data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Each of the natural numbers $1, 2, 3, \\ldots, 377$ is painted either red or blue (both colors are present). It is known that the number of red numbers is equal to the smallest red number, and the number of blue numbers is equal to the largest blue number. What is the smallest ... | math | {
"style": "rule",
"ground_truth": "189"
} | {
"index": 8200,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the nine dots in this figure is to be colored red, white or blue. No two dots connected by a segment (with no other dots between) may be the same color. How many ways are there to color the dots of this figure?\n\n[asy]\ndraw((-75,0)--(-45,0)--(-60,26)--cycle);\ndraw((0... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"index": 8201,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the nine letters in \"STATISTICS\" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word \"TEST\"? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 8202,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the numbers $a_1,$ $a_2,$ $\\dots,$ $a_{95}$ is $\\pm 1.$ Find the smallest possible positive value of\n\\[\\sum_{1 \\le i < j \\le 95} a_i a_j.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 8203,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the numbers 1, 2, 3, and 4 is substituted, in some order, for \\( p, q, r \\), and \\( s \\). Find the greatest possible value of \\( p^q + r^s \\)."
}
] | math | {
"style": "rule",
"ground_truth": "83"
} | {
"index": 8204,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the numbers \\( m \\) and \\( n \\) is the square of an integer. The difference \\( m - n \\) is a prime number. Which of the following could be \\( n \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "900"
} | {
"index": 8205,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the numbers \\(1, 2, 3, 4, 5, 6\\) is to be placed in the cells of a \\(2 \\times 3\\) table, with one number in each cell. In how many ways can this be done so that in each row and in each column the sum of the numbers is divisible by 3?"
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 8206,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the points $A,B,C,D,E,$ and $F$ in the figure below represents a different digit from $1$ to $6.$ Each of the five lines shown passes through some of these points. The digits along each line are added to produce five sums, one for each line. The total of the five sums i... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 8207,
"solution": "1. **Identify the sums along each line**: Given the points $A, B, C, D, E, F$, each representing a unique digit from 1 to 6, and the lines connecting them, we can write the sums for each line as:\n - Line through $A, B, C$: $A + B + C$\n - Line through $A, E, F$: $A + E + F$\n - ... |
DeepScaleR | [
{
"role": "user",
"content": "Each of the sides of a square $S_1$ with area $16$ is bisected, and a smaller square $S_2$ is constructed using the bisection points as vertices. The same process is carried out on $S_2$ to construct an even smaller square $S_3$. What is the area of $S_3$?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 8208,
"solution": "1. **Understanding the problem**: We are given a square $S_1$ with area 16. Each side of $S_1$ is bisected, and a smaller square $S_2$ is constructed using these bisection points. This process is repeated to construct $S_3$ from $S_2$.\n\n2. **Calculating the side length of $S_1$**: \n... |
DeepScaleR | [
{
"role": "user",
"content": "Each of the squares of an $8 \\times 8$ board can be colored white or black. Find the number of colorings of the board such that every $2 \\times 2$ square contains exactly 2 black squares and 2 white squares."
}
] | math | {
"style": "rule",
"ground_truth": "8448"
} | {
"index": 8209,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the symbols $\\diamond$ and $\\circ$ represents an operation in the set $\\{+,-,\\times,\\div\\}$, and $\\frac{15 \\diamond 3}{8 \\circ 2} = 3$. What is the value of $\\frac{9 \\diamond 4}{14 \\circ 7}$? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{7}"
} | {
"index": 8210,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the symbols $\\star$ and $*$ represents an operation in the set $\\{+,-,\\times,\\div\\}$, and $\\frac{12\\star 2}{9*3}=2$. What is the value of $\\frac{7\\star 3}{12*6}$? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{6}"
} | {
"index": 8211,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the ten cards has a real number written on it. For every non-empty subset of these cards, the sum of all the numbers written on the cards in that subset is calculated. It is known that not all of the obtained sums turned out to be integers. What is the largest possible ... | math | {
"style": "rule",
"ground_truth": "511"
} | {
"index": 8212,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the ten letters in ``MATHCOUNTS'' is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word ``TEACH''? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 8213,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the ten volumes of the collected works of Theodore Sturgeon is available in paperback for $\\$$15 or in hardcover for $\\$$25. Theresa buys a copy of each of the ten volumes for a total of $\\$$220. How many hardcover volumes did she buy?"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 8214,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the three aluminum cans holds an integer number of liters of water. The second can holds 1.5 times more than the first, and the volume of the third can is equal to \\(\\frac{4^3}{3}\\) times the volume of the first can. What is the total number of liters of water that t... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 8215,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the three cutlets needs to be fried on a pan on both sides for five minutes each side. The pan can hold only two cutlets at a time. Is it possible to fry all three cutlets in less than 20 minutes (ignoring the time for flipping and transferring the cutlets)?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 8216,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the twelve letters in \"STATISTICS\" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word \"TEST\"? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 8217,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the twelve letters in ``MATHEMATICS'' is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word ``CALM''? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{12}"
} | {
"index": 8218,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the variables $a, b, c, d$, and $e$ represents a positive integer with the properties that $b+d>a+d$, $c+e>b+e$, $b+d=c$, $a+c=b+e$. Which of the variables has the greatest value?"
