data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Given 2017 lines separated into three sets such that lines in the same set are parallel to each other, what is the largest possible number of triangles that can be formed with vertices on these lines?"
}
] | math | {
"style": "rule",
"ground_truth": "673 * 672^2"
} | {
"index": 12900,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 2414 cards, each with a unique natural number from 1 to 2414. We need to choose two cards such that the sum of the numbers on them is divisible by 100. In how many ways can this be done?"
}
] | math | {
"style": "rule",
"ground_truth": "29112"
} | {
"index": 12901,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 3 zeros and 2 ones are randomly arranged in a row, calculate the probability that the 2 ones are not adjacent."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 12902,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 365 cards, in which distinct numbers are written. We may ask for any three cards, the order of numbers written in them. Is it always possible to find out the order of all 365 cards by 2000 such questions?"
}
] | math | {
"style": "rule",
"ground_truth": "1691"
} | {
"index": 12903,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 4 distinct books that are to be distributed evenly between two students, find the probability that the Chinese language book and the Mathematics book are given to the same student."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 12904,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 40 feet of fencing, what is the greatest possible number of square feet in the area of a rectangular pen enclosed by the fencing?"
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 12905,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 5 balls with 2 identical black balls and one each of red, white, and blue, calculate the number of different arrangements of 4 balls in a row."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 12906,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 5 different letters from the word \"equation\", find the total number of different arrangements that contain \"qu\" where \"qu\" are consecutive and in the same order."
}
] | math | {
"style": "rule",
"ground_truth": "480"
} | {
"index": 12907,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 5 differently colored balls and 3 different boxes, with the requirement that no box is empty, calculate the total number of different ways to place 4 balls into the boxes."
}
] | math | {
"style": "rule",
"ground_truth": "180"
} | {
"index": 12908,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 5 distinct real numbers, any two of which are summed to yield 10 sums. Among these sums, the smallest three are 32, 36, and 37, and the largest two are 48 and 51. What is the largest of these 5 numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "27.5"
} | {
"index": 12909,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 5 people stand in a row, and there is exactly 1 person between person A and person B, determine the total number of possible arrangements."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 12910,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 5 points \\( A, B, C, D, E \\) on a plane, with no three points being collinear. How many different ways can one connect these points with 4 segments such that each point is an endpoint of at least one segment?"
}
] | math | {
"style": "rule",
"ground_truth": "135"
} | {
"index": 12911,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 50 feet of fencing, where 5 feet is used for a gate that does not contribute to the enclosure area, what is the greatest possible number of square feet in the area of a rectangular pen enclosed by the remaining fencing?"
}
] | math | {
"style": "rule",
"ground_truth": "126.5625"
} | {
"index": 12912,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 500 points inside a convex 1000-sided polygon, along with the polygon's vertices (a total of 1500 points), none of which are collinear, the polygon is divided into triangles with these 1500 points as the vertices of the triangles. There are no other vertices apart from th... | math | {
"style": "rule",
"ground_truth": "1998"
} | {
"index": 12913,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 6 digits: \\(0, 1, 2, 3, 4, 5\\). Find the sum of all four-digit even numbers that can be written using these digits (the same digit can be repeated in a number)."
}
] | math | {
"style": "rule",
"ground_truth": "1769580"
} | {
"index": 12914,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 6 parking spaces in a row and 3 cars that need to be parked such that no two cars are next to each other, calculate the number of different parking methods."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 12915,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 6 people $A$, $B$, $C$, $D$, $E$, $F$, they are randomly arranged in a line. The probability of the event \"$A$ is adjacent to $B$ and $A$ is not adjacent to $C$\" is $\\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{15}"
} | {
"index": 12916,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 6 persons, with the restriction that person A and person B cannot visit Paris, calculate the total number of distinct selection plans for selecting 4 persons to visit Paris, London, Sydney, and Moscow, where each person visits only one city."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 12917,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 6 teachers who will be allocated to two classes, where the maximum number of teachers in each class is 4, determine the number of different arrangements."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 12918,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 60 feet of fencing, what is the greatest possible number of square feet in the area of a pen, if the pen is designed as a rectangle subdivided evenly into two square areas?"
