data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Four new students are to be assigned to three classes: A, B, and C, with at least one student in each class. Student A cannot be assigned to class A. How many different assignment plans are there?"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 12200,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four normal students, A, B, C, and D, are to be assigned to work at three schools, School A, School B, and School C, with at least one person at each school. It is known that A is assigned to School A. What is the probability that B is assigned to School B?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{5}{12}"
} | {
"index": 12201,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four numbers, 2613, 2243, 1503, and 985, when divided by the same positive integer, yield the same remainder (which is not zero). Find the divisor and the remainder."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 12202,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four of the eight vertices of a cube are the vertices of a regular tetrahedron. Find the ratio of the surface area of the cube to the surface area of the tetrahedron."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 12203,
"solution": "1. **Assume the side length of the cube**: Let the side length of the cube be $s = 1$.\n\n2. **Identify the vertices of the tetrahedron**: Four vertices of a cube that form a regular tetrahedron can be chosen such that they are not all on the same face and no three are collinear on an... |
DeepScaleR | [
{
"role": "user",
"content": "Four of the five vertices of a polygon shaped as a right trapezoid are (5, 11), (16, 11), (16, -2), and (5, -2); it includes a semicircle with a diameter along the bottom base, centered at (10.5, -2) and radius 5.5. Determine the total area bounded by the lower base of the trap... | math | {
"style": "rule",
"ground_truth": "15.125\\pi"
} | {
"index": 12204,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four of the following test scores are Henry's and the other four are Julia's: 88, 90, 92, 94, 95, 97, 98, 99. Henry's mean score is 94. What is Julia's mean score?"
}
] | math | {
"style": "rule",
"ground_truth": "94.25"
} | {
"index": 12205,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four of the six numbers 1867, 1993, 2019, 2025, 2109, and 2121 have a mean (average) of 2008. What is the mean (average) of the other two numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "2051"
} | {
"index": 12206,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four packages are delivered to four houses, one to each house. If these packages are randomly delivered, what is the probability that exactly two of them are delivered to the correct houses? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 12207,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four pairs of socks in different colors are randomly selected from a wardrobe, and it is known that two of them are from the same pair. Calculate the probability that the other two are not from the same pair."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{9}"
} | {
"index": 12208,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four pens and three pencils cost $\\$2.24$. Two pens and five pencils cost $\\$1.54$. No prices include tax. In cents, what is the cost of a pencil?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 12209,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four people (A, B, C, D) are practicing passing a ball. The ball is initially passed by A, and each person who receives the ball has an equal probability of passing it to one of the other three people. Let \\( p_{n} \\) represent the probability that the ball returns to A after... | math | {
"style": "rule",
"ground_truth": "\\frac{61}{243}"
} | {
"index": 12210,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four 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. Calculate the probability that no two ... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{16}"
} | {
"index": 12211,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four people are sitting at four sides of a table, and they are dividing a 32-card Hungarian deck equally among themselves. If one selected player does not receive any aces, what is the probability that the player sitting opposite them also has no aces among their 8 cards?"
}
... | math | {
"style": "rule",
"ground_truth": "130/759"
} | {
"index": 12212,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four people can mow a lawn in 6 hours. How many more people will be needed to mow the lawn in 4 hours, assuming each person mows at the same rate?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 12213,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four people can paint a house in six hours. How many hours would it take three people to paint the same house, assuming everyone works at the same rate?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 12214,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four people sit around a circular table, and each person will roll a standard six-sided die. What is the probability that no two people sitting next to each other will roll the same number after they each roll the die once? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{35}{72}"
} | {
"index": 12215,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four people, A, B, C, and D, participated in an exam. The combined scores of A and B are 17 points higher than the combined scores of C and D. A scored 4 points less than B, and C scored 5 points more than D. How many points higher is the highest score compared to the lowest sc... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 12216,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four people, A, B, C, and D, stand in a line from left to right and are numbered 1, 2, 3, and 4 respectively. They have the following conversation:\n\nA: Both people to my left and my right are taller than me.\nB: Both people to my left and my right are shorter than me.\nC: I a... | math | {
"style": "rule",
"ground_truth": "2314"
} | {
"index": 12217,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four people, A, B, C, and D, stand on a staircase with 7 steps. If each step can accommodate up to 3 people, and the positions of people on the same step are not distinguished, then the number of different ways they can stand is (answer in digits)."