}
] | math | {
"style": "rule",
"ground_truth": "c"
} | {
"index": 8219,
"solution": "Since $b+d>a+d$, then $b>a$. This means that $a$ does not have the greatest value. Since $c+e>b+e$, then $c>b$. This means that $b$ does not have the greatest value. Since $b+d=c$ and each of $b, c, d$ is positive, then $d<c$, which means that $d$ does not have the greatest value. Cons... |
DeepScaleR | [
{
"role": "user",
"content": "Each of two baskets contains white and black balls such that the total number of balls in both baskets is 25. One ball is randomly drawn from each basket. It is known that the probability that both drawn balls are white is 0.54. Find the probability that both drawn balls are bl... | math | {
"style": "rule",
"ground_truth": "0.04"
} | {
"index": 8220,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of two boxes contains both black and white marbles, and the total number of marbles in the two boxes is $25.$ One marble is taken out of each box randomly. The probability that both marbles are black is $27/50,$ and the probability that both marbles are white is $m/n,$ whe... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 8221,
"solution": "We know that $\\frac{27}{50} = \\frac{b_1}{t_1} \\cdot \\frac{b_2}{t_2}$, where $b_1$ and $b_2$ are the number of black marbles in the first and the second box respectively, and $t_1$ and $t_2$ is the total number of marbles in the first and the second boxes respectively. So, $t_1 + t_... |
DeepScaleR | [
{
"role": "user",
"content": "Each of two boxes contains four chips numbered $1$, $2$, $3$, and $4$. Calculate the probability that the product of the numbers on the two chips is a multiple of $4$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 8222,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of two boxes contains three chips numbered $1$, $2$, $3$. A chip is drawn randomly from each box and the numbers on the two chips are multiplied. What is the probability that their product is even?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{9}"
} | {
"index": 8223,
"solution": "\nTo solve this problem, we need to determine the probability that the product of two randomly drawn chips, one from each box, is even. Each box contains chips numbered $1$, $2$, and $3$. \n\n#### Step 1: Determine Total Outcomes\nEach box has $3$ chips, and a chip is drawn from each b... |
DeepScaleR | [
{
"role": "user",
"content": "Each of two sides of a triangle is divided into six equal parts; the corresponding division points are connected by line segments. Find the sum of the lengths of these segments if the third side of the triangle is equal to 12."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 8224,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of two teams, Team A and Team B, sends 7 players in a predetermined order to participate in a Go contest. The players from both teams compete sequentially starting with Player 1 from each team. The loser of each match is eliminated, and the winner continues to compete with... | math | {
"style": "rule",
"ground_truth": "3432"
} | {
"index": 8225,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of two wheels contains numbers from 1 to 8. When the wheels are spun, a number is selected from each wheel. Find the probability that the sum of the two selected numbers is divisible by 4."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 8226,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each one of 2009 distinct points in the plane is coloured in blue or red, so that on every blue-centered unit circle there are exactly two red points. Find the gratest possible number of blue points."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 8227,
"solution": "\nConsider that there are 2009 distinct points in the plane, and each point is colored either blue or red. The objective is to determine the greatest possible number of blue points under the condition that every blue-centered unit circle contains exactly two red points.\n\nTo solve thi... |
DeepScaleR | [
{
"role": "user",
"content": "Each outcome on the spinner below has equal probability. If you spin the spinner four times and form a four-digit number from the four outcomes, such that the first outcome is the thousand digit, the second outcome is the hundreds digit, the third outcome is the tens digit, and... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 8228,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each outcome on the spinner below has equal probability. If you spin the spinner three times and form a three-digit number from the three outcomes, such that the first outcome is the hundreds digit, the second outcome is the tens digit and the third outcome is the units digit, ... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{9}"
} | {
"index": 8229,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each pack of hot dogs contains 10 hot dogs (no buns), but each pack of hot dog buns contains 8 buns. Phil buys a number of these packs for a barbecue. After the barbecue, Phil finds that he has 4 hot dogs left over. What is the SECOND smallest number of packs of hot dogs he ... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 8230,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each page number of a 488-page book is printed one time in the book. The first page is page 1 and the last page is page 488. When printing all of the page numbers, how many more 4's are printed than 8's?"
