that students with ID numbers 3, 29, and 42 are in the sample. What is the ID number of the fourth student in the sample... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 1314,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A class has prepared 6 programs to participate in the Xiamen No.1 Middle School Music Square event. The order of the programs has the following requirements: Programs A and B must be adjacent, and Programs C and D cannot be adjacent. How many possible arrangements of the progra... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 1315,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A class is scheduled to have 6 classes in one day: Chinese, Mathematics, Politics, English, PE, and Art. It is required that the Mathematics class is scheduled within the first 3 periods, and the PE class cannot be scheduled in the first period. The number of different scheduli... | math | {
"style": "rule",
"ground_truth": "312"
} | {
"index": 1316,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A class leader is planning to invite graduates from the class of 2016 to give speeches. Out of 8 people, labeled A, B, ..., H, the leader wants to select 4 to speak. The conditions are: (1) at least one of A and B must participate; (2) if both A and B participate, there must be... | math | {
"style": "rule",
"ground_truth": "1080"
} | {
"index": 1317,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A class of 10 students took a math test. Each problem was solved by exactly 7 of the students. If the first nine students each solved 4 problems, how many problems did the tenth student solve?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 1318,
"solution": "Suppose the last student solved $n$ problems, and the total number of problems on the test was $p$. Then the total number of correct solutions written was $7 p$ (seven per problem), and also equal to $36+n$ (the sum of the students' scores), so $p=(36+n) / 7$. The smallest $n \\geq 0$ ... |
DeepScaleR | [
{
"role": "user",
"content": "A class of 30 students was asked what they did on their winter holiday. 20 students said that they went skating. 9 students said that they went skiing. Exactly 5 students said that they went skating and went skiing. How many students did not go skating and did not go skiing?"
... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 1319,
"solution": "Since 20 students went skating and 5 students went both skating and skiing, then \\( 20-5=15 \\) students went skating only. Since 9 students went skiing and 5 students went both skating and skiing, then \\( 9-5=4 \\) students went skiing only. The number of students who went skating o... |
DeepScaleR | [
{
"role": "user",
"content": "A class of 54 students in the fifth grade took a group photo. The fixed price is 24.5 yuan for 4 photos. Additional prints cost 2.3 yuan each. If every student in the class wants one photo, how much money in total needs to be paid?"
}
] | math | {
"style": "rule",
"ground_truth": "139.5"
} | {
"index": 1320,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A class participates in a tree-planting event, divided into three groups. The first group plants 5 trees per person, the second group plants 4 trees per person, and the third group plants 3 trees per person. It is known that the number of people in the second group is one-third... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 1321,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A class prepared 5 programs to participate in the Xiamen No.1 Middle School Music Square event (this event only has 5 programs), and the order of the programs has the following requirements: Program A must be in the first two positions, Program B cannot be in the first position... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 1322,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A classroom consists of a $5 \\times 5$ array of desks, to be filled by anywhere from 0 to 25 students, inclusive. No student will sit at a desk unless either all other desks in its row or all others in its column are filled (or both). Considering only the set of desks that are... | math | {
"style": "rule",
"ground_truth": "962"
} | {
"index": 1323,
"solution": "The set of empty desks must be of the form (non-full rows) $\\times$ (non-full columns): each empty desk is in a non-full column and a non-full row, and the given condition implies that each desk in such a position is empty. So if there are fewer than 25 students, then both of these se... |
DeepScaleR | [
{
"role": "user",
"content": "A classroom has 10 chairs arranged in a row. Tom and Jerry choose their seats at random, but they are not allowed to sit on the first and last chairs. What is the probability that they don't sit next to each other?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 1324,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A classroom has 9 desks arranged in a row. Two students, Alex and Jessie, choose their desks at random. What is the probability that they do not sit next to each other?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{9}"
} | {
"index": 1325,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A classroom is paved with cubic bricks that have an edge length of 0.3 meters, requiring 600 bricks. If changed to cubic bricks with an edge length of 0.5 meters, how many bricks are needed? (Solve using proportions.)"