}
] | math | {
"style": "rule",
"ground_truth": "2394"
} | {
"index": 12218,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four planes divide space into $n$ parts at most. Calculate $n$."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 12219,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four players stand at distinct vertices of a square. They each independently choose a vertex of the square (which might be the vertex they are standing on). Then, they each, at the same time, begin running in a straight line to their chosen vertex at 10 mph, stopping when they ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 12220,
"solution": "Observe that no two players can choose the same vertex, and no two players can choose each others vertices. Thus, if two players choose their own vertices, then the remaining two also must choose their own vertices (because they can't choose each others vertices), thus all 4 players m... |
DeepScaleR | [
{
"role": "user",
"content": "Four points $B,$ $A,$ $E,$ and $L$ are on a straight line, as shown. The point $G$ is off the line so that $\\angle BAG = 120^\\circ$ and $\\angle GEL = 80^\\circ.$ If the reflex angle at $G$ is $x^\\circ,$ then what does $x$ equal?\n\n[asy]\ndraw((0,0)--(30,0),black+linewidth(... | math | {
"style": "rule",
"ground_truth": "340"
} | {
"index": 12221,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four points \\( B, A, E, L \\) are on a straight line. \\( G \\) is a point off the line such that \\(\\angle B A G = 120^\\circ\\) and \\(\\angle G E L = 80^\\circ\\). If the reflex angle at \\( G \\) is \\( x^\\circ \\), then \\( x \\) equals:"
}
] | math | {
"style": "rule",
"ground_truth": "340"
} | {
"index": 12222,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four points are chosen uniformly and independently at random in the interior of a given circle. Find the probability that they are the vertices of a convex quadrilateral."
}
] | math | {
"style": "rule",
"ground_truth": "1 - \\frac{35}{12 \\pi^2}"
} | {
"index": 12223,
"solution": "\\textbf{First solution:} (by Daniel Kane) The probability is $1 - \\frac{35}{12\\pi^2}$. We start with some notation and simplifications. For simplicity, we assume without loss of generality that the circle has radius 1. Let $E$ denote the expected value of a random variable over all... |
DeepScaleR | [
{
"role": "user",
"content": "Four points are independently chosen uniformly at random from the interior of a regular dodecahedron. What is the probability that they form a tetrahedron whose interior contains the dodecahedron's center?"
}
] | math | {
"style": "rule",
"ground_truth": "\\[\n\\frac{1}{8}\n\\]"
} | {
"index": 12224,
"solution": "Situate the origin $O$ at the dodecahedron's center, and call the four random points $P_{i}$, where $1 \\leq i \\leq 4$. To any tetrahedron $P_{1} P_{2} P_{3} P_{4}$ we can associate a quadruple $\\left(\\epsilon_{(i j k)}\\right)$, where $(i j k)$ ranges over all conjugates of the cy... |
DeepScaleR | [
{
"role": "user",
"content": "Four points are on a line segment.\n\nIf \\( A B : B C = 1 : 2 \\) and \\( B C : C D = 8 : 5 \\), then \\( A B : B D \\) equals"
}
] | math | {
"style": "rule",
"ground_truth": "4 : 13"
} | {
"index": 12225,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four points are randomly chosen from the vertices of a regular 12-sided polygon. Find the probability that the four chosen points form a rectangle (including square)."
