data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Five integers are written on a board. By summing them in pairs, the following set of 10 numbers was obtained: $-1, 5, 8, 9, 11, 12, 14, 18, 20, 24$. Determine which numbers are written on the board. Provide their product as the answer."
}
] | math | {
"style": "rule",
"ground_truth": "-2002"
} | {
"index": 11200,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five integers have an average of 69. The middle integer (the median) is 83. The most frequently occurring integer (the mode) is 85. The range of the five integers is 70. What is the second smallest of the five integers?"
}
] | math | {
"style": "rule",
"ground_truth": "77"
} | {
"index": 11201,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five marbles are distributed at a random among seven urns. What is the expected number of urns with exactly one marble?"
}
] | math | {
"style": "rule",
"ground_truth": "6480/2401"
} | {
"index": 11202,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five men and nine women stand equally spaced around a circle in random order. The probability that every man stands diametrically opposite a woman is $\\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$"
}
] | math | {
"style": "rule",
"ground_truth": "191"
} | {
"index": 11203,
"solution": "We get around the condition that each man can't be opposite to another man by simply considering all $7$ diagonals, and choosing $5$ where there will be a single man. For each diagonal, the man can go on either side, and there are $\\binom{14}{5}$ ways to arrange the men and the women... |
DeepScaleR | [
{
"role": "user",
"content": "Five million times eight million equals"
}
] | math | {
"style": "rule",
"ground_truth": "40,000,000,000,000"
} | {
"index": 11204,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five monkeys share a pile of peanuts. The first monkey divides the peanuts into five piles, leaving one peanut which it eats, and takes away one pile. The second monkey then divides the remaining peanuts into five piles, leaving exactly one peanut, eats it, and takes away one p... | math | {
"style": "rule",
"ground_truth": "3121"
} | {
"index": 11205,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five numbers, $a_1$, $a_2$, $a_3$, $a_4$, are drawn randomly and without replacement from the set $\\{1, 2, 3, \\dots, 50\\}$. Four other numbers, $b_1$, $b_2$, $b_3$, $b_4$, are then drawn randomly and without replacement from the remaining set of 46 numbers. Let $p$ be the pr... | math | {
"style": "rule",
"ground_truth": "71"
} | {
"index": 11206,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five packages are delivered to five different houses, with each house receiving one package. If these packages are randomly delivered, what is the probability that exactly three of them are delivered to their correct houses? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"index": 11207,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five packages are delivered to five houses, one to each house. If the packages are randomly delivered, what is the probability that exactly three of them are delivered to their correct houses?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"index": 11208,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five packages are delivered to five houses, one to each house. If these packages are randomly delivered, what is the probability that exactly three of them are delivered to the correct houses?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"index": 11209,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five packages are delivered to five houses, one to each house. If these packages are randomly delivered, what is the probability that exactly three of them are delivered to the correct houses? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"index": 11210,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five people are at a party. Each pair of them are friends, enemies, or frenemies (which is equivalent to being both friends and enemies). It is known that given any three people $A, B, C$ : - If $A$ and $B$ are friends and $B$ and $C$ are friends, then $A$ and $C$ are friends; ... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 11211,
"solution": "If $A$ and $B$ are frenemies, then regardless of whether another person $C$ is friends or enemies with $A$, $C$ will have to be frenemies with $B$ and vice versa. Therefore, if there is one pair of frenemies then all of them are frenemies with each other, and there is only one possibi... |
DeepScaleR | [
{
"role": "user",
"content": "Five people are crowding into a booth against a wall at a noisy restaurant. If at most three can fit on one side, how many seating arrangements accommodate them all?"
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 11212,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five people are gathered in a meeting. Some pairs of people shakes hands. An ordered triple of people $(A,B,C)$ is a *trio* if one of the following is true:\n\n\n\n- A shakes hands with B, and B shakes hands with C, or\n- A doesn't shake hands with B, and B doesn't shake hand... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 11213,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five people are sitting around a round table, with identical coins placed in front of each person. Everyone flips their coin simultaneously. If the coin lands heads up, the person stands up; if it lands tails up, the person remains seated. Determine the probability that no two ... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{32}"
} | {
"index": 11214,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five people are sitting at a round table. Let $f\\geq 0$ be the number of people sitting next to at least 1 female and $m\\geq0$ be the number of people sitting next to at least one male. The number of possible ordered pairs $(f,m)$ is\n$\\mathrm{(A) \\ 7 } \\qquad \\mathrm{(B)... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 11215,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five people can paint a house in 10 hours. How many hours would it take four people to paint the same house and mow the lawn if mowing the lawn takes an additional 3 hours per person, assuming that each person works at the same rate for painting and different rate for mowing?"