}
] | math | {
"style": "rule",
"ground_truth": "216"
} | {
"index": 1326,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A client of a brokerage company deposited 12,000 rubles into a brokerage account at a rate of 60 rubles per dollar with instructions to the broker to invest the amount in bonds of foreign banks, which have a guaranteed return of 12% per annum in dollars.\n\n(a) Determine the am... | math | {
"style": "rule",
"ground_truth": "39.52\\%"
} | {
"index": 1327,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A clock chimes 2 times at 2 o'clock, taking 2 seconds to complete. How many seconds does it take to chime 12 times at 12 o'clock?"
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 1328,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A clock currently shows the time $10:10$ . The obtuse angle between the hands measures $x$ degrees. What is the next time that the angle between the hands will be $x$ degrees? Round your answer to the nearest minute."
}
] | math | {
"style": "rule",
"ground_truth": "11:15"
} | {
"index": 1329,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A clock has a hour hand $OA$ and a minute hand $OB$ with lengths of $3$ and $4$ respectively. If $0$ hour is represented as $0$ time, then the analytical expression of the area $S$ of $\\triangle OAB$ with respect to time $t$ (unit: hours) is ______, and the number of times $S$... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"index": 1330,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A clock has a second, minute, and hour hand. A fly initially rides on the second hand of the clock starting at noon. Every time the hand the fly is currently riding crosses with another, the fly will then switch to riding the other hand. Once the clock strikes midnight, how man... | math | {
"style": "rule",
"ground_truth": "245"
} | {
"index": 1331,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A clock has an hour hand of length 3 and a minute hand of length 4. From 1:00 am to 1:00 pm of the same day, find the number of occurrences when the distance between the tips of the two hands is an integer."
}
] | math | {
"style": "rule",
"ground_truth": "132"
} | {
"index": 1332,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A clock takes $7$ seconds to strike $9$ o'clock starting precisely from $9:00$ o'clock. If the interval between each strike increases by $0.2$ seconds as time progresses, calculate the time it takes to strike $12$ o'clock."
}
] | math | {
"style": "rule",
"ground_truth": "12.925"
} | {
"index": 1333,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrappi... | math | {
"style": "rule",
"ground_truth": "2(w+h)^2"
} | {
"index": 1334,
"solution": "1. **Understanding the Problem Setup**: We have a square sheet of wrapping paper and a box with a square base of side $w$ and height $h$. The box is placed such that its base vertices lie on the midlines of the wrapping paper. The wrapping paper is folded up to meet at a point $A$ at t... |
DeepScaleR | [
{
"role": "user",
"content": "A clothing retailer offered a discount of $\\frac{1}{4}$ on all jackets tagged at a specific price. If the cost of the jackets was $\\frac{2}{3}$ of the price they were actually sold for and considering this price included a sales tax of $\\frac{1}{10}$, what would be the ratio... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{30}"
} | {
"index": 1335,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A clothing store buys 600 pairs of gloves at a price of 12 yuan per pair. They sell 470 pairs at a price of 14 yuan per pair, and the remaining gloves are sold at a price of 11 yuan per pair. What is the total profit made by the clothing store from selling this batch of gloves?... | math | {
"style": "rule",
"ground_truth": "810"
} | {
"index": 1336,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A club consisting of $11$ men and $12$ women needs to choose a committee from among its members so that the number of women on the committee is one more than the number of men on the committee. The committee could have as few as $1$ member or as many as $23$ members. Let $N$ be... | math | {
"style": "rule",
"ground_truth": "81"
} | {
"index": 1337,
"solution": "Let $k$ be the number of women selected. Then, the number of men not selected is $11-(k-1)=12-k$. Note that the sum of the number of women selected and the number of men not selected is constant at $12$. Each combination of women selected and men not selected corresponds to a committee... |
DeepScaleR | [
{
"role": "user",
"content": "A club consists initially of 20 total members, which includes eight leaders. Each year, all the current leaders leave the club, and each remaining member recruits three new members. Afterwards, eight new leaders are elected from outside. How many total members will the club hav... | math | {
"style": "rule",
"ground_truth": "980"
} | {
"index": 1338,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A club consists of five leaders and some regular members. Each year, all leaders leave the club and each regular member recruits three new people to join as regular members. Subsequently, five new leaders are elected from outside the club to join. Initially, there are eighteen ... | math | {
"style": "rule",
"ground_truth": "3164"
} | {
"index": 1339,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A club consists of three board members and a certain number of regular members. Every year, the board members retire and are not replaced. Each regular member recruits one new person to join as a regular member. Initially, there are nine people in the club total. How many peopl... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"index": 1340,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A club has $5$ members from each of $3$ different schools, for a total of $15$ members. How many possible ways are there to arrange a presidency meeting under the following conditions:\n\ni. The club must choose one of the $3$ schools at which to host the meeting, and\n\nii. Th... | math | {