}
] | math | {
"style": "rule",
"ground_truth": "1/33"
} | {
"index": 12226,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four points in the order \\( A, B, C, D \\) lie on a circle with the extension of \\( AB \\) meeting the extension of \\( DC \\) at \\( E \\) and the extension of \\( AD \\) meeting the extension of \\( BC \\) at \\( F \\). Let \\( EP \\) and \\( FQ \\) be tangents to this circ... | math | {
"style": "rule",
"ground_truth": "87"
} | {
"index": 12227,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four points, $A, B, C$, and $D$, are chosen randomly on the circumference of a circle with independent uniform probability. What is the expected number of sides of triangle $A B C$ for which the projection of $D$ onto the line containing the side lies between the two vertices?"... | math | {
"style": "rule",
"ground_truth": "3/2"
} | {
"index": 12228,
"solution": "By linearity of expectations, the answer is exactly 3 times the probability that the orthogonal projection of $D$ onto $A B$ lies interior to the segment. This happens exactly when either $\\angle D A B$ or $\\angle D B A$ is obtuse, which is equivalent to saying that $A$ and $B$ lie ... |
DeepScaleR | [
{
"role": "user",
"content": "Four positive integers $A$, $B$, $C$ and $D$ have a sum of 36. If $A+2 = B-2 = C \\times 2 = D \\div 2$, what is the value of the product $A \\times B \\times C \\times D$?"
}
] | math | {
"style": "rule",
"ground_truth": "3840"
} | {
"index": 12229,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four positive integers $A$, $B$, $C$ and $D$ have a sum of 64. If $A+3 = B-3 = C \\times 3 = D \\div 3$, what is the value of the product $A \\times B \\times C \\times D$?"
}
] | math | {
"style": "rule",
"ground_truth": "19440"
} | {
"index": 12230,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four positive integers $a$, $b$, $c$, and $d$ have a product of $8!$ and satisfy:\n\\[\\begin{array}{rl} ab + a + b & = 524 \\\\ bc + b + c & = 146 \\\\ cd + c + d & = 104 \\end{array}\\]\nWhat is $a-d$?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 12231,
"solution": "1. **Rewrite the given equations using Simon's Favorite Factoring Trick**: \n The equations given are:\n \\[\n \\begin{align*}\n ab + a + b &= 524, \\\\\n bc + b + c &= 146, \\\\\n cd + c + d &= 104.\n \\end{align*}\n \\]\n Adding 1 to both sides of each equation, we... |
DeepScaleR | [
{
"role": "user",
"content": "Four positive integers $p$, $q$, $r$, $s$ satisfy $p \\cdot q \\cdot r \\cdot s = 9!$ and $p < q < r < s$. What is the smallest possible value of $s-p$?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 12232,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four positive integers are given. Select any three of these integers, find their arithmetic average, and add this result to the fourth integer. Thus the numbers $29, 23, 21$, and $17$ are obtained. One of the original integers is:"
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 12233,
"solution": "1. Let the original integers be $a, b, c,$ and $d$. We are given the following equations based on the problem statement:\n \\[\n \\frac{a+b+c}{3} + d = 29\n \\]\n \\[\n \\frac{a+b+d}{3} + c = 23\n \\]\n \\[\n \\frac{a+c+d}{3} + b = 21\n \\]\n \\[\n \\frac{b+c+d}{... |
DeepScaleR | [
{
"role": "user",
"content": "Four prime numbers are randomly selected without replacement from the first ten prime numbers. What is the probability that the sum of the four selected numbers is odd? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"index": 12234,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four prime numbers are randomly selected without replacement from the first twelve prime numbers. What is the probability that the sum of the four selected numbers is even?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 12235,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four primes $a$, $b$, $c$ and $d$ form an increasing arithmetic sequence with $a > 5$ and common difference 6. What is the ones digit of $a$?"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 12236,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four princesses each guessed a two-digit number, and Ivan guessed a four-digit number. After they wrote their numbers in a row in some order, they got the sequence 132040530321. Find Ivan's number."