... | math | {
"style": "rule",
"ground_truth": "15.5"
} | {
"index": 11216,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five people of different heights are standing in line from shortest to tallest. As it happens, the tops of their heads are all collinear; also, for any two successive people, the horizontal distance between them equals the height of the shorter person. If the shortest person is... | math | {
"style": "rule",
"ground_truth": "\\sqrt{21}"
} | {
"index": 11217,
"solution": "If $A, B$, and $C$ are the tops of the heads of three successive people and $D, E$, and $F$ are their respective feet, let $P$ be the foot of the perpendicular from $A$ to $B E$ and let $Q$ be the foot of the perpendicular from $B$ to $C F$. Then, by equal angles, $\\triangle A B P \\... |
DeepScaleR | [
{
"role": "user",
"content": "Five people of heights $65,66,67,68$, and 69 inches stand facing forwards in a line. How many orders are there for them to line up, if no person can stand immediately before or after someone who is exactly 1 inch taller or exactly 1 inch shorter than himself?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 11218,
"solution": "Let the people be $A, B, C, D, E$ so that their heights are in that order, with $A$ the tallest and $E$ the shortest. We will do casework based on the position of $C$. - Case 1: $C$ is in the middle. Then, $B$ must be on one of the two ends, for two choices. This leaves only one choic... |
DeepScaleR | [
{
"role": "user",
"content": "Five people take a true-or-false test with five questions. Each person randomly guesses on every question. Given that, for each question, a majority of test-takers answered it correctly, let $p$ be the probability that every person answers exactly three questions correctly. Sup... | math | {
"style": "rule",
"ground_truth": "25517"
} | {
"index": 11219,
"solution": "There are a total of $16^{5}$ ways for the people to collectively ace the test. Consider groups of people who share the same problems that they got incorrect. We either have a group of 2 and a group of 3 , or a group 5 . In the first case, we can pick the group of two in $\\binom{5}{2... |
DeepScaleR | [
{
"role": "user",
"content": "Five people, named A, B, C, D, and E, stand in a row. If A and B must be adjacent, and B must be to the left of A, what is the total number of different arrangements?"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 11220,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five persons wearing badges with numbers $1, 2, 3, 4, 5$ are seated on $5$ chairs around a circular table. In how many ways can they be seated so that no two persons whose badges have consecutive numbers are seated next to each other? (Two arrangements obtained by rotation ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 11221,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five points $A$, $B$, $C$, $D$, and $O$ lie on a flat field. $A$ is directly north of $O$, $B$ is directly west of $O$, $C$ is directly south of $O$, and $D$ is directly east of $O$. The distance between $C$ and $D$ is 140 m. A hot-air balloon is positioned in the air at $H$... | math | {
"style": "rule",
"ground_truth": "30\\sqrt{11}"
} | {
"index": 11222,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five points $A_1,A_2,A_3,A_4,A_5$ lie on a plane in such a way that no three among them lie on a same straight line. Determine the maximum possible value that the minimum value for the angles $\\angle A_iA_jA_k$ can take where $i, j, k$ are distinct integers between $1$ and $5$... | math | {
"style": "rule",
"ground_truth": "36^\\circ"
} | {
"index": 11223,
"solution": "\nGiven five points \\( A_1, A_2, A_3, A_4, \\) and \\( A_5 \\) in the plane such that no three are collinear, we are tasked with determining the maximum possible minimum value for the angles \\( \\angle A_i A_j A_k \\), where \\( i, j, k \\) are distinct integers between \\(1\\) and ... |
DeepScaleR | [
{
"role": "user",
"content": "Five points are chosen on a sphere of radius 1. What is the maximum possible volume of their convex hull?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"index": 11224,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five points are chosen uniformly at random on a segment of length 1. What is the expected distance between the closest pair of points?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{24}"
} | {
"index": 11225,
"solution": "Choose five points arbitrarily at $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ in increasing order. Then the intervals $\\left(a_{2}-x, a_{2}\\right),\\left(a_{3}-x, a_{3}\\right),\\left(a_{4}-x, a_{4}\\right),\\left(a_{5}-x, a_{5}\\right)$ must all be unoccupied. The probability that this hap... |
DeepScaleR | [
{
"role": "user",
"content": "Five points on a circle are numbered 1,2,3,4, and 5 in clockwise order. A bug jumps in a clockwise direction from one point to another around the circle; if it is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the b... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 11226,
"solution": "1. **Identify the movement pattern**: The bug starts at point 5. According to the rules, since 5 is odd, it moves one point to point 1.\n\n2. **Continue the pattern**:\n - From point 1 (odd), it moves one point to point 2.\n - From point 2 (even), it moves two points to point 4.\n... |
DeepScaleR | [
{
"role": "user",
"content": "Five points, no three of which are collinear, are given. Calculate the least possible value of the number of convex polygons whose some corners are formed by these five points."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 11227,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five points, no three of which are collinear, are given. What is the least possible value of the numbers of convex polygons whose some corners are from these five points?"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 11228,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five positive consecutive integers starting with $a$ have average $b$. What is the average of $5$ consecutive integers that start with $b$?"