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 8231,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each pair of vertices of a regular $67$ -gon is joined by a line segment. Suppose $n$ of these segments are selected, and each of them is painted one of ten available colors. Find the minimum possible value of $n$ for which, regardless of which $n$ segments were selected... | math | {
"style": "rule",
"ground_truth": "2011"
} | {
"index": 8232,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either 12 pieces of red candy, 14 pieces of green candy, 15 pieces of blue candy, or $n$ pieces of purple candy. A piece of purple candy costs 20 cents. What is the smallest pos... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 8233,
"solution": "1. **Identify the total amount of money Casper has:** \n Let $r$, $g$, $b$, and $p$ be the costs of one piece of red, green, blue, and purple candy respectively. We know that:\n - $12r$ = total cost for 12 pieces of red candy\n - $14g$ = total cost for 14 pieces of green candy\n... |
DeepScaleR | [
{
"role": "user",
"content": "Each point in the hexagonal lattice shown is one unit from its nearest neighbor. How many equilateral triangles have all three vertices in the lattice? [asy]size(75);\ndot(origin);\ndot(dir(0));\ndot(dir(60));\ndot(dir(120));\ndot(dir(180));\ndot(dir(240));\ndot(dir(300));\n[/a... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 8234,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each positive integer $a$ undergoes the following procedure in order to obtain the number $d = d\\left(a\\right)$:\n\n(i) move the last digit of $a$ to the first position to obtain the numb er $b$;\n(ii) square $b$ to obtain the number $c$;\n(iii) move the first digit of $c$ to... | math | {
"style": "rule",
"ground_truth": "a = \\underbrace{2\\dots2}_{n \\ge 0}1, \\qquad a = 2, \\qquad a = 3."
} | {
"index": 8235,
"solution": "\nGiven the problem, we want to find all positive integers \\( a \\) such that the procedure outlined results in \\( d(a) = a^2 \\). Let's break down the steps of the procedure and solve for \\( a \\).\n\n### Procedure Analysis\n\n1. **Step (i):** Move the last digit of \\( a \\) to th... |
DeepScaleR | [
{
"role": "user",
"content": "Each positive integer number $n \\ ge 1$ is assigned the number $p_n$ which is the product of all its non-zero digits. For example, $p_6 = 6$ , $p_ {32} = 6$ , $p_ {203} = 6$ . Let $S = p_1 + p_2 + p_3 + \\dots + p_ {999}$ . Find the largest prime that divides $S $ ."