}
] | math | {
"style": "rule",
"ground_truth": "450"
} | {
"index": 12919,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 6000 cards, each with a unique natural number from 1 to 6000 written on it. It is required to choose two cards such that the sum of the numbers on them is divisible by 100. In how many ways can this be done?"
}
] | math | {
"style": "rule",
"ground_truth": "179940"
} | {
"index": 12920,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 8 people are sitting around a circular table for a meeting, including one leader, one vice leader, and one recorder, and the recorder is seated between the leader and vice leader, determine the number of different seating arrangements possible, considering that arrangemen... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 12921,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 8 planes in space, for each pair of planes, the line of their intersection is noted. For each pair of these noted lines, the point of their intersection is noted (if the lines intersect). What is the maximum number of noted points that could be obtained?"
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 12922,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 8 teams, of which 3 are weak teams, they are randomly divided into two groups, A and B, with 4 teams in each group. Find:\n\n(1) The probability that one of the groups A or B has exactly 2 weak teams.\n(2) The probability that group A has at least 2 weak teams."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 12923,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 8 volunteer positions to be allocated to 3 schools, with each school receiving at least one position and the allocations being unequal, find the number of ways to distribute the positions."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 12924,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given 95 numbers \\( a_{1}, a_{2}, \\cdots, a_{95} \\) where each number can only be +1 or -1, find the minimum value of the sum of the products of each pair of these numbers, \\( \\sum_{1 \\leq i<j \\leq 95} a_{i} a_{j} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 12925,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Abby finished the softball season with a total of 45 hits, among which were 2 home runs, 3 triples, and 7 doubles, calculate the percentage of her hits that were singles."
}
] | math | {
"style": "rule",
"ground_truth": "73.33\\%"
} | {
"index": 12926,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Alex, Jamie, and Casey play a game over 8 rounds, and for each round, the probability Alex wins is $\\frac{1}{3}$, and Jamie is three times as likely to win as Casey, calculate the probability that Alex wins four rounds, Jamie wins three rounds, and Casey wins one round."... | math | {
"style": "rule",
"ground_truth": "\\frac{35}{486}"
} | {
"index": 12927,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Allison's birthday cake is in the form of a $5 \\times 5 \\times 3$ inch rectangular prism with icing on the top, front, and back sides but not on the sides or bottom, calculate the number of $1 \\times 1 \\times 1$ inch smaller prisms that will have icing on exactly two ... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 12928,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Angela has 4 files that are each 1.2 MB, 8 files that are 0.9 MB each, and 10 files that are 0.6 MB each, calculate the minimum number of disks Angela will need to store all her files."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 12929,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Angie and Carlos are seated at a round table with three other people, determine the probability that Angie and Carlos are seated directly across from each other."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 12930,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Aniyah has a $5 \\times 7$ index card and if she shortens the length of one side of this card by $2$ inches, the card would have an area of $21$ square inches, determine the area of the card in square inches if she shortens the length of the other side by $2$ inches."
}... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 12931,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Ben's test scores $95, 85, 75, 65,$ and $90$, and his goal to increase his average by at least $5$ points and score higher than his lowest score of $65$ with his next test, calculate the minimum test score he would need to achieve both goals."
}
] | math | {
"style": "rule",
"ground_truth": "112"
} | {
"index": 12932,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Chelsea leads by 60 points halfway through a 120-shot archery tournament, scores at least 5 points per shot, and scores at least 10 points for each of her next n shots, determine the minimum number of shots, n, she must get as bullseyes to guarantee her victory."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"index": 12933,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Daphne's four friends visit her every 4, 6, 8, and 10 days respectively, all four friends visited her yesterday, calculate the number of days in the next 365-day period when exactly two friends will visit her."