}
] | math | {
"style": "rule",
"ground_truth": "$a+4$"
} | {
"index": 11229,
"solution": "1. **Define the sequence and calculate the average $b$:** \n The five consecutive integers starting with $a$ are $a, a+1, a+2, a+3, a+4$. The average of these integers, $b$, is calculated as follows:\n \\[\n b = \\frac{a + (a+1) + (a+2) + (a+3) + (a+4)}{5} = \\frac{5a + 10}{5} ... |
DeepScaleR | [
{
"role": "user",
"content": "Five positive consecutive integers starting with $a$ have average $b$. What is the average of 5 consecutive integers that start with $b$?"
}
] | math | {
"style": "rule",
"ground_truth": "$a+4$"
} | {
"index": 11230,
"solution": "1. **Define the sequence and calculate the average $b$:** \n The five consecutive integers starting with $a$ are $a, a+1, a+2, a+3, a+4$. The average of these integers, $b$, is calculated as follows:\n \\[\n b = \\frac{a + (a+1) + (a+2) + (a+3) + (a+4)}{5} = \\frac{5a + 10}{5} ... |
DeepScaleR | [
{
"role": "user",
"content": "Five positive integers (not necessarily all different) are written on five cards. Boris calculates the sum of the numbers on every pair of cards. He obtains only three different totals: 57, 70, and 83. What is the largest integer on any card?"
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 11231,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five positive integers (not necessarily all different) are written on five cards. Boris calculates the sum of the numbers on every pair of cards. He obtains only three different totals: 57, 70, and 83. What is the largest integer on any card?\n- A) 35\n- B) 42\n- C) 48\n- D) 53... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 11232,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five positive integers from $1$ to $15$ are chosen without replacement. What is the probability that their sum is divisible by $3$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"index": 11233,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five rays $\\overrightarrow{OA}$ , $\\overrightarrow{OB}$ , $\\overrightarrow{OC}$ , $\\overrightarrow{OD}$ , and $\\overrightarrow{OE}$ radiate in a clockwise order from $O$ forming four non-overlapping angles such that $\\angle EOD = 2\\angle COB$ , $\\angle COB = 2... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 11234,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five runners, $P$, $Q$, $R$, $S$, $T$, have a race, and $P$ beats $Q$, $P$ beats $R$, $Q$ beats $S$, and $T$ finishes after $P$ and before $Q$. Who could NOT have finished third in the race?"