... | math | {
"style": "rule",
"ground_truth": "103"
} | {
"index": 8236,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each principal of Lincoln High School serves exactly one $3$-year term. What is the maximum number of principals this school could have during an $8$-year period?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 8237,
"solution": "To determine the maximum number of principals that can serve during an 8-year period at Lincoln High School, where each principal serves a 3-year term, we need to consider how the terms can overlap with the 8-year period.\n\n1. **Understanding the Term Length**: Each principal serves f... |
DeepScaleR | [
{
"role": "user",
"content": "Each row of a $24 \\times 8$ table contains some permutation of the numbers $1, 2, \\cdots , 8.$ In each column the numbers are multiplied. What is the minimum possible sum of all the products?\n*(C. Wu)*"
}
] | math | {
"style": "rule",
"ground_truth": "8 * (8!)^3"
} | {
"index": 8238,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each row of a seating arrangement seats 7 or 8 people. Forty-six people are to be seated. How many rows seat exactly 8 people if every seat is occupied?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 8239,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each row of a seating arrangement seats either 9 or 10 people. A total of 100 people are to be seated. How many rows seat exactly 10 people if every seat is occupied?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 8240,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each segment whose ends are vertices of a regular 100-sided polygon is colored - in red if there are an even number of vertices between its ends, and in blue otherwise (in particular, all sides of the 100-sided polygon are red). Numbers are placed at the vertices, the sum of th... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 8241,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each segment with endpoints at the vertices of a regular 100-gon is colored red if there is an even number of vertices between its endpoints, and blue otherwise (in particular, all sides of the 100-gon are red). Numbers were placed at the vertices such that the sum of their squ... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"index": 8242,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each segment with endpoints at the vertices of a regular 100-gon is painted red if there is an even number of vertices between its endpoints, and blue otherwise (in particular, all sides of the 100-gon are red). Numbers are placed at the vertices such that the sum of their squa... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"index": 8243,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each segment with endpoints at the vertices of a regular 100-sided polygon is colored red if there is an even number of vertices between the endpoints, and blue otherwise (in particular, all sides of the 100-sided polygon are red). Numbers are placed at the vertices such that t... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"index": 8244,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each side of a cube has a stripe drawn diagonally from one vertex to the opposite vertex. The stripes can either go from the top-right to bottom-left or from top-left to bottom-right, chosen at random for each face. What is the probability that there is at least one continuous ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{64}"
} | {
"index": 8245,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each side of square $ABCD$ with side length of $4$ is divided into equal parts by three points. Choose one of the three points from each side, and connect the points consecutively to obtain a quadrilateral. Which numbers can be the area of this quadrilateral? Just write the num... | math | {
"style": "rule",
"ground_truth": "{6,7,7.5,8,8.5,9,10}"
} | {
"index": 8246,
"solution": "\nConsider square \\(ABCD\\) with a side length of \\(4\\). Each side of the square is divided into equal parts by three points, creating four sections of equal length, each measuring \\( \\frac{4}{4} = 1 \\) unit. Let's label these points for each side as follows:\n- On side \\(AB\\) ... |
DeepScaleR | [
{
"role": "user",
"content": "Each side of the large square in the figure is trisected (divided into three equal parts). The corners of an inscribed square are at these trisection points, as shown. The ratio of the area of the inscribed square to the area of the large square is\n[asy]\ndraw((0,0)--(3,0)--(3... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{9}"
} | {
"index": 8247,
"solution": "1. **Understanding the Problem:**\n The problem involves a large square whose sides are trisected, and an inscribed square is formed by connecting these trisection points. We need to find the ratio of the area of the inscribed square to the area of the large square.\n\n2. **Visualizi... |
DeepScaleR | [
{
"role": "user",
"content": "Each side of the square grid is 15 toothpicks long. Calculate the total number of toothpicks used to construct the square grid."
}
] | math | {
"style": "rule",
"ground_truth": "480"
} | {
"index": 8248,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each square in a $3 \\times 10$ grid is colored black or white. Let $N$ be the number of ways this can be done in such a way that no five squares in an 'X' configuration (as shown by the black squares below) are all white or all black. Determine $\\sqrt{N}$."
}
] | math | {
"style": "rule",
"ground_truth": "25636"
} | {
"index": 8249,
"solution": "Note that we may label half of the cells in our board the number 0 and the other half 1, in such a way that squares labeled 0 are adjacent only to squares labeled 1 and vice versa. In other words, we make this labeling in a 'checkerboard' pattern. Since cells in an 'X' formation are al... |
DeepScaleR | [
{
"role": "user",
"content": "Each square in an $8 \\times 8$ grid is to be painted either white or black. The goal is to ensure that for any $2 \\times 3$ or $3 \\times 2$ rectangle selected from the grid, there are at least two adjacent squares that are black. What is the minimum number of squares that ne... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 8250,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each square in the following hexomino has side length 1. Find the minimum area of any rectangle that contains the entire hexomino."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{21}{2}"
} | {
"index": 8251,
"solution": "If a rectangle contains the entire hexomino, it must also contain its convex hull, which is an origin-symmetric hexagon. It is fairly clear that the smallest rectangle that contains such a hexagon must share one set of parallel sides with the hexagon. There are three such rectangles, a... |
DeepScaleR | [
{
"role": "user",
"content": "Each square of a $33\\times 33$ square grid is colored in one of the three colors: red, yellow or blue, such that the numbers of squares in each color are the same. If two squares sharing a common edge are in different colors, call that common edge a separating edge. Find the... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 8252,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each square of an $n \\times n$ grid is coloured either blue or red, where $n$ is a positive integer. There are $k$ blue cells in the grid. Pat adds the sum of the squares of the numbers of blue cells in each row to the sum of the squares of the numbers of blue cells in e... | math | {
"style": "rule",
"ground_truth": "313"
} | {
"index": 8253,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each successive term in the sequence $2048, 512, 128, x,\ny, 2, \\frac 12, \\frac 18,...$ is obtained by multiplying the previous term by a constant. What is the value of $x + y$?"