}
] | math | {
"style": "rule",
"ground_truth": "129"
} | {
"index": 12934,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Elmer’s new car provides a 70% better fuel efficiency and uses a type of fuel that is 35% more expensive per liter, calculate the percentage by which Elmer will save money on fuel costs if he uses his new car for his journey."
}
] | math | {
"style": "rule",
"ground_truth": "20.6\\%"
} | {
"index": 12935,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Erin has 4 sisters and 6 brothers, determine the product of the number of sisters and the number of brothers of her brother Ethan."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 12936,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Farmer Euclid has a field in the shape of a right triangle with legs of lengths $5$ units and $12$ units, and he leaves a small unplanted square of side length S in the corner where the legs meet at a right angle, where $S$ is $3$ units from the hypotenuse, calculate the ... | math | {
"style": "rule",
"ground_truth": "\\frac{431}{480}"
} | {
"index": 12937,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Harry has 4 sisters and 6 brothers, and his sister Harriet has S sisters and B brothers, calculate the product of S and B."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 12938,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Jane lists the whole numbers $1$ through $50$ once and Tom copies Jane's numbers, replacing each occurrence of the digit $3$ by the digit $2$, calculate how much larger Jane's sum is than Tom's sum."
}
] | math | {
"style": "rule",
"ground_truth": "105"
} | {
"index": 12939,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Jasmine has two types of bottles, one that can hold 45 milliliters and another that can hold 675 milliliters, and a vase that can hold 95 milliliters, determine the total number of small bottles she must buy to fill the large bottle as much as possible and the vase."
}
... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 12940,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Jones traveled 100 miles on his first trip and 500 miles on a subsequent trip at a speed four times as fast, compare his new time to the old time."
}
] | math | {
"style": "rule",
"ground_truth": "1.25"
} | {
"index": 12941,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Josie makes lemonade by using 150 grams of lemon juice, 200 grams of sugar, and 300 grams of honey, and there are 30 calories in 100 grams of lemon juice, 386 calories in 100 grams of sugar, and 304 calories in 100 grams of honey, determine the total number of calories in... | math | {
"style": "rule",
"ground_truth": "665"
} | {
"index": 12942,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Joy has 50 thin rods, one each of every integer length from 1 cm through 50 cm, and rods with lengths 8 cm, 12 cm, and 25 cm are already placed on a table, determine the number of the remaining rods that can be chosen as the fourth rod to form a quadrilateral with positiv... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 12943,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Karl's rectangular garden measures \\(30\\) feet by \\(50\\) feet with a \\(2\\)-feet wide uniformly distributed pathway and Makenna's garden measures \\(35\\) feet by \\(55\\) feet with a \\(3\\)-feet wide pathway, compare the areas of their gardens, assuming the pathway... | math | {
"style": "rule",
"ground_truth": "225"
} | {
"index": 12944,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Laila had a 10-hour work day and attended three different meetings during this period, the first meeting took 50 minutes, the second meeting twice as long as the first, and the third meeting half as long as the second meeting, calculate the percentage of her work day that... | math | {
"style": "rule",
"ground_truth": "33.33\\%"
} | {
"index": 12945,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Lara ate $\\frac{1}{4}$ of a pie and Ryan ate $\\frac{3}{10}$ of the same pie, then Cassie ate $\\frac{2}{3}$ of the pie that was left. Calculate the fraction of the original pie that was not eaten."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{20}"
} | {
"index": 12946,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Liam has written one integer three times and another integer four times. The sum of these seven numbers is 131, and one of the numbers is 17, determine the value of the other number."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 12947,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Lucy starts with an initial term of 8 in her sequence, where each subsequent term is generated by either doubling the previous term and subtracting 2 if a coin lands on heads, or halving the previous term and subtracting 2 if a coin lands on tails, determine the probabili... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 12948,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given M be the greatest five-digit number whose digits have a product of 36, determine the sum of the digits of M."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 12949,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Madeline has 80 fair coins. She flips all the coins. Any coin that lands on tails is tossed again. Additionally, any coin that lands on heads in the first two tosses is also tossed again, but only once. What is the expected number of coins that are heads after these condi... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 12950,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Mindy made four purchases for $2.96, 6.57, 8.49, and 12.38. Each amount needs to be rounded up to the nearest dollar except the amount closest to a whole number, which should be rounded down. Calculate the total rounded amount."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 12951,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Mr. Thompson can choose between two routes to commute to his office: Route X, which is 8 miles long with an average speed of 35 miles per hour, and Route Y, which is 7 miles long with an average speed of 45 miles per hour excluding a 1-mile stretch with a reduced speed of... | math | {
"style": "rule",
"ground_truth": "1.71"
} | {
"index": 12952,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Ms. Grace Swift leaves her house at 7:30 AM to reach her office, she arrives 5 minutes late at an average speed of 50 miles per hour, and 5 minutes early at an average speed of 70 miles per hour, determine the average speed she should maintain to arrive exactly on time."