}
] | math | {
"style": "rule",
"ground_truth": "5303"
} | {
"index": 12237,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four princesses thought of two-digit numbers, and Ivan thought of a four-digit number. After they wrote their numbers in a row in some order, the result was 132040530321. Find Ivan's number."
}
] | math | {
"style": "rule",
"ground_truth": "5303"
} | {
"index": 12238,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four problems were attempted by 100 contestants in a Mathematics competition. The first problem was solved by 90 contestants, the second by 85 contestants, the third by 80 contestants, and the fourth by 75 contestants. What is the smallest possible number of contestants who sol... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 12239,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four rectangular strips each measuring $4$ by $16$ inches are laid out with two vertical strips crossing two horizontal strips forming a single polygon which looks like a tic-tack-toe pattern. What is the perimeter of this polygon?\n\n[asy]\nsize(100);\ndraw((1,0)--(2,0)--(... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 12240,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four red beads, two white beads, and one green bead are placed in a line in random order. What is the probability that no two neighboring beads are the same color?\nA) $\\frac{1}{15}$\nB) $\\frac{2}{15}$\nC) $\\frac{1}{7}$\nD) $\\frac{1}{30}$\nE) $\\frac{1}{21}$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{15}"
} | {
"index": 12241,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four red candies and three green candies can be combined to make many different flavors. Flavors are different if the percent red is different, so 3 red / 0 green is the same flavor as 2 red / 0 green; and likewise 4 red / 2 green is the same flavor as 2 red / 1 green. If a fla... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 12242,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four scores belong to Alex and the other three to Morgan: 78, 82, 90, 95, 98, 102, 105. Alex's mean score is 91.5. What is Morgan's mean score?"
}
] | math | {
"style": "rule",
"ground_truth": "94.67"
} | {
"index": 12243,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four siblings inherited a plot of land shaped like a convex quadrilateral. By connecting the midpoints of the opposite sides of the plot, they divided the inheritance into four quadrilaterals. The first three siblings received plots of $360 \\, \\mathrm{m}^{2}$, $720 \\, \\math... | math | {
"style": "rule",
"ground_truth": "540"
} | {
"index": 12244,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four spheres of radius 1 are placed so that each touches the other three. What is the radius of the smallest sphere that contains all four spheres?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{\\frac{3}{2}} + 1"
} | {
"index": 12245,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four spheres, each with a radius of 1, are placed on a horizontal table with each sphere tangential to its neighboring spheres (the centers of the spheres form a square). There is a cube whose bottom face is in contact with the table, and each vertex of the top face of the cube... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 12246,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four standard, six-sided dice are to be rolled. If the product of their values turns out to be an even number, what is the probability their sum is odd? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{15}"
} | {
"index": 12247,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four steel balls, each with a radius of 1, are completely packed into a container in the shape of a regular tetrahedron. Find the minimum height of this regular tetrahedron."
}
] | math | {
"style": "rule",
"ground_truth": "2+\\frac{2 \\sqrt{6}}{3}"
} | {
"index": 12248,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four steel balls, each with a radius of 1, are completely packed into a container in the shape of a regular tetrahedron. The minimum height of this regular tetrahedron is:"
}
] | math | {
"style": "rule",
"ground_truth": "2 + \\frac{2 \\sqrt{6}}{3}"
} | {
"index": 12249,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four steel balls, each with a radius of 1, are to be completely packed into a container shaped as a regular tetrahedron. What is the minimum height of the tetrahedron?"
}
] | math | {
"style": "rule",
"ground_truth": "2 + \\frac{2 \\sqrt{6}}{3}"
} | {
"index": 12250,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four students are admitted to three universities. Find the probability that each university admits at least one student."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"index": 12251,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four students from Harvard, one of them named Jack, and five students from MIT, one of them named Jill, are going to see a Boston Celtics game. However, they found out that only 5 tickets remain, so 4 of them must go back. Suppose that at least one student from each school must... | math | {
"style": "rule",
"ground_truth": "104"
} | {
"index": 12252,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four students participate in a competition where each chooses one question from two options, A and B. The rules result in the following point system: 21 points for correct A, -21 points for incorrect A, 7 points for correct B, and -7 points for incorrect B. If the total score o... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"index": 12253,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four students participate in a knowledge contest, each student must choose one of the two questions, A or B, to answer. Correctly answering question A earns 60 points, while an incorrect answer results in -60 points. Correctly answering question B earns 180 points, while an inc... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"index": 12254,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four students take an exam. Three of their scores are $70, 80,$ and $90$. If the average of their four scores is $70$, then what is the remaining score?"