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 8254,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each term of a geometric sequence $\\left\\{a_{n}\\right\\}$ is a real number, and the sum of the first $n$ terms is $S_{n}$. If $S_{10} = 10$ and $S_{30} = 70$, then find $S_{4n}$."
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 8255,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each term of a sequence, after the first term, is inversely proportional to the term preceding it, and the constant of proportionality stays the same. If the first term is 2 and the second term is 5, what is the 12th term?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 8256,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each third-grade classroom at Pearl Creek Elementary has $18$ students and $2$ pet rabbits. How many more students than rabbits are there in all $4$ of the third-grade classrooms?"
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 8257,
"solution": "We are given that each third-grade classroom at Pearl Creek Elementary has $18$ students and $2$ pet rabbits. We need to find the total number of students and rabbits in all $4$ classrooms and then determine how many more students there are than rabbits.\n\n#### Step 1: Calculate the t... |
DeepScaleR | [
{
"role": "user",
"content": "Each third-grade classroom at Pearl Creek Elementary has 18 students and 2 pet rabbits. How many more students than rabbits are there in all 4 of the third-grade classrooms?"
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 8258,
"solution": "1. **Calculate the total number of students in all classrooms**: \n Each classroom has $18$ students. Since there are $4$ classrooms, the total number of students is:\n \\[\n 18 \\times 4 = 72 \\text{ students}\n \\]\n\n2. **Calculate the total number of rabbits in all classroo... |
DeepScaleR | [
{
"role": "user",
"content": "Each time you click a toggle switch, the switch either turns from *off* to *on* or from *on* to *off*. Suppose that you start with three toggle switches with one of them *on* and two of them *off*. On each move you randomly select one of the three switches and click it. Let $m... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"index": 8259,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each triangle in a sequence is either a 30-60-90 triangle or a 45-45-90 triangle. The hypotenuse of each 30-60-90 triangle serves as the longer leg of the adjacent 30-60-90 triangle, except for the final triangle which is a 45-45-90 triangle. The hypotenuse of the largest trian... | math | {
"style": "rule",
"ground_truth": "\\frac{6\\sqrt{6}}{2}"
} | {
"index": 8260,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each triangle is a 30-60-90 triangle, and the hypotenuse of one triangle is the longer leg of an adjacent triangle. The hypotenuse of the largest triangle is 8 centimeters. What is the number of centimeters in the length of the longer leg of the smallest triangle? Express your ... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"index": 8261,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each twin from the first 4 sets shakes hands with all twins except his/her sibling and with one-third of the triplets; the remaining 8 sets of twins shake hands with all twins except his/her sibling but does not shake hands with any triplet; and each triplet shakes hands with a... | math | {
"style": "rule",
"ground_truth": "394"
} | {
"index": 8262,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each two-digit is number is coloured in one of $k$ colours. What is the minimum value of $k$ such that, regardless of the colouring, there are three numbers $a$ , $b$ and $c$ with different colours with $a$ and $b$ having the same units digit (second digit) and $b... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 8263,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each unit square of a $4 \\times 4$ square grid is colored either red, green, or blue. Over all possible colorings of the grid, what is the maximum possible number of L-trominos that contain exactly one square of each color?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 8264,
"solution": "Notice that in each $2 \\times 2$ square contained in the grid, we can form 4 L-trominoes. By the pigeonhole principle, some color appears twice among the four squares, and there are two trominoes which contain both. Therefore each $2 \\times 2$ square contains at most 2 L-trominoes wi... |
DeepScaleR | [
{
"role": "user",
"content": "Each unit square of a 3-by-3 unit-square grid is to be colored either blue or red. For each square, either color is equally likely to be used. The probability of obtaining a grid that does not have a 2-by-2 red square is $\\frac {m}{n}$, where $m$ and $n$ are relatively prime p... | math | {
"style": "rule",
"ground_truth": "929"
} | {
"index": 8265,
"solution": "We consider how many ways we can have 2*2 grid\n$(1)$: All the girds are red--$1$ case\n$(2)$: One unit square is blue--The blue lies on the center of the bigger square, makes no 2*2 grid $9-1=8$ cases\n$(3)$: Two unit squares are blue--one of the squares lies in the center of the bigg... |
DeepScaleR | [
{
"role": "user",
"content": "Each valve $A$, $B$, and $C$, when open, releases water into a tank at its own constant rate. With all three valves open, the tank fills in 1 hour, with only valves $A$ and $C$ open it takes 1.5 hours, and with only valves $B$ and $C$ open it takes 2 hours. The number of hours ... | math | {
"style": "rule",
"ground_truth": "1.2"
} | {
"index": 8266,
"solution": "1. **Define the rates and volume**: Let the rate of water flow through valve $A$ be $a$ liters per hour, through valve $B$ be $b$ liters per hour, and through valve $C$ be $c$ liters per hour. Assume the volume of the tank is 1 liter.\n\n2. **Set up the equations based on given conditi... |
DeepScaleR | [
{
"role": "user",
"content": "Each vertex of a convex hexagon $ABCDEF$ is to be assigned a color, where there are $7$ colors to choose from. Find the number of possible different colorings such that both the ends of each diagonal and each pair of adjacent vertices have different colors."