... | math | {
"style": "rule",
"ground_truth": "58"
} | {
"index": 12953,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Orvin goes to a store with just enough money to buy 40 balloons, and the store has a special promotion: for every balloon bought at full price, a second one can be bought at 1/2 off. Find the maximum number of balloons Orvin can buy."
}
] | math | {
"style": "rule",
"ground_truth": "52"
} | {
"index": 12954,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given P(A) = 0.65, P(B) = 0.2, and P(C) = 0.1, calculate the probability of the event \"the drawn product is not a first-class product\"."
}
] | math | {
"style": "rule",
"ground_truth": "0.35"
} | {
"index": 12955,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Professor Chen has ten different language books, including two Arabic, four German, and four Spanish books, arranged on the shelf so that the Arabic books are together, the Spanish books are together, and two of the German books are treated as indistinguishable, calculate... | math | {
"style": "rule",
"ground_truth": "576"
} | {
"index": 12956,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Professor Lee has ten different mathematics books on a shelf, consisting of three calculus books, four algebra books, and three statistics books, determine the number of ways to arrange the ten books on the shelf keeping all calculus books together and all statistics book... | math | {
"style": "rule",
"ground_truth": "25920"
} | {
"index": 12957,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Professor Lewis has twelve different language books lined up on a bookshelf: three Arabic, four German, three Spanish, and two French, calculate the number of ways to arrange the twelve books on the shelf keeping the Arabic books together, the Spanish books together, and ... | math | {
"style": "rule",
"ground_truth": "362,880"
} | {
"index": 12958,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Ricardo has $3000$ coins comprised of pennies ($1$-cent coins), nickels ($5$-cent coins), and dimes ($10$-cent coins), with at least one of each type of coin, calculate the difference in cents between the highest possible and lowest total value that Ricardo can have."
}... | math | {
"style": "rule",
"ground_truth": "26973"
} | {
"index": 12959,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given School A and School B each have 3 teachers signing up for volunteer teaching, with School A having 2 males and 1 female, and School B having 1 male and 2 females, calculate the probability that the selected 2 teachers have the same gender."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"index": 12960,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given Tom paid $180, Dorothy paid $200, Sammy paid $240, and Alice paid $280, and they agreed to split the costs evenly, calculate the amount that Tom gave to Sammy minus the amount that Dorothy gave to Alice."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 12961,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( 0 \\leq m-n \\leq 1 \\) and \\( 2 \\leq m+n \\leq 4 \\), when \\( m - 2n \\) reaches its maximum value, what is the value of \\( 2019m + 2020n \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "2019"
} | {