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 12255,
"solution": "\n#### Detailed Steps:\n\n1. **Understanding the Problem:**\n We are given the scores of three students as $70, 80,$ and $90$. We also know that the average score of all four students is $70$. We need to find the score of the fourth student, denoted as $r$.\n\n2. **Setting Up the Eq... |
DeepScaleR | [
{
"role": "user",
"content": "Four students, named A, B, C, and D, are divided into two volunteer groups to participate in two off-campus activities. The probability that students B and C participate in the same activity is ________."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 12256,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four teams play in a tournament in which each team plays exactly one game against each of the other three teams. At the end of each game, either the two teams tie or one team wins and the other team loses. A team is awarded 3 points for a win, 0 points for a loss, and 1 point f... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 12257,
"solution": "Suppose that the four teams in the league are called W, X, Y, and Z. Then there is a total of 6 games played: W against X, W against Y, W against Z, X against Y, X against Z, Y against Z. In each game that is played, either one team is awarded 3 points for a win and the other is award... |
DeepScaleR | [
{
"role": "user",
"content": "Four teams, including Quixajuba, are competing in a volleyball tournament where:\n\n- Each team plays against every other team exactly once;\n- Any match ends with one team winning;\n- In any match, the teams have an equal probability of winning;\n- At the end of the tournament... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"index": 12258,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four tour guides are leading eight tourists. Each tourist must choose one of the guides, however, each guide must take at least two tourists. How many different groupings of guides and tourists are possible?"
}
] | math | {
"style": "rule",
"ground_truth": "105"
} | {
"index": 12259,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four two-inch squares are placed with their bases on a line. The second square from the left is lifted out, rotated 45 degrees, then centered and lowered back until it touches its adjacent squares on both sides. Determine the distance, in inches, of point P, the top vertex of t... | math | {
"style": "rule",
"ground_truth": "1 + \\sqrt{2}"
} | {
"index": 12260,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four unit circles are centered at the vertices of a unit square, one circle at each vertex. What is the area of the region common to all four circles?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}+1-\\sqrt{3}"
} | {
"index": 12261,
"solution": "The desired region consists of a small square and four \"circle segments,\" i.e. regions of a circle bounded by a chord and an arc. The side of this small square is just the chord of a unit circle that cuts off an angle of $30^{\\circ}$, and the circle segments are bounded by that cho... |
DeepScaleR | [
{
"role": "user",
"content": "Four vehicles were traveling on the highway at constant speeds: a car, a motorcycle, a scooter, and a bicycle. The car passed the scooter at 12:00, encountered the bicyclist at 14:00, and met the motorcyclist at 16:00. The motorcyclist met the scooter at 17:00 and caught up wit... | math | {
"style": "rule",
"ground_truth": "15:20"
} | {
"index": 12262,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four vertices of a cube are given as \\(A=(1, 2, 3)\\), \\(B=(1, 8, 3)\\), \\(C=(5, 2, 3)\\), and \\(D=(5, 8, 3)\\). Calculate the surface area of the cube."
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"index": 12263,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four vertices of a rectangle include the points $(2, 3)$, $(2, 15)$, and $(13, 3)$. What is the area of the intersection of this rectangular region and the region inside the graph of the equation $(x - 13)^2 + (y - 3)^2 = 16$?"