"style": "rule",
"ground_truth": "750"
} | {
"index": 1341,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A club has 10 members, 5 boys and 5 girls. Two of the members are chosen at random. What is the probability that they are both girls?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{2}{9}"
} | {
"index": 1342,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A club has 12 members - 6 boys and 6 girls. Each member is also categorized either as a senior or junior with equal distribution among genders. Two of the members are chosen at random. What is the probability that they are both girls where one girl is a senior and the other is ... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{66}"
} | {
"index": 1343,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A club has 12 members, and wishes to pick a president, a vice-president, a secretary, and a treasurer. However, the president and vice-president must have been members of the club for at least 3 years. If 4 of the existing members meet this criterion, in how many ways can these... | math | {
"style": "rule",
"ground_truth": "1080"
} | {
"index": 1344,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A club has 15 members and needs to choose 2 members to be co-presidents. In how many ways can the club choose its co-presidents?"
}
] | math | {
"style": "rule",
"ground_truth": "105"
} | {
"index": 1345,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A club has 20 members and needs to choose 2 members to be co-presidents and 1 member to be a treasurer. In how many ways can the club select its co-presidents and treasurer?"
}
] | math | {
"style": "rule",
"ground_truth": "3420"
} | {
"index": 1346,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A club has 30 members, which includes 12 females and 18 males. In how many ways can a 5-person executive committee be formed such that at least one member must be female?"
}
] | math | {
"style": "rule",
"ground_truth": "133,938"
} | {
"index": 1347,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A club has 99 members. Find the smallest positive integer $n$ such that if the number of acquaintances of each person is greater than $n$, there must exist 4 people who all know each other mutually (here it is assumed that if $A$ knows $B$, then $B$ also knows $A$)."
}
] | math | {
"style": "rule",
"ground_truth": "66"
} | {
"index": 1348,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A club has between 150 and 250 members. Every month, all the members meet up for a group activity that requires the members to be divided into seven distinct groups. If one member is unable to attend, the remaining members can still be evenly divided into the seven groups. Calc... | math | {
"style": "rule",
"ground_truth": "2807"
} | {
"index": 1349,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A club has between 300 and 400 members. The members gather every weekend and are divided into eight distinct groups. If two members are absent, the groups can all have the same number of members. What is the sum of all possible numbers of members in the club?"
}
] | math | {
"style": "rule",
"ground_truth": "4200"
} | {
"index": 1350,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A club has increased its membership to 12 members and needs to elect a president, vice president, secretary, and treasurer. Additionally, they want to appoint two different advisory board members. Each member can hold only one position. In how many ways can these positions be f... | math | {
"style": "rule",
"ground_truth": "665,280"
} | {
"index": 1351,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A club is creating a special series of membership cards that includes a code consisting of five characters selected from the letters B, E, T, A and the digits in 2023. No character should appear in a code more often than it appears in \"BETA\" or \"2023\" (B, E, T, A appear onc... | math | {
"style": "rule",
"ground_truth": "312"
} | {
"index": 1352,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A clueless ant makes the following route: starting at point $ A $ goes $ 1$ cm north, then $ 2$ cm east, then $ 3$ cm south, then $ 4$ cm west, immediately $ 5$ cm north, continues $ 6$ cm east, and so on, finally $ 41$ cm north and ends in point $ B $ . Calcul... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 1353,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A coin collector has 100 identical-looking coins. Among them, there are 30 genuine coins and 70 counterfeit coins. The collector knows that all genuine coins have the same weight, all counterfeit coins have different weights, and all counterfeit coins are heavier than the genui... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 1354,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A coin is altered so that the probability that it lands on heads is less than $\\frac{1}{2}$ and when the coin is flipped four times, the probability of an equal number of heads and tails is $\\frac{1}{6}$. What is the probability that the coin lands on heads?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3-\\sqrt{3}}{6}"
} | {
"index": 1355,
"solution": "1. **Define the probability of heads**: Let $x$ be the probability that the coin lands on heads. Consequently, the probability that the coin lands on tails is $1 - x$.\n\n2. **Set up the probability equation for 2 heads and 2 tails in 4 flips**: The number of ways to choose 2 heads out... |
DeepScaleR | [
{
"role": "user",
"content": "A coin is flipped eight times, and the sequence of heads and tails occurring is recorded. How many distinct sequences are possible?"
