data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length $1$. The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. What is the length of one of the two congr... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"index": 35600,
"solution": "1. **Calculate the area of the equilateral triangle**:\n The formula for the area of an equilateral triangle with side length $s$ is $\\frac{\\sqrt{3}}{4}s^2$. For an equilateral triangle with side length $1$, the area is:\n \\[\n \\frac{\\sqrt{3}}{4} \\times 1^2 = \\frac{\\sqrt... |
DeepScaleR | [
{
"role": "user",
"content": "Three coplanar circles intersect as shown. What is the maximum number of points on the circles that a line passing through all three circles can touch?\n\n[asy]import graph;\ndraw(Circle((-9,9),15));\ndraw(Circle((0,-9),15));\ndraw(Circle((9,9),15));\n[/asy]"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 35601,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three coplanar squares with sides of lengths two, four and six units, respectively, are arranged side-by-side, as shown so that one side of each square lies on line $AB$ and a segment connects the bottom left corner of the smallest square to the upper right corner of the larges... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 35602,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three couples dine at the same restaurant every Saturday at the same table. The table is round and the couples agreed that:\n(a) under no circumstances should husband and wife sit next to each other; and\n(b) the seating arrangement of the six people at the table must be differ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 35603,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three couples sit for a photograph in $2$ rows of three people each such that no couple is sitting in the same row next to each other or in the same column one behind the other. How many such arrangements are possible?"
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"index": 35604,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three cubes are each formed from the pattern shown. They are then stacked on a table one on top of another so that the $13$ visible numbers have the greatest possible sum. What is that sum?"
}
] | math | {
"style": "rule",
"ground_truth": "164"
} | {
"index": 35605,
"solution": "To solve this problem, we need to maximize the sum of the visible numbers on three stacked cubes, each formed from a pattern with numbers 1, 2, 4, 8, 16, and 32. Each cube has six faces, but when stacked, some faces will not be visible.\n\n1. **Maximize the sum of visible numbers on e... |
DeepScaleR | [
{
"role": "user",
"content": "Three cubes, each with faces numbered from 1 to 6, are tossed. What is the probability that the sum of the numbers shown on the top faces of all three cubes is divisible by 3? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 35606,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three dice are thrown, and the sums of the points that appear on them are counted. In how many ways can you get a total of 5 points and 6 points?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 35607,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three dice with faces numbered 1 through 6 are placed vertically in a stack as described. The visible numbers on these dice are 2, 2, 3, 3, 4, 5, 6. Calculate the total number of dots that are NOT visible on these dice."
}
] | math | {
"style": "rule",
"ground_truth": "38"
} | {
"index": 35608,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three dice with faces numbered 1 through 6 are stacked as shown. Seven of the eighteen faces are visible, leaving eleven faces hidden (back, bottom, between). The total number of dots NOT visible in this view is"
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 35609,
"solution": "1. **Calculate the total number of dots on three dice**: Each die has faces numbered from 1 to 6. The sum of the numbers on one die is calculated as follows:\n \\[\n 1 + 2 + 3 + 4 + 5 + 6 = 21\n \\]\n Since there are three dice, the total number of dots on all three dice is:\n... |
DeepScaleR | [
{
"role": "user",
"content": "Three dice with faces numbered 1 through 6 are stacked as shown. Seven of the eighteen faces are visible, leaving eleven faces hidden(back, bottom, between). What is the total number of dots NOT visible in this view? [asy]\n/* AMC8 2000 #8 Problem */\ndraw((0,0)--(1,0)--(1.5,0.... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"index": 35610,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three different integers are randomly chosen from the set $$\\{ -6, -3, 0, 2, 5, 7 \\}$$. What is the probability that their sum is even? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{19}{20}"
} | {
"index": 35611,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three different natural numbers $x$, $y$ and $z$ each have exactly three natural-number factors. How many factors does $x^{2}y^{3}z^{4}$ have?"
}
] | math | {
"style": "rule",
"ground_truth": "315"
} | {
"index": 35612,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three different numbers are chosen at random from the list \\(1, 3, 5, 7, 9, 11, 13, 15, 17, 19\\). The probability that one of them is the mean of the other two is \\(p\\). What is the value of \\(\\frac{120}{p}\\) ?"