}
] | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"index": 12264,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four whole numbers, when added three at a time, give the sums $180, 197, 208$ and $222$. What is the largest of the four numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "89"
} | {
"index": 12265,
"solution": "Let the four whole numbers be $a$, $b$, $c$, and $d$. According to the problem, the sums of these numbers taken three at a time are given as follows:\n- $a + b + c = 180$\n- $a + b + d = 197$\n- $a + c + d = 208$\n- $b + c + d = 222$\n\n1. **Calculate the sum of all four numbers taken... |
DeepScaleR | [
{
"role": "user",
"content": "Four years ago you invested some money at $10\\%$ interest. You now have $\\$439.23$ in the account. If the interest was compounded yearly, how much did you invest 4 years ago?"
}
] | math | {
"style": "rule",
"ground_truth": "300"
} | {
"index": 12266,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Four-digit \"progressive numbers\" are arranged in ascending order, determine the 30th number."
}
] | math | {
"style": "rule",
"ground_truth": "1359"
} | {
"index": 12267,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Fran writes the numbers \\(1,2,3, \\ldots, 20\\) on a chalkboard. Then she erases all the numbers by making a series of moves; in each move, she chooses a number \\(n\\) uniformly at random from the set of all numbers still on the chalkboard, and then erases all of the divisors... | math | {
"style": "rule",
"ground_truth": "\\frac{131}{10}"
} | {
"index": 12268,
"solution": "For each \\(n, 1 \\leq n \\leq 20\\), consider the first time that Fran chooses one of the multiples of \\(n\\). It is in this move that \\(n\\) is erased, and all the multiples of \\(n\\) at most 20 are equally likely to be chosen for this move. Hence this is the only move in which F... |
DeepScaleR | [
{
"role": "user",
"content": "Francesca uses $100$ grams of lemon juice, $100$ grams of sugar, and $400$ grams of water to make lemonade. There are $25$ calories in $100$ grams of lemon juice and $386$ calories in $100$ grams of sugar. Water contains no calories. How many calories are in $200$ grams of her ... | math | {
"style": "rule",
"ground_truth": "137"
} | {
"index": 12269,
"solution": "1. **Calculate Total Calories in the Lemonade Mixture**:\n Francesca uses $100$ grams of lemon juice, $100$ grams of sugar, and $400$ grams of water. The calorie content of each ingredient is given as:\n - Lemon juice: $25$ calories per $100$ grams\n - Sugar: $386$ calories per ... |
DeepScaleR | [
{
"role": "user",
"content": "Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade?"
... | math | {
"style": "rule",
"ground_truth": "137"
} | {
"index": 12270,
"solution": "1. **Calculate the total weight of the lemonade**: \n Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water. The total weight of the lemonade is:\n \\[\n 100 \\text{ grams (lemon juice)} + 100 \\text{ grams (sugar)} + 400 \\text{ grams (water)} = 60... |
DeepScaleR | [
{
"role": "user",
"content": "Frank and Joe are playing ping pong. For each game, there is a $30 \\%$ chance that Frank wins and a $70 \\%$ chance Joe wins. During a match, they play games until someone wins a total of 21 games. What is the expected value of number of games played per match?"
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 12271,
"solution": "The expected value of the ratio of Frank's to Joe's score is 3:7, so Frank is expected to win 9 games for each of Frank's 21. Thus the expected number of games in a match is 30."