}
] | math | {
"style": "rule",
"ground_truth": "256"
} | {
"index": 1356,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A coin is tossed 10 times. Find the probability that no two heads appear consecutively."
}
] | math | {
"style": "rule",
"ground_truth": "9/64"
} | {
"index": 1357,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A coin is tossed 10 times. Find the probability that two heads never appear consecutively."
}
] | math | {
"style": "rule",
"ground_truth": "9/64"
} | {
"index": 1358,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A coin is tossed. If heads appear, point \\( P \\) moves +1 on the number line; if tails appear, point \\( P \\) does not move. The coin is tossed no more than 12 times, and if point \\( P \\) reaches coordinate +10, the coin is no longer tossed. In how many different ways can ... | math | {
"style": "rule",
"ground_truth": "66"
} | {
"index": 1359,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A coin that comes up heads with probability $p > 0$ and tails with probability $1 - p > 0$ independently on each flip is flipped $8$ times. Suppose that the probability of three heads and five tails is equal to $\\frac {1}{25}$ of the probability of five heads and three tails. ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 1360,
"solution": "Rewrite it as : $(P)^3$$(1-P)^5=\\frac {1}{25}$ $(P)^5$$(1-P)^3$\n This can be simplified as $24P^2 -50P + 25 = 0$\n This can be factored into $(4P-5)(6P-5)$\n This yields two solutions: $\\frac54$ (ignored because it would result in $1-p<0$ ) or $\\frac56$\n Therefore, the answer is $... |
DeepScaleR | [
{
"role": "user",
"content": "A collection $\\mathcal{S}$ of 10000 points is formed by picking each point uniformly at random inside a circle of radius 1. Let $N$ be the expected number of points of $\\mathcal{S}$ which are vertices of the convex hull of the $\\mathcal{S}$. (The convex hull is the smallest ... | math | {
"style": "rule",
"ground_truth": "72.8"
} | {
"index": 1361,
"solution": "Here is C++ code by Benjamin Qi to estimate the answer via simulation. It is known that the expected number of vertices of the convex hull of $n$ points chosen uniformly at random inside a circle is $O\\left(n^{1 / 3}\\right)$. See \"On the Expected Complexity of Random Convex Hulls\" ... |
DeepScaleR | [
{
"role": "user",
"content": "A collection of $n$ squares on the plane is called tri-connected if the following criteria are satisfied: (i) All the squares are congruent. (ii) If two squares have a point $P$ in common, then $P$ is a vertex of each of the squares. (iii) Each square touches exactly three othe... | math | {
"style": "rule",
"ground_truth": "501"
} | {
"index": 1362,
"solution": "We will prove that there is no tri-connected collection if $n$ is odd, and that tri-connected collections exist for all even $n \\geq 38$. Since there are 501 even numbers in the range from 2018 to 3018, this yields 501 as the answer. For any two different squares $A$ and $B$, let us w... |
DeepScaleR | [
{
"role": "user",
"content": "A collection of 8 cubes consists of one cube with edge-length $k$ for each integer $k, 1 \\le k \\le 8.$ A tower is to be built using all 8 cubes according to the rules:\nAny cube may be the bottom cube in the tower.\nThe cube immediately on top of a cube with edge-length $k$ m... | math | {
"style": "rule",
"ground_truth": "458"
} | {
"index": 1363,
"solution": "We proceed recursively. Suppose we can build $T_m$ towers using blocks of size $1, 2, \\ldots, m$. How many towers can we build using blocks of size $1, 2, \\ldots, m, m + 1$? If we remove the block of size $m + 1$ from such a tower (keeping all other blocks in order), we get a valid t... |
DeepScaleR | [
{
"role": "user",