}
] | math | {
"style": "rule",
"ground_truth": "P and S"
} | {
"index": 11235,
"solution": "1. **Analyze the given information**: We know that $P$ beats $Q$, $P$ beats $R$, $Q$ beats $S$, and $T$ finishes after $P$ but before $Q$.\n\n2. **Determine the possible positions for $P$**:\n - Since $P$ beats $Q$, $R$, and $S$ (by transitivity from $Q$ beating $S$), $P$ must be in... |
DeepScaleR | [
{
"role": "user",
"content": "Five six-sided dice are rolled. It is known that after the roll, there are two pairs of dice showing the same number, and one odd die. The odd die is rerolled. What is the probability that after rerolling the odd die, the five dice show a full house?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 11236,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five soccer teams play a match where each team plays every other team exactly once. Each match awards 3 points to the winner, 0 points to the loser, and 1 point to each team in the event of a draw. After all matches have been played, the total points of the five teams are found... | math | {
"style": "rule",
"ground_truth": "13213"
} | {
"index": 11237,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five socks, colored blue, brown, black, red, and purple are in a drawer. In how many different ways can we choose three socks from the drawer if the order of the socks does not matter?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 11238,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five squares and two right-angled triangles are positioned as shown. The areas of three squares are \\(3 \\, \\mathrm{m}^{2}, 7 \\, \\mathrm{m}^{2}\\), and \\(22 \\, \\mathrm{m}^{2}\\). What is the area, in \\(\\mathrm{m}^{2}\\), of the square with the question mark?\nA) 18\nB)... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 11239,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five standard six-sided dice are rolled. We are told there is no three-of-a-kind, but there is a pair of dice that show the same number. These two dice are set aside, and the other three dice are re-rolled. What is the probability that after re-rolling these three dice, at l... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"index": 11240,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five standard, six-sided dice are to be rolled. If the product of their values is an even number, what is the probability that their sum is divisible by 3?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 11241,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five students from a certain class participated in a speech competition and the order of appearance was determined by drawing lots, under the premise that student A must appear before student B. Calculate the probability of students A and B appearing adjacent to each other."
... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"index": 11242,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five students play chess matches against each other. Each student plays three matches against each of the other students. How many matches are played in total?"
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 11243,
"solution": "We label the players as A, B, C, D, and E. The total number of matches played will be equal to the number of pairs of players that can be formed times the number of matches that each pair plays. The possible pairs of players are AB, AC, AD, AE, BC, BD, BE, CD, CE, and DE. There are 10... |
DeepScaleR | [
{
"role": "user",
"content": "Five students, $A$, $B$, $C$, $D$, and $E$, entered the final of a school skills competition and the rankings from first to fifth were determined (with no ties). It is known that students $A$ and $B$ are neither first nor last. Calculate the number of different arrangements of ... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 11244,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five students, A, B, C, D, and E, participated in a labor skills competition. A and B asked about the results. The respondent told A, \"Unfortunately, neither you nor B got first place.\" To B, the respondent said, \"You certainly are not the worst.\" Determine the number of di... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"index": 11245,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five students, labeled as A, B, C, D, and E, are standing in a row to participate in a literary performance. If A does not stand at either end, calculate the number of different arrangements where C and D are adjacent."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 11246,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five students, labeled as A, B, C, D, and E, are standing in a row to participate in a literary performance. If student A does not stand at either end, calculate the number of different arrangements where students C and D are adjacent."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 11247,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five test scores have a mean (average score) of $90$, a median (middle score) of $91$ and a mode (most frequent score) of $94$. The sum of the two lowest test scores is"
}
] | math | {
"style": "rule",
"ground_truth": "171"
} | {
"index": 11248,
"solution": "1. **Identify the given values and their implications:**\n - The mean of the five test scores is $90$. This implies that the sum of all five scores is $90 \\times 5 = 450$.\n - The median of the scores is $91$, which is the middle value in the ordered list of scores.\n - The mod... |
DeepScaleR | [
{
"role": "user",
"content": "Five times $A$'s money added to $B$'s money is more than $51.00$. Three times $A$'s money minus $B$'s money is $21.00$.\nIf $a$ represents $A$'s money in dollars and $b$ represents $B$'s money in dollars, then:"
}
] | math | {
"style": "rule",
"ground_truth": "$a>9, b>6$"
} | {
"index": 11249,
"solution": "1. **Set up the equations** based on the problem statement:\n - The first condition states that five times A's money plus B's money is more than $51.00. This can be written as:\n \\[\n 5a + b > 51\n \\]\n - The second condition states that three times A's money minus B... |
DeepScaleR | [
{
"role": "user",
"content": "Five toppings for a pizza are available at Polina's Pizza. How many combinations of two different toppings are possible?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 11250,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passi... | math | {
"style": "rule",
"ground_truth": "544"
} | {
"index": 11251,
"solution": "As noted before, you can see that there are 2 ways that the condition cannot be met; it either has all roads leading into one city, or all roads leading out of one city. We then use complementary counting to count these cases, with PIE. Obviously there are $2^5*2^5$ ways to make roads... |
DeepScaleR | [
{
"role": "user",
"content": "Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from $(c,0)$ to $(3,3)$, divides the entire region into two regions of equal area. What is $c$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 11252,
"solution": "\nWe are given a configuration of five unit squares in the coordinate plane, and a line extending from $(c,0)$ to $(3,3)$ that divides the entire region into two regions of equal area. We need to find the value of $c$.\n\n#### Step-by-step Analysis:\n\n1. **Total Area of the Squares**... |
DeepScaleR | [
{
"role": "user",
"content": "Five volunteers and two elderly people are taking a photo, and they need to be arranged in a row. The two elderly people must stand next to each other but cannot be at either end of the row. How many different arrangements are possible?"