}
] | math | {
"style": "rule",
"ground_truth": "5040"
} | {
"index": 8267,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each vertex of a convex hexagon $ABCDEF$ is to be assigned a color. There are $7$ colors to choose from, and no two adjacent vertices can have the same color, nor can the vertices at the ends of each diagonal. Calculate the total number of different colorings possible."
}
] | math | {
"style": "rule",
"ground_truth": "5040"
} | {
"index": 8268,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each vertex of a cube is to be labeled with an integer 1 through 8, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for each face. Arrangements that can be obtained from each other through rotations of the ... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 8269,
"solution": "To solve this problem, we need to label each vertex of a cube with integers from $1$ to $8$ such that the sum of the numbers on the vertices of each face is the same. Additionally, we consider two arrangements the same if one can be obtained from the other by rotating the cube.\n\n####... |
DeepScaleR | [
{
"role": "user",
"content": "Each vertex of a regular dodecagon ($12$-gon) is to be colored either red or blue, and thus there are $2^{12}$ possible colorings. Find the number of these colorings with the property that no four vertices colored the same color are the four vertices of a rectangle."
}
] | math | {
"style": "rule",
"ground_truth": "928"
} | {
"index": 8270,
"solution": "Note that the condition is equivalent to stating that there are no 2 pairs of oppositely spaced vertices with the same color.\n Case 1: There are no pairs. This yields $2$ options for each vertices 1-6, and the remaining vertices 7-12 are set, yielding $2^6=64$ cases.\n Case 2: There i... |
DeepScaleR | [
{
"role": "user",
"content": "Each vertex of convex pentagon $ABCDE$ is to be assigned a color. There are $6$ colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?"
}
] | math | {
"style": "rule",
"ground_truth": "3120"
} | {
"index": 8271,
"solution": "To solve this problem, we need to consider the constraints given by the diagonals of the pentagon. Each diagonal connects two vertices, and the vertices at the ends of each diagonal must have different colors. We will analyze the possible colorings by considering different cases based ... |
DeepScaleR | [
{
"role": "user",
"content": "Each vertex of the parallelogram $ABCD$ lies on the same side of the plane $S$ such that the distances of the vertices $A, B$, and $C$ from the plane $S$ are 4 cm, 6 cm, and 8 cm, respectively. The area of the projection of the parallelogram onto the plane $S$, which forms the ... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 8272,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each vertex of this parallelogram has integer coordinates. The perimeter of the parallelogram is \\( p \\) units, and the area is \\( a \\) square units. If the parallelogram is defined by vertices \\((2, 3)\\), \\((7, 3)\\), \\((x, y)\\), and \\((x-5, y)\\), where \\(x\\) and ... | math | {
"style": "rule",
"ground_truth": "38"
} | {
"index": 8273,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each vertex of this parallelogram has integer coordinates. The perimeter of this parallelogram is $p$ units, while the area is $a$ square units. What is the value of the sum $p + a$?\n\n[asy]\nsize(5cm,5cm);\ndraw((-2,0)--(11,0));\ndraw((0,-1)--(0,6));\ndraw((0,0)--(3,4)--(10,4... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"index": 8274,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Earl and Bob start their new jobs on the same day. Earl's work schedule is to work for 3 days followed by 1 day off, while Bob's work schedule is to work for 7 days followed by 3 days off. In the first 1000 days, how many days off do they have in common?"