"content": "A collection of 8 cubes consists of one cube with edge-length $k$ for each integer $k, 1 \\le k \\le 8.$ A tower is to be built using all 8 cubes according to the rules:\nAny cube may be the bottom cube in the tower.\nThe cube immediately on top of a cube with edge-length $k$ m... | math | {
"style": "rule",
"ground_truth": "458"
} | {
"index": 1364,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A collection of circles in the upper half-plane, all tangent to the $x$-axis, is constructed in layers as follows. Layer $L_0$ consists of two circles of radii $70^2$ and $73^2$ that are externally tangent. For $k \\ge 1$, the circles in $\\bigcup_{j=0}^{k-1}L_j$ are ordered ac... | math | {
"style": "rule",
"ground_truth": "\\frac{143}{14}"
} | {
"index": 1365,
"solution": "1. **Identify the radii of initial circles**: Let the two circles from $L_0$ be of radius $r_1 = 70^2$ and $r_2 = 73^2$, with $r_1 < r_2$. Let the circle of radius $r_1$ be circle $A$ and the circle of radius $r_2$ be circle $B$.\n\n2. **Constructing the circle in $L_1$**: A new circle... |
DeepScaleR | [
{
"role": "user",
"content": "A collection of five positive integers has mean 4.4, unique mode 3 and median 4. If an 8 is added to the collection, what is the new median? Express your answer as a decimal to the nearest tenth."
}
] | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"index": 1366,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A collection of seven positive integers has a mean of 6, a unique mode of 4, and a median of 6. If a 12 is added to this collection, what is the new median?"
}
] | math | {
"style": "rule",
"ground_truth": "6.5"
} | {
"index": 1367,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A collector has \\( N \\) precious stones. If he takes away the three heaviest stones, then the total weight of the stones decreases by \\( 35\\% \\). From the remaining stones, if he takes away the three lightest stones, the total weight further decreases by \\( \\frac{5}{13} ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 1368,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A college student drove his compact car $120$ miles home for the weekend and averaged $30$ miles per gallon. On the return trip the student drove his parents' SUV and averaged only $20$ miles per gallon. What was the average gas mileage, in miles per gallon, for the round trip?... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 1369,
"solution": "1. **Calculate the total distance of the round trip**: \n The student drove $120$ miles to his home and $120$ miles back, so the total distance for the round trip is:\n \\[\n 120 \\text{ miles} + 120 \\text{ miles} = 240 \\text{ miles}\n \\]\n\n2. **Calculate the amount of gaso... |
DeepScaleR | [
{
"role": "user",
"content": "A coloring of all plane points with coordinates belonging to the set $S=\\{0,1,\\ldots,99\\}$ into red and white colors is said to be *critical* if for each $i,j\\in S$ at least one of the four points $(i,j),(i + 1,j),(i,j + 1)$ and $(i + 1, j + 1)$ $(99 + 1\\equiv0)$ ... | math | {
"style": "rule",
"ground_truth": "5000"
} | {
"index": 1370,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A column of infantry stretched out over 1 km. Sergeant Kim, riding on a hoverboard from the end of the column, reached its front and returned to the end. During this time, the infantrymen walked 2 km 400 meters. What distance did the sergeant cover during this time?"
}
] | math | {
"style": "rule",
"ground_truth": "3.6"
} | {
"index": 1371,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A committee of 5 is to be chosen from a group of 9 people. How many ways can it be chosen, if Bill and Karl must serve together or not at all, and Alice and Jane refuse to serve with each other?"