}
] | math | {
"style": "rule",
"ground_truth": "720"
} | {
"index": 35613,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three different numbers are chosen from the set $\\{-4, -3, -1, 3, 5, 8\\}$ and multiplied. Find the largest possible positive product."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 35614,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three different numbers from the list $2, 3, 4, 6$ have a sum of 11. What is the product of these numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 35615,
"solution": "The sum of 2, 3 and 6 is $2 + 3 + 6 = 11$. Their product is $2 \\cdot 3 \\cdot 6 = 36$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Three distinct diameters are drawn on a unit circle such that chords are drawn as shown. If the length of one chord is \\(\\sqrt{2}\\) units and the other two chords are of equal lengths, what is the common length of these chords?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2-\\sqrt{2}}"
} | {
"index": 35616,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three distinct integers $a, b,$ and $c$ satisfy the following three conditions: $abc=17955$, $a, b,$ and $c$ form an arithmetic sequence in that order, and $(3a+b), (3b+c),$ and $(3c+a)$ form a geometric sequence in that order. What is the value of $a+b+c$?"
}
] | math | {
"style": "rule",
"ground_truth": "-63"
} | {
"index": 35617,
"solution": "Since $a, b$ and $c$ form an arithmetic sequence in this order, then $a=b-d$ and $c=b+d$ for some real number $d$. We note that $d \\neq 0$, since otherwise we would have $a=b=c$ and then $abc=17955$ would tell us that $b^{3}=17955$ or $b=\\sqrt[3]{17955}$, which is not an integer. Wr... |
DeepScaleR | [
{
"role": "user",
"content": "Three distinct integers are chosen uniformly at random from the set $$ \\{2021, 2022, 2023, 2024, 2025, 2026, 2027, 2028, 2029, 2030\\}. $$ Compute the probability that their arithmetic mean is an integer."
}
] | math | {
"style": "rule",
"ground_truth": "7/20"
} | {
"index": 35618,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three distinct integers, $x$, $y$, and $z$, are randomly chosen from the set $\\{1, 2, 3, \\dots, 12\\}$. What is the probability that $xyz - x - y - z$ is even?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{11}"
} | {
"index": 35619,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three distinct integers, $x$, $y$, and $z$, are randomly chosen from the set $\\{1,2,3,4,5,6,7,8,9,10,11,12\\}$. What is the probability that $xyz-xy-xz-yz$ is even?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10}{11}"
} | {
"index": 35620,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three distinct numbers are selected simultaneously and at random from the set $\\{1, 2, 3, 4, 5, 6, 7, 8, 9\\}$. What is the probability that the smallest positive difference between any two of those numbers is $3$ or greater? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{14}"
} | {
"index": 35621,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three distinct numbers are selected simultaneously and at random from the set $\\{1, 2, 3, 4, 6\\}$. What is the probability that at least one number divides another among the selected numbers? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{10}"
} | {
"index": 35622,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three distinct real numbers form (in some order) a 3-term arithmetic sequence, and also form (in possibly a different order) a 3-term geometric sequence. Compute the greatest possible value of the common ratio of this geometric sequence."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 35623,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{223}{286}"
} | {
"index": 35624,
"solution": "1. **Setup and Definitions**:\n - Consider a regular 12-gon with vertices labeled from 1 to 12.\n - The segments between vertices are the sides and diagonals of the 12-gon.\n - The total number of segments is $\\binom{12}{2} = 66$ since each segment is defined by two distinct ve... |
DeepScaleR | [
{
"role": "user",
"content": "Three distinct vertices are chosen at random from the vertices of a given regular polygon of $(2n+1)$ sides. If all such choices are equally likely, what is the probability that the center of the given polygon lies in the interior of the triangle determined by the three chosen ... | math | {
"style": "rule",
"ground_truth": "\\frac{n}{2n+1}"
} | {
"index": 35625,
"solution": "1. **Counting Total Ways to Choose Vertices**: \n The total number of ways to choose three vertices from a regular polygon with $(2n+1)$ sides is given by the combination formula:\n \\[\n \\binom{2n+1}{3} = \\frac{(2n+1)(2n)(2n-1)}{6}\n \\]\n\n2. **Counting Ways that Do Not C... |
DeepScaleR | [
{
"role": "user",
"content": "Three distinct vertices are randomly selected among the five vertices of a regular pentagon. Let $p$ be the probability that the triangle formed by the chosen vertices is acute. Compute $10 p$."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 35626,
"solution": "The only way for the three vertices to form an acute triangle is if they consist of two adjacent vertices and the vertex opposite their side. Since there are 5 ways to choose this and $\\binom{5}{3}=10$ ways to choose the three vertices, we have $p=\\frac{5}{10}=\\frac{1}{2}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Three distinct vertices of a cube are chosen at random. What is the probability that the plane determined by these three vertices contains points inside the cube?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{7}"
} | {
"index": 35627,
"solution": "\nTo solve this problem, we need to determine the probability that three randomly chosen vertices of a cube will form a plane that intersects the interior of the cube. We will use a combinatorial approach to count the favorable and total outcomes.\n\n#### Step 1: Total number of ways ... |
DeepScaleR | [
{
"role": "user",
"content": "Three distinct vertices of a regular 2020-gon are chosen uniformly at random. The probability that the triangle they form is isosceles can be expressed as $\\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a+b$."