}
] | math | {
"style": "rule",
"ground_truth": "960"
} | {
"index": 11253,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five volunteers and two elderly people need to line up in a row, with the two elderly people next to each other but not at the ends. How many different ways can they arrange themselves?"
}
] | math | {
"style": "rule",
"ground_truth": "960"
} | {
"index": 11254,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five volunteers participate in community service for two days, Saturday and Sunday. Each day, two people are selected from the group to serve. Determine the number of ways to select exactly one person to serve for both days."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 11255,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five volunteers participate in community service for two days, Saturday and Sunday. Each day, two people are selected to serve. Calculate the number of ways to select exactly one person to serve for both days."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 11256,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five volunteers participate in community service for two days, Saturday and Sunday. Each day, two people are selected to serve. Find the number of ways to select exactly one person to serve for both days."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 11257,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five women of different heights are standing in a line at a social gathering. Each woman decides to only shake hands with women taller than herself. How many handshakes take place?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 11258,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five years ago, Tim was three times as old as his sister Sarah, and three years before that, Tim was five times as old as Sarah. Determine the number of years it will take for the ratio of their ages to be 3 : 2."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 11259,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five years ago, there were 25 trailer homes on Maple Street with an average age of 12 years. Since then, a group of brand new trailer homes was added, and 5 old trailer homes were removed. Today, the average age of all the trailer homes on Maple Street is 11 years. How many new... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 11260,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Five years from now, Billy's age will be twice Joe's current age. Currently, the sum of their ages is 60. How old is Billy right now?"
}
] | math | {
"style": "rule",
"ground_truth": "38\\frac{1}{3}"
} | {
"index": 11261,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Fix a sequence $ a_1,a_2,a_3,... $ of integers satisfying the following condition:for all prime numbers $ p $ and all positive integers $ k $ , we have $ a_{pk+1}=pa_k-3a_p+13 $ .Determine all possible values of $ a_{2013} $ ."
}
] | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"index": 11262,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Fix a sequence $a_1,a_2,a_3\\ldots$ of integers satisfying the following condition:for all prime numbers $p$ and all positive integers $k$ ,we have $a_{pk+1}=pa_k-3a_p+13$ .Determine all possible values of $a_{2013}$ ."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 11263,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Fix positive integers $k,n$. A candy vending machine has many different colours of candy, where there are $2n$ candies of each colour. A couple of kids each buys from the vending machine $2$ candies of different colours. Given that for any $k+1$ kids there are two kids who have... | math | {
"style": "rule",
"ground_truth": "n(3k)"
} | {
"index": 11264,
"solution": "\nFix positive integers \\( k \\) and \\( n \\). Consider a candy vending machine that has many different colors of candy, with \\( 2n \\) candies of each color. A couple of kids each buys from the vending machine 2 candies of different colors. We are to find the maximum number of kid... |
DeepScaleR | [
{
"role": "user",
"content": "Flat Albert and his buddy Mike are watching the game on Sunday afternoon. Albert is drinking lemonade from a two-dimensional cup which is an isosceles triangle whose height and base measure 9 cm and 6 cm; the opening of the cup corresponds to the base, which points upwards. Eve... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 11265,
"solution": "Let $A_{0}=\\frac{1}{2}(6)(9)=27$ denote the area of Albert's cup; since area varies as the square of length, at time $n$ Mike adds $$A\\left(1-\\left(1-\\frac{1}{9 n^{2}}\\right)^{2}\\right)$$ whence in all, he adds $$A_{0} \\sum_{n=1}^{\\infty}\\left(\\frac{2}{9 n^{2}}-\\frac{1}{81 ... |
DeepScaleR | [
{
"role": "user",
"content": "Flea Kuzya can make jumps along a line \\(L\\). The starting point for the jumps is point \\(A\\) on line \\(L\\), the length of each jump is \\(h\\), and the direction of each jump is chosen randomly and equally likely. Find the probability that after making between two and fi... | math | {
"style": "rule",
"ground_truth": "5/8"
} | {
"index": 11266,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Flights are arranged between 13 countries. For $ k\\ge 2$ , the sequence $ A_{1} ,A_{2} ,\\ldots A_{k}$ is said to a cycle if there exist a flight from $ A_{1}$ to $ A_{2}$ , from $ A_{2}$ to $ A_{3}$ , $ \\ldots$ , from $ A_{k \\minus{} 1}$ to $ A_{k}$ , and from ... | math | {
"style": "rule",
"ground_truth": "79"
} | {
"index": 11267,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Floyd looked at a standard $12$ hour analogue clock at $2\\!:\\!36$ . When Floyd next looked at the clock, the angles through which the hour hand and minute hand of the clock had moved added to $247$ degrees. How many minutes after $3\\!:\\!00$ was that?"