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"index": 1372,
"solution": "If Bill and Karl are on the committee, there are $\\binom{7}{3}=35$ ways for the other group members to be chosen. However, if Alice and Jane are on the committee with Bill and Karl, there are $\\binom{5}{1}=5$ ways for the last member to be chosen, yielding 5 unacceptable committees. ... |
DeepScaleR | [
{
"role": "user",
"content": "A community organization begins with twenty members, among which five are leaders. The leaders are replaced annually. Each remaining member persuades three new members to join the organization every year. Additionally, five new leaders are elected from outside the community eac... | math | {
"style": "rule",
"ground_truth": "15365"
} | {
"index": 1373,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A community plans to organize three activities, \"Book Club,\" \"Fun Sports,\" and \"Environmental Theme Painting,\" to enrich the lives of residents. A total of 120 people have signed up for the activities, with each resident participating in at most two activities. It is know... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 1374,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A company allocates 5 employees to 3 different departments, with each department being allocated at least one employee. Among them, employees A and B must be allocated to the same department. Calculate the number of different allocation methods."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 1375,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A company buys an assortment of 150 pens from a catalog for \\$15.00. Shipping costs an additional \\$5.50. Furthermore, they receive a 10% discount on the total price due to a special promotion. What is the average cost, in cents, for each pen?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 1376,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A company conducts quality checks on a batch of products using systematic sampling. From 100 products, 5 are to be sampled for testing. The products are randomly numbered and divided into 5 groups: Group 1 contains numbers 1–20, Group 2 contains numbers 21–40, and so on up to G... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 1377,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A company decides to increase the price of a product by 20%. If they aim to increase their total income by 10% despite the decrease in demand, by what proportion must the demand decrease to meet this new income goal?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"index": 1378,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A company gathered at a meeting. Let's call a person sociable if, in this company, they have at least 20 acquaintances, with at least two of those acquaintances knowing each other. Let's call a person shy if, in this company, they have at least 20 non-acquaintances, with at lea... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 1379,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A company has 200 employees randomly assigned numbers from 1 to 200, and they are evenly divided into 40 groups. If employee number 22 is selected from the 5th group, find the number of the employee to be selected from the 10th group."
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"index": 1380,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A company has 45 male employees and 15 female employees. A 4-person research and development team was formed using stratified sampling. \n(1) Calculate the probability of an employee being selected and the number of male and female employees in the research and development tea... | math | {
"style": "rule",
"ground_truth": "\\frac {1}{2}"
} | {
"index": 1381,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A company has a total of 60 employees. In order to carry out club activities, a questionnaire survey was conducted among all employees. There are 28 people who like sports, 26 people who like literary and artistic activities, and 12 people who do not like either sports or liter... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 1382,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A company has calculated that investing x million yuan in project A will yield an economic benefit y that satisfies the relationship: $y=f(x)=-\\frac{1}{4}x^{2}+2x+12$. Similarly, the economic benefit y from investing in project B satisfies the relationship: $y=h(x)=-\\frac{1}{... | math | {
"style": "rule",
"ground_truth": "6.5"
} | {
"index": 1383,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A company has recruited 8 new employees, who are to be evenly distributed between two sub-departments, A and B. There are restrictions that the two translators cannot be in the same department, and the three computer programmers cannot all be in the same department. How many di... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 1384,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A company is planning to increase the annual production of a product by implementing technical reforms in 2013. According to the survey, the product's annual production volume $x$ (in ten thousand units) and the technical reform investment $m$ (in million yuan, where $m \\ge 0$... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 1385,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A company needs 500 tons of raw materials to produce a batch of Product A, and each ton of raw material can generate a profit of 1.2 million yuan. Through equipment upgrades, the raw materials required to produce this batch of Product A were reduced by $x (x > 0)$ tons, and the... | math | {
"style": "rule",
"ground_truth": "5.5"
} | {
"index": 1386,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A company needs to renovate its new office building. If the renovation is done solely by Team A, it would take 18 weeks, and if done solely by Team B, it would take 12 weeks. The result of the bidding is that Team A will work alone for the first 3 weeks, and then both Team A an... | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"index": 1387,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A company needs to transport two types of products, $A$ and $B$, with the following volumes and masses per unit as shown in the table:\n\n| | Volume $(m^{3}/$unit) | Mass (tons$/$unit) |\n|----------|-----------------------|--------------------|\n| $A$ type | $0.8$ ... | math | {
"style": "rule",
"ground_truth": "2100"
} | {
"index": 1388,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A company offers its employees a salary increase, provided they increase their work productivity by 2% per week. If the company operates 5 days a week, by what percentage per day must employees increase their productivity to achieve the desired goal?"