}
] | math | {
"style": "rule",
"ground_truth": "773"
} | {
"index": 35628,
"solution": "The number of isosceles triangles that share vertices with the 2020-gon is $2020 \\cdot 1009$, since there are 2020 ways to choose the apex of the triangle and then 1009 ways to choose the other two vertices. (Since 2020 is not divisible by 3, there are no equilateral triangles, so no... |
DeepScaleR | [
{
"role": "user",
"content": "Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34$. What is the distance between two adjacent parallel lines?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 35629,
"solution": "1. **Identify the setup and apply Stewart's Theorem**: We are given three equally spaced parallel lines intersecting a circle, creating three chords of lengths 38, 38, and 34. Let's denote the center of the circle as $O$, and the points where the chords intersect the circle as $C, D, ... |
DeepScaleR | [
{
"role": "user",
"content": "Three equally spaced parallel lines intersect a circle, creating three chords of lengths 38, 38, and 34. What is the distance between two adjacent parallel lines?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 35630,
"solution": "1. **Identify the setup and apply Stewart's Theorem**: We are given three equally spaced parallel lines intersecting a circle, creating three chords of lengths 38, 38, and 34. Let's denote the center of the circle as $O$, and the points where the chords intersect the circle as $C, D, ... |
DeepScaleR | [
{
"role": "user",
"content": "Three equilateral triangles $ABC$, $BCD$, and $CDE$ are positioned such that $B$, $C$, and $D$ are collinear, and $C$ is the midpoint of $BD$. Triangle $CDE$ is positioned such that $E$ is on the same side of line $BD$ as $A$. What is the value of $AE \\div BC$ when expressed i... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 35631,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three faces $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$ of a unit cube share a common vertex. Suppose the projections of $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$ onto a fixed plane $\\mathcal{P}$ have areas $x, y, z$, respectively. If $x: y: z=6: 10: 15$, then $x+y+z$ can be w... | math | {
"style": "rule",
"ground_truth": "3119"
} | {
"index": 35632,
"solution": "Introduce coordinates so that $\\mathcal{X}, \\mathcal{Y}, \\mathcal{Z}$ are normal to $(1,0,0),(0,1,0)$, and $(0,0,1)$, respectively. Also, suppose that $\\mathcal{P}$ is normal to unit vector $(\\alpha, \\beta, \\gamma)$ with $\\alpha, \\beta, \\gamma \\geq 0$. Since the area of $\\... |
DeepScaleR | [
{
"role": "user",
"content": "Three faces of a rectangular box meet at a corner, and the centers of these faces form the vertices of a triangle with side lengths of 4 cm, 5 cm, and 6 cm. What is the volume of the box, in cm^3?"
}
] | math | {
"style": "rule",
"ground_truth": "90 \\sqrt{6}"
} | {
"index": 35633,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three faces of a right rectangular prism have areas of 48, 49 and 50 square units. What is the volume of the prism, in cubic units? Express your answer to the nearest whole number."
}
] | math | {
"style": "rule",
"ground_truth": "343"
} | {
"index": 35634,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three faces of a right rectangular prism have areas of 54, 56, and 60 square units. Calculate the volume of the prism in cubic units and round it to the nearest whole number."
}
] | math | {
"style": "rule",
"ground_truth": "426"
} | {
"index": 35635,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three fair coins are to be tossed once. For each head that results, one fair die is to be rolled. Calculate the probability that the sum of the die rolls is odd."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{16}"
} | {
"index": 35636,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three fair coins are to be tossed once. For each head that results, one fair die is to be rolled. If all three coins show heads, roll an additional fourth die. Determine the probability that the sum of the die rolls is odd."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{16}"
} | {
"index": 35637,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three fair coins are tossed once. For each head that results, one fair die is rolled. Determine the probability that the sum of the results of the die rolls is odd."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{16}"
} | {
"index": 35638,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three fair dice are tossed at random (i.e., all faces have the same probability of coming up). What is the probability that the three numbers turned up can be arranged to form an arithmetic progression with common difference one?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"index": 35639,
"solution": "1. **Total Outcomes**: When three fair dice are tossed, each die has 6 faces, and each face is equally likely to come up. Therefore, the total number of outcomes when three dice are tossed is $6 \\times 6 \\times 6 = 6^3 = 216$.\n\n2. **Favorable Outcomes**: We need to find the number... |
DeepScaleR | [
{
"role": "user",
"content": "Three fair six-sided dice are rolled (meaning all outcomes are equally likely). What is the probability that the numbers on the top faces form an arithmetic sequence with a common difference of 1?"