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 8275,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eastbound traffic flows at 80 miles per hour and westbound traffic flows at 60 miles per hour. An eastbound driver observes 30 westbound vehicles in a 10-minute period. Calculate the number of westbound vehicles in a 150-mile section of the highway."
}
] | math | {
"style": "rule",
"ground_truth": "193"
} | {
"index": 8276,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Ed and Ann both have lemonade with their lunch. Ed orders the regular size. Ann gets the large lemonade, which is 50% more than the regular. After both consume $\\frac{3}{4}$ of their drinks, Ann gives Ed a third of what she has left, and 2 additional ounces. When they finish t... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 8277,
"solution": "1. **Define the variables:**\n Let the size of Ed's drink be $x$ ounces. Since Ann's drink is 50% more than Ed's, her drink size is $\\frac{3}{2}x$ ounces.\n\n2. **Calculate the amount consumed and remaining:**\n Both Ed and Ann consume $\\frac{3}{4}$ of their drinks. Therefore, th... |
DeepScaleR | [
{
"role": "user",
"content": "Ed and Sue bike at equal and constant rates. Similarly, they jog at equal and constant rates, and they swim at equal and constant rates. Ed covers $74$ kilometers after biking for $2$ hours, jogging for $3$ hours, and swimming for $4$ hours, while Sue covers $91$ kilometers aft... | math | {
"style": "rule",
"ground_truth": "314"
} | {
"index": 8278,
"solution": "Let the biking rate be $b$, swimming rate be $s$, jogging rate be $j$, all in km/h.\n We have $2b + 3j + 4s = 74,2j + 3s + 4b = 91$. Subtracting the second from twice the first gives $4j + 5s = 57$. Mod 4, we need $s\\equiv1\\pmod{4}$. Thus, $(j,s) = (13,1),(8,5),(3,9)$.\n$(13,1)$ and ... |
DeepScaleR | [
{
"role": "user",
"content": "Ed has five identical green marbles, and a large supply of identical red marbles. He arranges the green marbles and some of the red ones in a row and finds that the number of marbles whose right hand neighbor is the same color as themselves is equal to the number of marbles who... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 8279,
"solution": "We are limited by the number of marbles whose right hand neighbor is not the same color as the marble. By surrounding every green marble with red marbles - RGRGRGRGRGR. That's 10 \"not the same colors\" and 0 \"same colors.\" Now, for every red marble we add, we will add one \"same col... |
DeepScaleR | [
{
"role": "user",
"content": "Eddy draws $6$ cards from a standard $52$ -card deck. What is the probability that four of the cards that he draws have the same value?"
}
] | math | {
"style": "rule",
"ground_truth": "3/4165"
} | {
"index": 8280,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Edward stopped to rest at a place 1,875 feet from the prison and was spotted by a guard with a crossbow. The guard fired an arrow with an initial velocity of \\( 100 \\, \\mathrm{ft/s} \\). At the same time, Edward started running away with an acceleration of \\( 1 \\, \\mathrm... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"index": 8281,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Egor wrote a number on the board and encoded it according to the rules of letter puzzles (different letters correspond to different digits, and identical letters correspond to identical digits). The word \"GUATEMALA\" was the result. How many different numbers could Egor have o... | math | {
"style": "rule",
"ground_truth": "20160"
} | {
"index": 8282,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Egor wrote a number on the board and encoded it according to the rules of letter puzzles (different letters correspond to different digits, the same letters to the same digits). The result was the word \"ГВАТЕМАЛА\". How many different numbers could Egor have originally written... | math | {
"style": "rule",
"ground_truth": "21600"
} | {
"index": 8283,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Egor wrote a number on the board and encrypted it according to the rules of letter puzzles (different letters correspond to different digits, identical letters - identical digits). The result was the word \"GWATEMALA\". How many different numbers could Egor have originally writ... | math | {
"style": "rule",
"ground_truth": "114240"
} | {
"index": 8284,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Egor, Nikita, and Innokentiy took turns playing chess with each other (two play, one watches). After each game, the loser gave up their place to the spectator (there were no draws). As a result, Egor participated in 13 games, and Nikita participated in 27 games. How many games ... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 8285,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Egor, Nikita, and Innokentiy took turns playing chess with each other (two play, one watches). After each game, the loser gave up their spot to the spectator (there were no draws). It turned out that Egor participated in 13 games, and Nikita in 27 games. How many games did Inno... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 8286,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Egorov decided to open a savings account to buy a car worth 900,000 rubles. The initial deposit is 300,000 rubles. Every month, Egorov plans to add 15,000 rubles to his account. The bank offers a monthly interest rate of $12\\%$ per annum. The interest earned each month is adde... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 8287,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eight balls are randomly and independently painted either black or white with equal probability. What is the probability that each ball is painted a color such that it is different from the color of at least half of the other 7 balls?\nA) $\\frac{21}{128}$\nB) $\\frac{28}{128}$... | math | {
"style": "rule",
"ground_truth": "\\frac{35}{128}"
} | {
"index": 8288,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eight celebrities meet at a party. It so happens that each celebrity shakes hands with exactly two others. A fan makes a list of all unordered pairs of celebrities who shook hands with each other. If order does not matter, how many different lists are possible?"