"index": 12962,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( 1991 = 2^{\\alpha_{1}} + 2^{\\alpha_{2}} + \\cdots + 2^{\\alpha_{n}} \\), where \\( \\alpha_{1}, \\alpha_{2}, \\cdots, \\alpha_{n} \\) are distinct non-negative integers, find the sum \\( \\alpha_{1} + \\alpha_{2} + \\cdots + \\alpha_{n} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 12963,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( 5 \\sin 2 \\alpha = \\sin 2^\\circ \\), find the value of \\( \\frac{\\tan (\\alpha + 1^\\circ)}{\\tan (\\alpha - 1^\\circ)} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"index": 12964,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( A \\cup B = \\left\\{a_{1}, a_{2}, a_{3}\\right\\} \\) and \\( A \\neq B \\), where \\((A, B)\\) and \\((B, A)\\) are considered different pairs, find the number of such pairs \\((A, B)\\)."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 12965,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( A \\subseteq \\{1, 2, \\ldots, 25\\} \\) such that \\(\\forall a, b \\in A\\), \\(a \\neq b\\), then \\(ab\\) is not a perfect square. Find the maximum value of \\( |A| \\) and determine how many such sets \\( A \\) exist where \\( |A| \\) achieves this maximum value.... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 12966,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( A, B, C, D \\in\\{1, 2, \\cdots, 6\\} \\), and each is distinct from the others. If the curves \\( y = Ax^{2} + B \\) and \\( y = Cx^{2} + D \\) intersect, then there are \\(\\quad\\) different ways to choose \\( A, B, C, D \\) (the intersection of the curves is indep... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 12967,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( A=\\left\\{x \\mid \\log _{3}\\left(x^{2}-2 x\\right) \\leqslant 1\\right\\}, B=(-\\infty, a] \\cup(b,+\\infty) \\), where \\( a < b \\), if \\( A \\cup B=\\mathbf{R} \\), what is the minimum value of \\( a - b \\) ?"
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 12968,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( AB \\) as the diameter of the smallest radius circle centered at \\( C(0,1) \\) that intersects the graph of \\( y = \\frac{1}{|x|-1} \\), where \\( O \\) is the origin. Find the value of \\( \\overrightarrow{OA} \\cdot \\overrightarrow{OB} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 12969,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( A_{1}, A_{2}, \\cdots, A_{n} \\) are \\( n \\) non-empty subsets of the set \\( A=\\{1,2,3, \\cdots, 10\\} \\), if for any \\( i, j \\in \\{1,2,3, \\cdots, n\\} \\), we have \\( A_{i} \\cup A_{j} \\neq A \\), then the maximum value of \\( n \\) is \\(\\qquad\\)."
}
... | math | {
"style": "rule",
"ground_truth": "511"
} | {
"index": 12970,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( DE = 9 \\), \\( EB = 6 \\), \\( AB = \\frac{1}{3}AD \\), and \\( FD = \\frac{3}{4}AD \\), find \\( FC \\). Assume triangle \\( ADE \\) is similar to triangle \\( AFB \\) and triangle \\( AFC \\)."
}
] | math | {
"style": "rule",
"ground_truth": "16.875"
} | {
"index": 12971,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( P = 3659893456789325678 \\) and \\( 342973489379256 \\), the product \\( P \\) is calculated. The number of digits in \\( P \\) is:"
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"index": 12972,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( P \\) is the product of \\( 3,659,893,456,789,325,678 \\) and \\( 342,973,489,379,256 \\), find the number of digits of \\( P \\)."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"index": 12973,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( S = [\\sqrt{1}] + [\\sqrt{2}] + \\cdots + [\\sqrt{1988}] \\), find \\( [\\sqrt{S}] \\)."
}
] | math | {
"style": "rule",
"ground_truth": "241"
} | {
"index": 12974,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( S = x^{2} + y^{2} - 2(x + y) \\), where \\( x \\) and \\( y \\) satisfy \\( \\log_{2} x + \\log_{2} y = 1 \\), find the minimum value of \\( S \\)."