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{1}{9}$"
} | {
"index": 35640,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three fair six-sided dice are rolled. What is the probability that the values shown on two of the dice sum to the value shown on the remaining die?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{24}"
} | {
"index": 35641,
"solution": "1. **Total Outcomes**: When three six-sided dice are rolled, each die has 6 possible outcomes. Therefore, the total number of outcomes when rolling three dice is:\n \\[\n 6 \\times 6 \\times 6 = 216\n \\]\n\n2. **Favorable Outcomes**: We need to count the number of ways two dice... |
DeepScaleR | [
{
"role": "user",
"content": "Three fair six-sided dice, each numbered 1 through 6 , are rolled. What is the probability that the three numbers that come up can form the sides of a triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "37/72"
} | {
"index": 35642,
"solution": "Denote this probability by $p$, and let the three numbers that come up be $x, y$, and $z$. We will calculate $1-p$ instead: $1-p$ is the probability that $x \\geq y+z, y \\geq z+x$, or $z \\geq x+y$. Since these three events are mutually exclusive, $1-p$ is just 3 times the probabilit... |
DeepScaleR | [
{
"role": "user",
"content": "Three fair, six-sided dice are rolled. What is the probability that the sum of the three numbers showing is less than 16?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{103}{108}"
} | {
"index": 35643,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three fair, standard six-sided dice are rolled. What is the probability that the sum of the numbers on the top faces is 18? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{216}"
} | {
"index": 35644,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three faucets fill a 100-gallon tub in 6 minutes. How long, in seconds, does it take six faucets to fill a 25-gallon tub? Assume that all faucets dispense water at the same rate."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 35645,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three flower beds overlap as shown. Bed A has 500 plants, bed B has 450 plants, and bed C has 350 plants. Beds A and B share 50 plants, while beds A and C share 100. The total number of plants is"
}
] | math | {
"style": "rule",
"ground_truth": "1150"
} | {
"index": 35646,
"solution": "\n#### Method 1: Using the Principle of Inclusion-Exclusion\nWe are given:\n- Total plants in Bed A, $|A| = 500$\n- Total plants in Bed B, $|B| = 450$\n- Total plants in Bed C, $|C| = 350$\n- Plants shared between Bed A and Bed B, $|A \\cap B| = 50$\n- Plants shared between Bed A and ... |
DeepScaleR | [
{
"role": "user",
"content": "Three flower beds, X, Y, and Z, contain X = 600, Y = 480, and Z = 420 plants, respectively. The plants are shared as follows: Beds X and Y share 60 plants, Beds Y and Z share 70 plants, and Beds X and Z share 80 plants. Also, there are 30 plants common to all three beds. Find t... | math | {
"style": "rule",
"ground_truth": "1320"
} | {
"index": 35647,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three fourths of a pitcher is filled with pineapple juice. The pitcher is emptied by pouring an equal amount of juice into each of $5$ cups. What percent of the total capacity of the pitcher did each cup receive?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 35648,
"solution": "1. **Assume the total capacity of the pitcher**: Let's denote the total capacity of the pitcher as $C$ ounces.\n\n2. **Calculate the amount of juice in the pitcher**: Since the pitcher is filled to three-fourths of its capacity with pineapple juice, the amount of juice in the pitcher ... |
DeepScaleR | [
{
"role": "user",
"content": "Three friends are driving cars on a road in the same direction. At a certain moment, they are positioned relative to each other as follows: Andrews is at a certain distance behind Brooks, and Carter is at a distance twice the distance from Andrews to Brooks, ahead of Brooks. Ea... | math | {
"style": "rule",
"ground_truth": "6.666666666666667"
} | {
"index": 35649,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three friends are driving to New York City and splitting the gas cost equally. At the last minute, 2 more friends joined the trip. The cost of gas was then evenly redistributed among all of the friends. The cost for each of the original 3 decreased by $\\$$11.00. What was the t... | math | {
"style": "rule",
"ground_truth": "82.50"
} | {
"index": 35650,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three friends are in the park. Bob and Clarise are standing at the same spot and Abe is standing 10 m away. Bob chooses a random direction and walks in this direction until he is 10 m from Clarise. What is the probability that Bob is closer to Abe than Clarise is to Abe?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 35651,
"solution": "We call Clarise's spot $C$ and Abe's spot $A$. Consider a circle centred at $C$ with radius 10 m. Since $A$ is 10 m from $C$, then $A$ is on this circle. Bob starts at $C$ and picks a direction to walk, with every direction being equally likely to be chosen. We model this by having Bo... |
DeepScaleR | [
{
"role": "user",