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 11268,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Fluffball and Shaggy the squirrels ate a basket of berries and a pack of seeds containing between 50 and 65 seeds, starting and finishing at the same time. Initially, Fluffball ate berries while Shaggy ate seeds. Later, they swapped tasks. Shaggy ate berries six times faster th... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"index": 11269,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Fnargs are either red or blue and have 2, 3, or 4 heads. A group of six Fnargs consisting of one of each possible form (one red and one blue for each number of heads) is made to line up such that no immediate neighbors are the same color nor have the same number of heads. How m... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 11270,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Fold a 10m long rope in half 5 times, then cut it in the middle with scissors. How many segments is the rope cut into?"
}
] | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 11271,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Fold a piece of graph paper once so that the point (0, 2) coincides with the point (4, 0), and the point (9, 5) coincides with the point (m, n). The value of m+n is ______."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 11272,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Following the concept of a healthy, low-carbon lifestyle, an increasing number of people are renting bicycles for cycling tours. A particular bicycle rental point charges no fee for rentals that do not exceed two hours, and for rentals that exceed two hours, the charging standa... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{2}"
} | {
"index": 11273,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Following the directions of the arrows, how many different paths are there from $A$ to $C$?\n\n[asy]\npair A,B,C;\nA=(0,0);\nB=(5,0);\nC=(10,0);\ndot(A);\ndot(B);\ndot(C);\nlabel(\"$A$\",A,S);\nlabel(\"$B$\",B,S);\nlabel(\"$C$\",C,S);\ndraw((0,0)--(2.5,1)--(5,0),Arrow);\ndraw((... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 11274,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Foma and Erema were traveling along a straight road to Moscow in a cart at a constant speed.\n\n- At 12:00, Foma asked: \"How many versts to Moscow?\"\n- Erema replied: \"82.\"\n- At 13:00, Foma asked: \"How many versts to Moscow?\"\n- Erema replied: \"71.\"\n- At 15:00, Foma a... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"index": 11275,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Food safety issues are increasingly attracting people's attention. The abuse of pesticides and chemical fertilizers poses certain health risks to the public. To provide consumers with safe vegetables, a rural cooperative invests 2 million yuan each year to build two pollution-f... | math | {
"style": "rule",
"ground_truth": "282"
} | {
"index": 11276,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Football tickets are normally priced at $15 each. After buying 5 tickets, any additional tickets are sold at a discounted price of $12 each. If Jane has $150, what is the maximum number of tickets she can buy?"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 11277,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Football tickets cost $\\$13.50$ each. What is the maximum number of tickets Jane can buy with $\\$100.00$?"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 11278,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For $ n \\in \\mathbb{N}$ , let $ f(n)\\equal{}1^n\\plus{}2^{n\\minus{}1}\\plus{}3^{n\\minus{}2}\\plus{}...\\plus{}n^1$ . Determine the minimum value of: $ \\frac{f(n\\plus{}1)}{f(n)}.$"
}
] | math | {
"style": "rule",
"ground_truth": "8/3"
} | {
"index": 11279,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For $a>0$ , denote by $S(a)$ the area of the part bounded by the parabolas $y=\\frac 12x^2-3a$ and $y=-\\frac 12x^2+2ax-a^3-a^2$ .\nFind the maximum area of $S(a)$ ."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8\\sqrt{2}}{3}"
} | {
"index": 11280,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For $a>0$ , let $f(a)=\\lim_{t\\to\\+0} \\int_{t}^1 |ax+x\\ln x|\\ dx.$ Let $a$ vary in the range $0 <a< +\\infty$ , find the minimum value of $f(a)$ ."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\ln 2}{2}"
} | {
"index": 11281,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For $k\\ge 1$ , define $a_k=2^k$ . Let $$ S=\\sum_{k=1}^{\\infty}\\cos^{-1}\\left(\\frac{2a_k^2-6a_k+5}{\\sqrt{(a_k^2-4a_k+5)(4a_k^2-8a_k+5)}}\\right). $$ Compute $\\lfloor 100S\\rfloor$ ."