}
] | math | {
"style": "rule",
"ground_truth": "4 - 4\\sqrt{2}"
} | {
"index": 12975,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( S_{n} = \\sum_{i=1}^{10} i|n-i| \\left( n \\in \\mathbf{Z}_{+} \\right) \\), find the minimum value of \\( S_{n} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "112"
} | {
"index": 12976,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( \\alpha, \\beta \\in (0, \\pi) \\), and \\( \\tan \\alpha, \\tan \\beta \\) are the roots of the equation \\( x^{2} + 3x + 1 = 0 \\), find the value of \\( \\cos(\\alpha - \\beta) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 12977,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( \\cos \\left( \\frac {\\pi}{2}+\\alpha \\right)=3\\sin \\left(\\alpha+ \\frac {7\\pi}{6}\\right) \\), find the value of \\( \\tan \\left( \\frac {\\pi}{12}+\\alpha \\right) = \\) ______."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt {3} - 4"
} | {
"index": 12978,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( \\frac{1}{3} \\leqslant a \\leqslant 1 \\), if \\( f(x)=a x^{2}-2 x+1 \\) attains its maximum value \\( M(a) \\) and minimum value \\( N(a) \\) on the interval \\([1,3]\\), and let \\( g(a)=M(a)-N(a) \\), then the minimum value of \\( g(a) \\) is \\(\\quad\\) ."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 12979,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( \\frac{\\pi}{4} < \\theta < \\frac{\\pi}{2} \\), find the maximum value of \\( S = \\sin 2\\theta - \\cos^2 \\theta \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5} - 1}{2}"
} | {
"index": 12980,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( \\theta_{1}, \\theta_{2}, \\theta_{3}, \\theta_{4} \\in \\mathbf{R}^{+} \\) and \\( \\theta_{1} + \\theta_{2} + \\theta_{3} + \\theta_{4} = \\pi \\), find the minimum value of \\( \\left(2 \\sin^{2} \\theta_{1} + \\frac{1}{\\sin^{2} \\theta_{1}}\\right)\\left(2 \\sin^... | math | {
"style": "rule",
"ground_truth": "81"
} | {
"index": 12981,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( a > b \\), the quadratic inequality \\( ax^{2}+2x+b \\geqslant 0 \\) holds for all real numbers \\( x \\), and there exists \\( x_{0} \\in \\mathbb{R} \\) such that \\( ax_{0}^{2}+2x_{0}+b=0 \\) is satisfied. Find the minimum value of \\( 2a^{2}+b^{2} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"index": 12982,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( a \\in \\mathbb{R} \\), the complex number \\( z = \\frac{(a-i)(1+i)}{i} \\), if \\( \\bar{z} = z \\), then find the value of \\( a \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 12983,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( a, b > 0 \\). Find the maximum values of \\( \\sqrt{\\frac{a}{2a+b}} + \\sqrt{\\frac{b}{2b+a}} \\) and \\( \\sqrt{\\frac{a}{a+2b}} + \\sqrt{\\frac{b}{b+2a}} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"index": 12984,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( a, b, c \\geq 0 \\), \\( t \\geq 1 \\), and satisfying\n\\[\n\\begin{cases}\na + b + c = \\frac{1}{2}, \\\\\n\\sqrt{a + \\frac{1}{2}(b - c)^{2}} + \\sqrt{b} + \\sqrt{c} = \\frac{\\sqrt{6t}}{2},\n\\end{cases}\n\\]\n\nfind \\( a^{2t} + b^{2t} + c^{2t} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"index": 12985,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( a=\\underset{2016 \\uparrow}{55 \\cdots 5} \\), determine the remainder when \\( a \\) is divided by 84."
}
] | math | {
"style": "rule",
"ground_truth": "63"
} | {
"index": 12986,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( a=\\underset{2016 \\uparrow}{55 \\cdots 5} \\), what is the remainder when \\( a \\) is divided by 84?"
}
] | math | {
"style": "rule",
"ground_truth": "63"
} | {
"index": 12987,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( a_{0}=1, a_{1}=2 \\), and \\( n(n+1) a_{n+1}=n(n-1) a_{n}-(n-2) a_{n-1} \\) for \\( n=1, 2, 3, \\ldots \\), find \\( \\frac{a_{0}}{a_{1}}+\\frac{a_{1}}{a_{2}}+\\frac{a_{2}}{a_{3}}+\\cdots+\\frac{a_{50}}{a_{51}} \\)."