"content": "Three friends have a total of 6 identical pencils, and each one has at least one pencil. In how many ways can this happen?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 35652,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three friends, Rowan, Sara, and Tim, are playing a monetary game. Each starts with $3. A bell rings every 20 seconds, and with each ring, any player with money chooses one of the other two players independently at random and gives them $1. The game continues for 2020 rounds. Wh... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 35653,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three frogs in a swamp jumped one after another. Each one landed exactly in the middle of the segment between the other two. The jump length of the second frog is 60 cm. Find the jump length of the third frog."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 35654,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three generations of the Wen family are going to the movies, two from each generation. The two members of the youngest generation receive a $50$% discount as children. The two members of the oldest generation receive a $25\\%$ discount as senior citizens. The two members of the... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 35655,
"solution": "1. **Calculate the cost of a regular ticket**: \n Given that a senior ticket (with a 25% discount) costs $6.00, we need to find the original price of the ticket before the discount. The discount implies that seniors pay 75% (or $\\frac{3}{4}$) of the original price. Let $x$ be the ... |
DeepScaleR | [
{
"role": "user",
"content": "Three generous friends, each with some money, redistribute the money as followed:\nAmy gives enough money to Jan and Toy to double each amount has.\nJan then gives enough to Amy and Toy to double their amounts. \nFinally, Toy gives enough to Amy and Jan to double their amounts.... | math | {
"style": "rule",
"ground_truth": "252"
} | {
"index": 35656,
"solution": "1. **Initial Setup**: Let's denote the initial amounts of money that Amy, Jan, and Toy have as $a$, $j$, and $t$ respectively. According to the problem, Toy starts with $t = 36$ dollars.\n\n2. **After Amy's Redistribution**: Amy gives enough money to Jan and Toy to double their amount... |
DeepScaleR | [
{
"role": "user",
"content": "Three hexagons of increasing size are shown below. Suppose the dot pattern continues so that each successive hexagon contains one more band of dots. How many dots are in the next hexagon?\n[asy] // diagram by SirCalcsALot, edited by MRENTHUSIASM size(250); path p = scale(0.8)*u... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"index": 35657,
"solution": "To solve this problem, we need to understand the pattern of how the dots are arranged in each hexagon and how the number of dots increases as the hexagons grow in size.\n\n1. **Observing the pattern:**\n - The first hexagon has 1 dot.\n - The second hexagon has 7 dots: 1 central d... |
DeepScaleR | [
{
"role": "user",
"content": "Three identical rectangles are put together to form rectangle $ABCD$, as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is 5 feet, what is the area in square feet of rectangle $ABCD$?"
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 35658,
"solution": "\n1. **Identify the dimensions of the smaller rectangles**: \n Each smaller rectangle has a shorter side of 5 feet. Since the problem suggests that the rectangles are placed such that two smaller rectangles' shorter sides are aligned vertically, the longer side of each smaller recta... |
DeepScaleR | [
{
"role": "user",
"content": "Three identical rods each have a piece broken off at a random point. What is the probability that the three resulting pieces can form a triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"index": 35659,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three identical square sheets of paper each with side length $6$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\\circ$ about its center and the top sheet is rotated clockwise $60^\\circ$ about its center, resulting in the $24$-sided polygon shown i... | math | {
"style": "rule",
"ground_truth": "147"
} | {
"index": 35660,
"solution": "1. **Understanding the Problem:**\n We have three identical square sheets of paper, each with side length $6$. The middle sheet is rotated $30^\\circ$ clockwise, and the top sheet is rotated $60^\\circ$ clockwise. We need to find the area of the resulting $24$-sided polygon.\n\n2. *... |
DeepScaleR | [
{
"role": "user",
"content": "Three identical square sheets of paper each with side length \\(8\\) are stacked on top of each other. The middle sheet is rotated clockwise \\(20^\\circ\\) about its center and the top sheet is rotated clockwise \\(50^\\circ\\) about its center. Determine the area of the resul... | math | {
"style": "rule",
"ground_truth": "192"
} | {
"index": 35661,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three integers from the list $1,2,4,8,16,20$ have a product of 80. What is the sum of these three integers?"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 35662,
"solution": "The three integers from the list whose product is 80 are 1, 4, and 20, since $1 \\times 4 \\times 20=80$. The sum of these integers is $1+4+20=25$. \n (Since 80 is a multiple of 5 and 20 is the only integer in the list that is a multiple of 5, then 20 must be included in the product. ... |
DeepScaleR | [
{
"role": "user",