} |
DeepScaleR | [
{
"role": "user",
"content": "Franklin has four bags, numbered 1 through 4. Initially, the first bag contains fifteen balls, numbered 1 through 15 , and the other bags are empty. Franklin randomly pulls a pair of balls out of the first bag, throws away the ball with the lower number, and moves the ball with... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 12272,
"solution": "Pretend there is a 16 th ball numbered 16. This process is equivalent to randomly drawing a tournament bracket for the 16 balls, and playing a tournament where the higher ranked ball always wins. The probability that a ball is left in a bag at the end is the probability that it loses ... |
DeepScaleR | [
{
"role": "user",
"content": "Fred the Four-Dimensional Fluffy Sheep is walking in 4 -dimensional space. He starts at the origin. Each minute, he walks from his current position $\\left(a_{1}, a_{2}, a_{3}, a_{4}\\right)$ to some position $\\left(x_{1}, x_{2}, x_{3}, x_{4}\\right)$ with integer coordinates ... | math | {
"style": "rule",
"ground_truth": "\\binom{40}{10}\\binom{40}{20}^{3}"
} | {
"index": 12273,
"solution": "The possible moves correspond to the vectors $\\pm\\langle 2,0,0,0\\rangle, \\pm\\langle 1,1,1,-1\\rangle$, and their permutations. It's not hard to see that these vectors form the vertices of a 4-dimensional hypercube, which motivates the change of coordinates $$\\left(x_{1}, x_{2}, ... |
DeepScaleR | [
{
"role": "user",
"content": "Freddy the frog is jumping around the coordinate plane searching for a river, which lies on the horizontal line $y = 24$. A fence is located at the horizontal line $y = 0$. On each jump Freddy randomly chooses a direction parallel to one of the coordinate axes and moves one uni... | math | {
"style": "rule",
"ground_truth": "273"
} | {
"index": 12274,
"solution": "Clearly Freddy's $x$-coordinate is irrelevant, so we let $E(y)$ be the expected value of the number of jumps it will take him to reach the river from a given $y$-coordinate. Observe that $E(24)=0$, and \\[E(y)=1+\\frac{E(y+1)+E(y-1)+2E(y)}{4}\\] for all $y$ such that $1\\le y\\le 23$.... |
DeepScaleR | [
{
"role": "user",
"content": "Frederik wants to make a special seven-digit password. Each digit of his password occurs exactly as many times as its digit value. The digits with equal values always occur consecutively, e.g., 4444333 or 1666666. How many possible passwords can he make?\n\nA) 6 \nB) 7 \nC) 1... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 12275,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "French mathematician Poincaré is a person who likes to eat bread. He goes to the same bakery every day to buy a loaf of bread. The baker at the bakery claims that the average weight of the bread he sells is $1000g$, with a fluctuation of no more than $50g$. In mathematical term... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{24}"
} | {
"index": 12276,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Freshmen go for a long walk in the suburbs after the start of school. They arrive at point \\( A \\) 6 minutes later than the originally planned time of 10:10, and they arrive at point \\( C \\) 6 minutes earlier than the originally planned time of 13:10. There is exactly one p... | math | {
"style": "rule",
"ground_truth": "11:40"
} | {
"index": 12277,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Frieda the frog begins a sequence of hops on a $3 \\times 3$ grid of squares, moving one square on each hop and choosing at random the direction of each hop-up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she \"... | math | {
"style": "rule",
"ground_truth": "\\frac{25}{32}"
} | {
"index": 12278,
"solution": "To solve this problem, we will calculate the probability that Frieda reaches a corner square within four hops, starting from the center of a $3 \\times 3$ grid. We will use a state-based approach to model Frieda's possible positions and transitions.\n\n#### Definitions:\n- **State**: ... |
DeepScaleR | [
{
"role": "user",
"content": "From $A$ to $B$ it is 999 km. Along the road, there are kilometer markers with distances written to $A$ and to $B$:\n\n$0|999,1|998, \\ldots, 999|0$.\n\nHow many of these markers have only two different digits?"
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 12279,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From 1 to 100, take a pair of integers (repetitions allowed) so that their sum is greater than 100. How many ways are there to pick such pairs?"
}
] | math | {
"style": "rule",
"ground_truth": "5050"
} | {
"index": 12280,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From 1000 to 9999, a four-digit number is randomly chosen. The probability that all the digits in this number are different and the units digit is an odd number is ( )."
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{56}{225}$"
} | {
"index": 12281,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From 125 sugar cubes, a $5 \\times 5 \\times 5$ cube was made. Ponchik picked all the cubes that have an odd number of neighbors and ate them (neighbors are those cubes that share a face). How many cubes did Ponchik eat in total?"