}
] | math | {
"style": "rule",
"ground_truth": "3507"
} | {
"index": 8289,
"solution": "Let the celebrities get into one or more circles so that each circle has at least three celebrities, and each celebrity shook hands precisely with his or her neighbors in the circle. Let's consider the possible circle sizes: - There's one big circle with all 8 celebrities. Depending on... |
DeepScaleR | [
{
"role": "user",
"content": "Eight chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least four adjacent chairs."
}
] | math | {
"style": "rule",
"ground_truth": "288"
} | {
"index": 8290,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eight circles of diameter 1 are packed in the first quadrant of the coordinate plane as shown. Let region $\\mathcal{R}$ be the union of the eight circular regions. Line $l,$ with slope 3, divides $\\mathcal{R}$ into two regions of equal area. Line $l$'s equation can be express... | math | {
"style": "rule",
"ground_truth": "65"
} | {
"index": 8291,
"solution": "The line passing through the tangency point of the bottom left circle and the one to its right and through the tangency of the top circle in the middle column and the one beneath it is the line we are looking for: a line passing through the tangency of two circles cuts congruent areas,... |
DeepScaleR | [
{
"role": "user",
"content": "Eight circles of diameter 1 are packed in the first quadrant of the coordinate plane as shown. Let region $\\mathcal{R}$ be the union of the eight circular regions. Line $l,$ with slope 3, divides $\\mathcal{R}$ into two regions of equal area. Line $l$'s equation can be express... | math | {
"style": "rule",
"ground_truth": "65"
} | {
"index": 8292,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eight coins are arranged in a circle heads up. A move consists of flipping over two adjacent coins. How many different sequences of six moves leave the coins alternating heads up and tails up?"
}
] | math | {
"style": "rule",
"ground_truth": "7680"
} | {
"index": 8293,
"solution": "Imagine we flip over two adjacent coins by pushing a button halfway between them. Then the outcome depends only on the parities of the number of times that each button is pushed. To flip any coin, we must push the two buttons adjacent to that coin a total of an odd number of times. To ... |
DeepScaleR | [
{
"role": "user",
"content": "Eight congruent copies of the parabola \\( y = x^2 \\) are arranged symmetrically around a circle such that each vertex is tangent to the circle, and each parabola is tangent to its two neighbors. Find the radius of the circle. Assume that one of the tangents to the parabolas c... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 8294,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eight congruent disks are arranged around a square with side length 2 in such a manner that each disk is tangent to two others, with four disks centered on the midpoints of the square’s sides and four at the corners. The disks completely cover the boundary of the square. Calcul... | math | {
"style": "rule",
"ground_truth": "4\\pi"
} | {
"index": 8295,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like the other.... | math | {
"style": "rule",
"ground_truth": "1680"
} | {
"index": 8296,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eight consecutive three-digit positive integers have the following property: each of them is divisible by its last digit. What is the sum of the digits of the smallest of these eight integers?"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 8297,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eight distinct integers are picked at random from $\\{1,2,3,\\ldots,15\\}$. What is the probability that, among those selected, the third smallest is $5$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{72}{307}"
} | {
"index": 8298,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eight distinct pieces of candy are to be distributed among three bags: red, blue, and white, with each bag receiving at least one piece of candy. Determine the total number of arrangements possible."
}
] | math | {
"style": "rule",
"ground_truth": "846720"
} | {
"index": 8299,
"solution": ""
} |
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