}
] | math | {
"style": "rule",
"ground_truth": "157"
} | {
"index": 11282,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For $n > 1$ , let $a_n$ be the number of zeroes that $n!$ ends with when written in base $n$ . Find the maximum value of $\\frac{a_n}{n}$ ."
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"index": 11283,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For $n \\geq 1$ , let $a_n$ be the number beginning with $n$ $9$ 's followed by $744$ ; eg., $a_4=9999744$ . Define $$ f(n)=\\text{max}\\{m\\in \\mathbb{N} \\mid2^m ~ \\text{divides} ~ a_n \\} $$ , for $n\\geq 1$ . Find $f(1)+f(2)+f(3)+ \\cdots + f(10)$ ."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"index": 11284,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For $n$ measured in degrees, let $T(n) = \\cos^2(30^\\circ -n) - \\cos(30^\\circ -n)\\cos(30^\\circ +n) +\\cos^2(30^\\circ +n)$ . Evaluate $$ 4\\sum^{30}_{n=1} n \\cdot T(n). $$"
}
] | math | {
"style": "rule",
"ground_truth": "1395"
} | {
"index": 11285,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For $-1<r<1$, let $S(r)$ denote the sum of the geometric series \\[12+12r+12r^2+12r^3+\\cdots .\\] Let $a$ between $-1$ and $1$ satisfy $S(a)S(-a)=2016$. Find $S(a)+S(-a)$."
}
] | math | {
"style": "rule",
"ground_truth": "336"
} | {
"index": 11286,
"solution": "The sum of an infinite geometric series is $\\frac{a}{1-r}\\rightarrow \\frac{12}{1\\mp a}$. The product $S(a)S(-a)=\\frac{144}{1-a^2}=2016$. $\\frac{12}{1-a}+\\frac{12}{1+a}=\\frac{24}{1-a^2}$, so the answer is $\\frac{2016}{6}=\\boxed{336}$.\n ~ pi_is_3.14"
} |
DeepScaleR | [
{
"role": "user",
"content": "For $-1<r<1$, let $S(r)$ denote the sum of the geometric series \\[12+12r+12r^2+12r^3+\\cdots .\\]Let $a$ between $-1$ and $1$ satisfy $S(a)S(-a)=2016$. Find $S(a)+S(-a)$."
}
] | math | {
"style": "rule",
"ground_truth": "336"
} | {
"index": 11287,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For $-1<r<1$, let $T(r)$ denote the sum of the geometric series \\[20 + 10r + 10r^2 + 10r^3 + \\cdots.\\] Let $b$ between $-1$ and $1$ satisfy $T(b)T(-b)=5040$. Find $T(b)+T(-b)$."
}
] | math | {
"style": "rule",
"ground_truth": "504"
} | {
"index": 11288,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For $-1<s<1$, let $T(s)$ denote the sum of the geometric series \\[20 + 20s + 20s^2 + 20s^3+\\cdots .\\] Let $b$ between $-1$ and $1$ satisfy $T(b)T(-b) = 4800$. Find $T(b) + T(-b)$."
}
] | math | {
"style": "rule",
"ground_truth": "480"
} | {
"index": 11289,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For $-25 \\le x \\le 25,$ find the maximum value of $\\sqrt{25 + x} + \\sqrt{25 - x}.$"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 11290,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For $0 \\leq p \\leq 1/2$, let $X_1, X_2, \\dots$ be independent random variables such that \\[ X_i = \\begin{cases} 1 & \\mbox{with probability $p$,} \\\\ -1 & \\mbox{with probability $p$,} \\\\ 0 & \\mbox{with probability $1-2p$,} \\end{cases} \\] for all $i \\geq 1$. Given a... | math | {
"style": "rule",
"ground_truth": "p \\leq 1/4"
} | {
"index": 11291,
"solution": "The answer is $p \\leq 1/4$. We first show that $p >1/4$ does not satisfy the desired condition. For $p>1/3$, $P(0,1) = 1-2p < p = P(1,1)$. For $p=1/3$, it is easily calculated (or follows from the next calculation) that $P(0,1,2) = 1/9 < 2/9 = P(1,1,2)$. Now suppose $1/4 < p < 1/3$, ... |
DeepScaleR | [
{
"role": "user",
"content": "For $1 \\le n \\le 100$, how many integers are there such that $\\frac{n}{n+1}$ is a repeating decimal?"