}
] | math | {
"style": "rule",
"ground_truth": "62"
} | {
"index": 12282,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From 3 male students and 2 female students, calculate the number of different election results in which at least one female student is elected."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 12283,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From 5 cards labeled with the numbers $1, 2, 3, 4, 5$, 3 cards are randomly drawn to form a three-digit number. How many such three-digit numbers cannot be divided by 3?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 12284,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From 5 differently colored balls, select 4 balls to place into 3 distinct boxes, with the requirement that no box is left empty. The total number of different ways to do this is ______. (Answer with a number)"
}
] | math | {
"style": "rule",
"ground_truth": "180"
} | {
"index": 12285,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From 5 male doctors and 4 female doctors, 3 doctors are to be selected to form a medical team, with the requirement that both male and female doctors must be included. Calculate the total number of different team formation plans."
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 12286,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From 5 students, 4 are to be selected to participate in competitions in four subjects: mathematics, physics, chemistry, and biology, with each competition having only 1 participant. If student A does not participate in the biology competition, then the total number of different... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"index": 12287,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From 50 products, 10 are selected for inspection. The total number of items is \\_\\_\\_\\_\\_\\_\\_, and the sample size is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 12288,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From 51 consecutive odd numbers $1, 3, 5, \\cdots, 101$, select $\\mathrm{k}$ numbers such that their sum is 1949. What is the maximum value of $\\mathrm{k}$?"
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"index": 12289,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From 6 athletes, 4 are to be chosen to participate in a 4×100 meters relay race, given that athletes A and B both cannot run the first leg, calculate the number of different methods to select the participants."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 12290,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From 6 sprinters, 4 are to be selected to participate in a 4×100 m relay. If among them, Athlete A cannot run the first leg, and Athlete B cannot run the fourth leg, how many different ways are there to form the team?"
}
] | math | {
"style": "rule",
"ground_truth": "252"
} | {
"index": 12291,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From 6 students, 4 are to be selected to undertake four different tasks labeled A, B, C, and D. If two of the students, named A and B, cannot be assigned to task A, calculate the total number of different assignment plans."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 12292,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From 8 female students and 4 male students, 3 students are to be selected to participate in a TV program. Determine the number of different selection methods when the selection is stratified by gender."
}
] | math | {
"style": "rule",
"ground_truth": "112"
} | {
"index": 12293,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From June to August 1861, a total of 1026 inches of rain fell in Cherrapunji, India, heavily influenced by the monsoon season, and the total duration of these summer months is 92 days and 24 hours per day. Calculate the average rainfall in inches per hour."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1026}{2208}"
} | {
"index": 12294,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From May 1st to May 3rd, the provincial hospital plans to schedule 6 doctors to be on duty, with each person working 1 day and 2 people scheduled per day. Given that doctor A cannot work on the 2nd and doctor B cannot work on the 3rd, how many different scheduling arrangements ... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 12295,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From Moscow to city \\( N \\), a passenger can travel by train, taking 20 hours. If the passenger waits for a flight (waiting will take more than 5 hours after the train departs), they will reach city \\( N \\) in 10 hours, including the waiting time. By how many times is the p... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 12296,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From a 12 × 12 grid, a 4 × 4 square has been cut out, located at the intersection of horizontals from the fourth to the seventh and the same verticals. What is the maximum number of rooks that can be placed on this board such that no two rooks attack each other, given that rook... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 12297,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From a \\(6 \\times 6\\) square grid, gray triangles were cut out. What is the area of the remaining shape? The length of each side of the cells is 1 cm. Provide your answer in square centimeters."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 12298,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "From a barrel, 4 liters of wine are drawn, and this is replaced with 4 liters of water. From the resulting mixture, 4 liters are drawn again and replaced with 4 liters of water. This operation is repeated a total of three times, and the final result is that there are 2.5 liters... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 12299,
"solution": ""
} |
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