}
] | math | {
"style": "rule",
"ground_truth": "0.4"
} | {
"index": 1389,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A company organizes its employees into 7 distinct teams for a cycling event. The employee count is between 200 and 300. If one employee takes a day off, the teams are still equally divided among all present employees. Determine the total number of possible employees the company... | math | {
"style": "rule",
"ground_truth": "3493"
} | {
"index": 1390,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A company plans to advertise on TV stations A and B for a total time of no more than 300 minutes in 2011, with an advertising budget of no more than 90,000 yuan. The advertising rates for TV stations A and B are 500 yuan/minute and 200 yuan/minute, respectively. Assuming that f... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 1391,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A company plans to invest in 3 different projects among 5 candidate cities around the Bohai Economic Rim, which are Dalian, Yingkou, Panjin, Jinzhou, and Huludao. The number of projects invested in the same city cannot exceed 2. How many different investment plans can the compa... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 1392,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A company plans to promote the same car in two locations, A and B. It is known that the relationship between the sales profit (unit: ten thousand yuan) and the sales volume (unit: cars) in the two locations is $y_1=5.06t-0.15t^2$ and $y_2=2t$, respectively, where $t$ is the sal... | math | {
"style": "rule",
"ground_truth": "45.6"
} | {
"index": 1393,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A company purchases 400 tons of a certain type of goods annually. Each purchase is of $x$ tons, and the freight cost is 40,000 yuan per shipment. The annual total storage cost is 4$x$ million yuan. To minimize the sum of the annual freight cost and the total storage cost, find ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 1394,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A company sells a brand of cars in two places, A and B, with profits (in units of 10,000 yuan) of $L_{1}=-x^{2}+21x$ and $L_{2}=2x$ respectively, where $x$ is the sales volume (in units). If the company sells a total of 15 cars in both places, what is the maximum profit it can ... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 1395,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A company that sells widgets has to pay $\\$500$ in maintenance fees each day and then it pays each worker $\\$15$ per hour. Each worker makes 5 widgets per hour, which are sold at $\\$3.10$ each. What is the least number of workers the company has to hire in order to make a pr... | math | {
"style": "rule",
"ground_truth": "126"
} | {
"index": 1396,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A company's capital increases by a factor of two each year compared to the previous year after dividends have been paid, with a fixed dividend of 50 million yuan paid to shareholders at the end of each year. The company's capital after dividends were paid at the end of 2010 was... | math | {
"style": "rule",
"ground_truth": "2017"
} | {
"index": 1397,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A complex quartic polynomial $Q$ is quirky if it has four distinct roots, one of which is the sum of the other three. There are four complex values of $k$ for which the polynomial $Q(x)=x^{4}-k x^{3}-x^{2}-x-45$ is quirky. Compute the product of these four values of $k$."
}
] | math | {
"style": "rule",
"ground_truth": "720"
} | {
"index": 1398,
"solution": "Let the roots be $a, b, c, d$ with $a+b+c=d$. Since $a+b+c=k-d$ by Vieta's formulas, we have $d=k / 2$. Hence $$0=P\\left(\\frac{k}{2}\\right)=\\left(\\frac{k}{2}\\right)^{4}-k\\left(\\frac{k}{2}\\right)^{3}-\\left(\\frac{k}{2}\\right)^{2}-\\left(\\frac{k}{2}\\right)-45=-\\frac{k^{4}}{... |
DeepScaleR | [
{
"role": "user",
"content": "A component is made up of 3 identical electronic components in parallel. The component works normally if at least one of the electronic components works normally. It is known that the service life $\\xi$ (in years) of this type of electronic component follows a normal distribut... | math | {
"style": "rule",
"ground_truth": "0.488"
} | {
"index": 1399,
"solution": ""
} |
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