}
] | math | {
"style": "rule",
"ground_truth": "86"
} | {
"index": 11292,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For $1 \\le n \\le 200$, how many integers are there such that $\\frac{n}{n+1}$ is a repeating decimal?"
}
] | math | {
"style": "rule",
"ground_truth": "182"
} | {
"index": 11293,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For $1 \\leq i \\leq 215$ let $a_i = \\dfrac{1}{2^{i}}$ and $a_{216} = \\dfrac{1}{2^{215}}$. Let $x_1, x_2, ..., x_{216}$ be positive real numbers such that $\\sum_{i=1}^{216} x_i=1$ and $\\sum_{1 \\leq i < j \\leq 216} x_ix_j = \\dfrac{107}{215} + \\sum_{i=1}^{216} \\dfrac{a_i... | math | {
"style": "rule",
"ground_truth": "863"
} | {
"index": 11294,
"solution": "Note that \\begin{align*}\\sum_{1 \\leq i < j \\leq 216} x_ix_j &= \\frac12\\left(\\left(\\sum_{i=1}^{216} x_i\\right)^2-\\sum_{i=1}^{216} x_i^2\\right)\\\\&=\\frac12\\left(1-\\sum x_i^2\\right).\\end{align*} Substituting this into the second equation and collecting $x_i^2$ terms, we ... |
DeepScaleR | [
{
"role": "user",
"content": "For $\\mathbf{v} = \\begin{pmatrix} -10 \\\\ 6 \\end{pmatrix}$ and $\\mathbf{w} = \\begin{pmatrix} 15 \\\\ -9 \\end{pmatrix}$, compute $\\text{proj}_{\\mathbf{w}} \\mathbf{v}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -10 \\\\ 6 \\end{pmatrix}"
} | {
"index": 11295,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For $\\mathbf{v} = \\begin{pmatrix} 1 \\\\ y \\end{pmatrix}$ and $\\mathbf{w} = \\begin{pmatrix} 9 \\\\ 3 \\end{pmatrix}$,\n\\[\\text{proj}_{\\mathbf{w}} \\mathbf{v} = \\begin{pmatrix} -6 \\\\ -2 \\end{pmatrix}.\\]Find $y$."
}
] | math | {
"style": "rule",
"ground_truth": "-23"
} | {
"index": 11296,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For $\\pi \\le \\theta < 2\\pi$, let\n\\begin{align*} P &= \\frac12\\cos\\theta - \\frac14\\sin 2\\theta - \\frac18\\cos 3\\theta + \\frac{1}{16}\\sin 4\\theta + \\frac{1}{32} \\cos 5\\theta - \\frac{1}{64} \\sin 6\\theta - \\frac{1}{128} \\cos 7\\theta + \\cdots \\end{align*}\... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 11297,
"solution": "Use sum to product formulas to rewrite $P$ and $Q$\n\\[P \\sin\\theta\\ + Q \\cos\\theta\\ = \\cos \\theta\\ - \\frac{1}{4}\\cos \\theta + \\frac{1}{8}\\sin 2\\theta + \\frac{1}{16}\\cos 3\\theta - \\frac{1}{32}\\sin 4\\theta + ...\\]\n Therefore, \\[P \\sin \\theta - Q \\cos \\theta ... |
DeepScaleR | [
{
"role": "user",
"content": "For $\\{1, 2, 3, ..., n\\}$ and each of its non-empty subsets, a unique alternating sum is defined as follows: Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. Find the sum of all such a... | math | {
"style": "rule",
"ground_truth": "5120"
} | {
"index": 11298,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For $\\{1, 2, 3, \\ldots, 10\\}$ and each of its non-empty subsets, a unique alternating sum is defined as follows: Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. Find the sum of all s... | math | {
"style": "rule",
"ground_truth": "5120"
} | {
"index": 11299,
"solution": ""
} |
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