"content": "Three lathes \\( A, B, C \\) each process the same type of standard parts at a certain work efficiency. Lathe \\( A \\) starts 10 minutes earlier than lathe \\( C \\), and lathe \\( C \\) starts 5 minutes earlier than lathe \\( B \\). After lathe \\( B \\) has been working for ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 35663,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three lines are drawn parallel to each of the three sides of $\\triangle ABC$ so that the three lines intersect in the interior of $ABC$ . The resulting three smaller triangles have areas $1$ , $4$ , and $9$ . Find the area of $\\triangle ABC$ .\n\n[asy]\ndefaultpen(line... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 35664,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three lines are drawn parallel to the sides of a triangle through a point inside it, dividing the triangle into six parts: three triangles and three quadrilaterals. The areas of all three inner triangles are equal. Determine the range within which the ratio of the area of each ... | math | {
"style": "rule",
"ground_truth": "1/9"
} | {
"index": 35665,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three lines were drawn through the point $X$ in space. These lines crossed some sphere at six points. It turned out that the distances from point $X$ to some five of them are equal to $2$ cm, $3$ cm, $4$ cm, $5$ cm, $6$ cm. What can be the distance from point $X$... | math | {
"style": "rule",
"ground_truth": "2.4"
} | {
"index": 35666,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three machines $\\text{P, Q, and R,}$ working together, can do a job in $x$ hours. When working alone, $\\text{P}$ needs an additional $6$ hours to do the job; $\\text{Q}$, one additional hour; and $\\text{R}$, $x$ additional hours. The value of $x$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 35667,
"solution": "1. **Identify individual working times**: \n - Machine P takes $x+6$ hours to complete the job alone.\n - Machine Q takes $x+1$ hours to complete the job alone.\n - Machine R takes $2x$ hours to complete the job alone.\n\n2. **Set up the equation for combined work**:\n Since P... |
DeepScaleR | [
{
"role": "user",
"content": "Three male students and three female students, a total of six students, stand in a row. If female students do not stand at the end of the row, and female students A and B are not adjacent to female student C, then find the number of different arrangements."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 35668,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three male students and three female students, a total of six students, stand in a row. If male student A does not stand at either end, and exactly two of the three female students stand next to each other, then the number of different arrangements is ______."
}
] | math | {
"style": "rule",
"ground_truth": "288"
} | {
"index": 35669,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three male students and two female students stand in a row. The total number of arrangements where the female students do not stand at the ends is given by what total count."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 35670,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three marbles are randomly selected, without replacement, from a bag containing two red, two blue and two green marbles. What is the probability that one marble of each color is selected? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"index": 35671,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three members of the Euclid Middle School girls' softball team had the following conversation.\nAshley: I just realized that our uniform numbers are all $2$-digit primes.\nBethany : And the sum of your two uniform numbers is the date of my birthday earlier this month.\nCaitlin:... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 35672,
"solution": "Let's denote the uniform numbers of Ashley, Bethany, and Caitlin as $a$, $b$, and $c$ respectively. According to the problem, all these numbers are 2-digit primes. The smallest 2-digit primes are $11, 13, 17, 19, 23, \\ldots$.\n\nFrom the conversation:\n- Bethany's birthday earlier th... |
DeepScaleR | [
{
"role": "user",
"content": "Three men, Alpha, Beta, and Gamma, working together, do a job in 6 hours less time than Alpha alone, in 1 hour less time than Beta alone, and in one-half the time needed by Gamma when working alone. Let $h$ be the number of hours needed by Alpha and Beta, working together, to d... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 35673,
"solution": "1. **Define Variables:**\n Let $A$, $B$, and $C$ be the number of hours needed by Alpha, Beta, and Gamma, respectively, to complete the job alone. Their respective rates of work are $\\frac{1}{A}$, $\\frac{1}{B}$, and $\\frac{1}{C}$ jobs per hour.\n\n2. **Set Up Equations:**\n - W... |
DeepScaleR | [
{
"role": "user",
"content": "Three merchants - Sosipatra Titovna, Olympiada Karpovna, and Poliksena Uvarovna - sat down to drink tea. Olympiada Karpovna and Sosipatra Titovna together drank 11 cups, Poliksena Uvarovna and Olympiada Karpovna drank 15 cups, and Sosipatra Titovna and Poliksena Uvarovna drank ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 35674,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three musicians, Janek, Mikeš, and Vávra usually divide their shared fee in the ratio $4: 5: 6$, with Janek receiving the least and Vávra the most. This time, Vávra did not perform well, so he gave up his portion. Janek suggested that Vávra's share should be divided equally bet... | math | {
"style": "rule",
"ground_truth": "1800"
} | {
"index": 35675,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three natural numbers are written on the board: two ten-digit numbers \\( a \\) and \\( b \\), and their sum \\( a + b \\). What is the maximum number of odd digits that could be written on the board?"