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"index": 12988,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( a_{n} = 4^{2n - 1} + 3^{n - 2} \\) (for \\( n = 1, 2, 3, \\cdots \\)), where \\( p \\) is the smallest prime number dividing infinitely many terms of the sequence \\( a_{1}, a_{2}, a_{3}, \\cdots \\), and \\( q \\) is the smallest prime number dividing every term of t... | math | {
"style": "rule",
"ground_truth": "5 \\times 13"
} | {
"index": 12989,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( a_{n} = 50 + n^{2} \\) for \\( n = 1, 2, \\cdots \\), find the maximum value of the greatest common divisor \\( d_{n} = \\gcd(a_{n}, a_{n+1}) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "201"
} | {
"index": 12990,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( a_{n} = \\mathrm{C}_{200}^{n} \\cdot (\\sqrt[3]{6})^{200-n} \\cdot \\left( \\frac{1}{\\sqrt{2}} \\right)^{n} \\) for \\( n = 1, 2, \\ldots, 95 \\), find the number of integer terms in the sequence \\(\\{a_{n}\\}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 12991,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( f(u) = u^{2} + au + (b-2) \\), where \\( u = x + \\frac{1}{x} \\) (with \\( x \\in \\mathbb{R} \\) and \\( x \\neq 0 \\)). If \\( a \\) and \\( b \\) are real numbers such that the equation \\( f(u) = 0 \\) has at least one real root, find the minimum value of \\( a^{... | math | {
"style": "rule",
"ground_truth": "4/5"
} | {
"index": 12992,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( f(x) = 2^x \\) and \\( g(x) = \\log_{\\sqrt{2}} (8x) \\), find the value of \\( x \\) that satisfies \\( f[g(x)] = g[f(x)] \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1 + \\sqrt{385}}{64}"
} | {
"index": 12993,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( f(x) = \\max \\left| x^3 - a x^2 - b x - c \\right| \\) for \\( 1 \\leq x \\leq 3 \\), find the minimum value of \\( f(x) \\) as \\( a, b, \\) and \\( c \\) range over all real numbers."
}
] | math | {
"style": "rule",
"ground_truth": "1/4"
} | {
"index": 12994,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( f(x) = \\sum_{k=1}^{2017} \\frac{\\cos k x}{\\cos^k x} \\), find \\( f\\left(\\frac{\\pi}{2018}\\right) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 12995,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( f(x) = a \\sin x + b \\sqrt[3]{x} + c \\ln \\left(x + \\sqrt{x^{2} + 1}\\right) + 1003 \\) (where \\( a \\), \\( b \\), and \\( c \\) are real numbers), and \\( f\\left(\\lg^{2} 10\\right) = 1 \\), what is \\( f(\\lg \\lg 3) \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "2005"
} | {
"index": 12996,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( f(x) = x^2 + 3x + 2 \\) and \\( S = \\{0, 1, 2, 3, \\cdots, 100\\} \\), if \\( a \\in S \\) and \\( f(a) \\) is divisible by 6, how many such \\( a \\) exist?"
}
] | math | {
"style": "rule",
"ground_truth": "67"
} | {
"index": 12997,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( f(x) = x^{5} + a_{1} x^{4} + a_{2} x^{3} + a_{3} x^{2} + a_{4} x + a_{5} \\), and \\( f(m) = 2017m \\) for \\( m = 1, 2, 3, 4 \\), calculate \\( f(10) - f(-5) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "75615"
} | {
"index": 12998,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Given \\( f(x) = x^{5} + a_{1} x^{4} + a_{2} x^{3} + a_{3} x^{2} + a_{4} x + a_{5} \\), and \\( f(m) = 2017m \\) for \\( m = 1, 2, 3, 4 \\), find \\( f(10) - f(-5) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "75615"
} | {
"index": 12999,
"solution": ""
} |
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