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 35676,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three non-collinear lattice points $A,B,C$ lie on the plane $1+3x+5y+7z=0$ . The minimal possible area of triangle $ABC$ can be expressed as $\\frac{\\sqrt{m}}{n}$ where $m,n$ are positive integers such that there does not exists a prime $p$ dividing $n$ with $p^2... | math | {
"style": "rule",
"ground_truth": "8302"
} | {
"index": 35677,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $1$. The polygons meet at a point $A$ in such a way that the sum of the three interior angles at $A$ is $360^{\\circ}$. Thus the three polygons form a new polygon with $A... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 35678,
"solution": "1. **Identify the Problem Requirements:**\n We need to find three non-overlapping regular polygons, at least two of which are congruent, that meet at a point $A$ such that the sum of their interior angles at $A$ is $360^\\circ$. The sides of each polygon are of length $1$. We aim to... |
DeepScaleR | [
{
"role": "user",
"content": "Three noncollinear points and a line $\\ell$ are given in the plane. Suppose no two of the points lie on a line parallel to $\\ell$ (or $\\ell$ itself). There are exactly $n$ lines perpendicular to $\\ell$ with the following property: the three circles with centers at the given... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 35679,
"solution": "The condition for the line is that each of the three points lies at an equal distance from the line as from some fixed point; in other words, the line is the directrix of a parabola containing the three points. Three noncollinear points in the coordinate plane determine a quadratic po... |
DeepScaleR | [
{
"role": "user",
"content": "Three not necessarily distinct positive integers between 1 and 99, inclusive, are written in a row on a blackboard. Then, the numbers, without including any leading zeros, are concatenated to form a new integer $N$. For example, if the integers written, in order, are 25, 6, and... | math | {
"style": "rule",
"ground_truth": "825957"
} | {
"index": 35680,
"solution": "We will divide this into cases based on the number of digits of $N$.\n- Case 1: 6 digits. Then each of the three numbers must have two digits, so we have 90 choices for each. So we have a total of $90^{3}=729000$ possibilities.\n- Case 2: 5 digits. Then, exactly one of the three numbe... |
DeepScaleR | [
{
"role": "user",
"content": "Three not necessarily integer numbers are given. If each of these numbers is increased by 1, the product of the numbers also increases by 1. If each of these numbers is increased by 2, their product also increases by 2. Find these numbers."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 35681,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three numbers $a, b,$ and $c$ were written on a board. They were erased and replaced with the numbers $a-1, b+1,$ and $c^2$. After this, it turned out that the same numbers were on the board as initially (possibly in a different order). What values can the number $a$ take, give... | math | {
"style": "rule",
"ground_truth": "1004"
} | {
"index": 35682,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three numbers are stored in a computer's memory. Every second, the following operation is performed: each number in this triplet is replaced by the sum of the other two numbers. For example, the triplet \\((1; 3; 7)\\) becomes \\((10; 8; 4)\\). What will be the difference betwe... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 35683,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three numbers, $a_1, a_2, a_3$, are drawn randomly and without replacement from the set $\\{1, 2, 3,\\ldots, 1000\\}$. Three other numbers, $b_1, b_2, b_3$, are then drawn randomly and without replacement from the remaining set of $997$ numbers. Let $p$ be the probability that,... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 35684,
"solution": "There is a total of $P(1000,6)$ possible ordered $6$-tuples $(a_1,a_2,a_3,b_1,b_2,b_3).$\n There are $C(1000,6)$ possible sets $\\{a_1,a_2,a_3,b_1,b_2,b_3\\}.$ We have five valid cases for the increasing order of these six elements:\n$aaabbb$\n$aababb$\n$aabbab$\n$abaabb$\n$ababab$\n ... |
DeepScaleR | [
{
"role": "user",
"content": "Three numbers, $a_1\\,$, $a_2\\,$, $a_3\\,$, are drawn randomly and without replacement from the set $\\{1, 2, 3, \\dots, 1000\\}\\,$. Three other numbers, $b_1\\,$, $b_2\\,$, $b_3\\,$, are then drawn randomly and without replacement from the remaining set of 997 numbers. Let $... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 35685,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three of the edges of a cube are $\\overline{AB}, \\overline{BC},$ and $\\overline{CD},$ and $\\overline{AD}$ is an interior diagonal. Points $P, Q,$ and $R$ are on $\\overline{AB}, \\overline{BC},$ and $\\overline{CD},$ respectively, so that $AP = 5, PB = 15, BQ = 15,$ and $CR... | math | {
"style": "rule",
"ground_truth": "525"
} | {
"index": 35686,
"solution": "\"For non-asymptote version of image, see Image:1998_AIME-11.png\"\n[asy] import three; size(280); defaultpen(linewidth(0.6)+fontsize(9)); currentprojection=perspective(30,-60,40); triple A=(0,0,0),B=(20,0,0),C=(20,0,20),D=(20,20,20); triple P=(5,0,0),Q=(20,0,15),R=(20,10,20),Pa=(15,2... |
DeepScaleR | [
{
"role": "user",
"content": "Three of the following test scores are Cyprian's and the other three are Margaret's: 85, 87, 92, 93, 94, 98. Cyprian's mean score is 90. What is Margaret's mean score?"
}
] | math | {
"style": "rule",
"ground_truth": "93"
} | {
"index": 35687,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three of the four endpoints of the axes of an ellipse are, in some order, \\[(-2, 4), \\; (3, -2), \\; (8, 4).\\]Find the distance between the foci of the ellipse."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{11}"
} | {
"index": 35688,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three of the four endpoints of the axes of an ellipse are, in some order, \\[(10, -3), \\; (15, 7), \\; (25, -3).\\] Find the distance between the foci of the ellipse."
}
] | math | {
"style": "rule",
"ground_truth": "11.18"
} | {
"index": 35689,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three of the four vertices of a rectangle are $(3, 7)$, $(12, 7)$, and $(12, -4)$. What is the area of the intersection of this rectangular region and the region inside the graph of the equation $(x - 3)^2 + (y + 4)^2 = 16$?"
}
] | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"index": 35690,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three of the four vertices of a rectangle are $(5, 11)$, $(16, 11)$ and $(16, -2)$. What is the area of the intersection of this rectangular region and the region inside the graph of the equation $(x - 5)^2 + (y + 2)^2 = 9$? Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac94\\pi"
} | {
"index": 35691,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three of the four vertices of a rectangle are \\((1, 7)\\), \\((14, 7)\\), and \\((14, -4)\\). What is the area of the intersection of this rectangular region and the region inside the graph of the equation \\((x - 1)^2 + (y + 4)^2 = 16\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"index": 35692,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three of the four vertices of a square are $(2, 8)$, $(13, 8)$, and $(13, -3)$. What is the area of the intersection of this square region and the region inside the graph of the equation $(x - 2)^2 + (y + 3)^2 = 16$?"
}
] | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"index": 35693,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three of the vertices of parallelogram $ABCD$ are $A = (3,-1,2),$ $B = (1,2,-4),$ and $C = (-1,1,2).$ Find the coordinates of $D.$"
}
] | math | {
"style": "rule",
"ground_truth": "(1,-2,8)"
} | {
"index": 35694,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated 45 degrees, as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point $B$ from the l... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}+\\frac{1}{2}"
} | {
"index": 35695,
"solution": "1. **Understanding the Problem Setup**: We have three 1-inch squares aligned horizontally. The middle square is rotated by 45 degrees and then lowered until it touches the other two squares. We need to find the vertical distance from point $B$ (the top vertex of the rotated square) to... |
DeepScaleR | [
{
"role": "user",
"content": "Three people are sitting in a row of eight seats. If there must be empty seats on both sides of each person, then the number of different seating arrangements is."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 35696,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three people jointly start a business with a total investment of 143 million yuan. The ratio of the highest investment to the lowest investment is 5:3. What is the maximum and minimum amount the third person could invest in millions of yuan?"
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"index": 35697,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three people, A, B, and C, are playing a game: each person chooses a real number from the interval $[0,1]$. The person whose chosen number is between the numbers chosen by the other two wins. A randomly chooses a number from the interval $[0,1]$, B randomly chooses a number fro... | math | {
"style": "rule",
"ground_truth": "$\\frac{13}{24}$"
} | {
"index": 35698,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Three people, A, B, and C, are taking an elevator from the 1st floor to the 3rd to 7th floors of a mall. Each floor can accommodate at most 2 people getting off the elevator. How many ways are there for them to get off the elevator?"
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 35699,
"solution": ""
} |
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