data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "A housewife saved $2.50 in buying a dress on sale. If she spent $25 for the dress, she saved about:"
}
] | math | {
"style": "rule",
"ground_truth": "9 \\%"
} | {
"index": 2000,
"solution": "1. **Calculate the original price of the dress**:\n The housewife saved $\\textdollar{2.50}$ and spent $\\textdollar{25}$ on the dress. Therefore, the original price of the dress before the discount was:\n \\[\n \\text{Original Price} = \\text{Amount Spent} + \\text{Amount Saved}... |
DeepScaleR | [
{
"role": "user",
"content": "A huge number $y$ is given by $2^33^24^65^57^88^39^{10}11^{11}$. What is the smallest positive integer that, when multiplied with $y$, results in a product that is a perfect square?"
}
] | math | {
"style": "rule",
"ground_truth": "110"
} | {
"index": 2001,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A hydra consists of several heads and several necks, where each neck joins two heads. When a hydra's head $A$ is hit by a sword, all the necks from head $A$ disappear, but new necks grow up to connect head $A$ to all the heads which weren't connected to $A$ . Heracle d... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 2002,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A hyperbola has its center at the origin O, with its foci on the x-axis and two asymptotes denoted as l₁ and l₂. A line perpendicular to l₁ passes through the right focus F intersecting l₁ and l₂ at points A and B, respectively. It is known that the magnitudes of vectors |OA|, ... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{2}"
} | {
"index": 2003,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A hyperbola has its two foci at $(5, 0)$ and $(9, 4).$ Find the coordinates of its center."
}
] | math | {
"style": "rule",
"ground_truth": "(7,2)"
} | {
"index": 2004,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A hyperbola in the coordinate plane passing through the points $(2,5)$ , $(7,3)$ , $(1,1)$ , and $(10,10)$ has an asymptote of slope $\\frac{20}{17}$ . The slope of its other asymptote can be expressed in the form $-\\frac{m}{n}$ , where $m$ and $n$ are relatively pr... | math | {
"style": "rule",
"ground_truth": "1720"
} | {
"index": 2005,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A hyperbola is centered at the origin and opens either horizontally or vertically. It passes through the points $(-3, 4),$ $(-2, 0),$ and $(t, 2).$ Find $t^2.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{21}{4}"
} | {
"index": 2006,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A hyperbola with its center shifted to $(1,1)$ passes through point $(4, 2)$. The hyperbola opens horizontally, with one of its vertices at $(3, 1)$. Determine $t^2$ if the hyperbola also passes through point $(t, 4)$."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 2007,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A jacket and a shirt originally sold for $80$ dollars and $40$ dollars, respectively. During a sale Chris bought the $80$ dollar jacket at a $40\\%$ discount and the $40$ dollar shirt at a $55\\%$ discount. The total amount saved was what percent of the total of the original pr... | math | {
"style": "rule",
"ground_truth": "45\\%"
} | {
"index": 2008,
"solution": "1. **Calculate the original total cost**: The jacket was originally priced at $80$ dollars and the shirt at $40$ dollars. Therefore, the total original cost is:\n \\[\n 80 + 40 = 120 \\text{ dollars}\n \\]\n\n2. **Calculate the savings on each item**: \n - **Jacket**: The disco... |
DeepScaleR | [
{
"role": "user",
"content": "A jacket was originally priced $\\textdollar 100$ . The price was reduced by $10\\%$ three times and increased by $10\\%$ four times in some order. To the nearest cent, what was the final price?"
}
] | math | {
"style": "rule",
"ground_truth": "106.73"
} | {
"index": 2009,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A jar contains $2$ yellow candies, $4$ red candies, and $6$ blue candies. Candies are randomly drawn out of the jar one-by-one and eaten. The probability that the $2$ yellow candies will be eaten before any of the red candies are eaten is given by the fraction $\\frac{... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 2010,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A jar contains $29\\frac{5}{7}$ tablespoons of peanut butter. If one serving of peanut butter is 2 tablespoons, how many servings of peanut butter does the jar contain? Express your answer as a mixed number."
}
] | math | {
"style": "rule",
"ground_truth": "14\\frac{6}{7}"
} | {
"index": 2011,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A jar contains $5$ different colors of gumdrops. $30\\%$ are blue, $20\\%$ are brown, $15\\%$ are red, $10\\%$ are yellow, and other $30$ gumdrops are green. If half of the blue gumdrops are replaced with brown gumdrops, how many gumdrops will be brown?"
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 2012,
"solution": "1. **Calculate the total number of gumdrops**:\n Given that $30\\%$ are blue, $20\\%$ are brown, $15\\%$ are red, $10\\%$ are yellow, and $30$ gumdrops are green. First, we calculate the percentage of gumdrops that are green:\n \\[\n 100\\% - (30\\% + 20\\% + 15\\% + 10\\%) = 100... |
DeepScaleR | [
{
"role": "user",
"content": "A jar contains 8 red balls and 2 blue balls. Every minute, a ball is randomly removed. The probability that there exists a time during this process where there are more blue balls than red balls in the jar can be expressed as $\\frac{a}{b}$ for relatively prime integers $a$ and... | math | {
"style": "rule",
"ground_truth": "209"
} | {
"index": 2013,
"solution": "One can show that the condition in the problem is satisfied if and only the last ball drawn is blue (which happens with probability $\\frac{1}{5}$), or the blue balls are drawn second-to-last and third-to-last (which happens with probability $\\frac{1}{\\binom{10}{2}}=\\frac{1}{45}$). ... |
DeepScaleR | [
{
"role": "user",
"content": "A jar contains 97 marbles that are either red, green, or blue. Neil draws two marbles from the jar without replacement and notes that the probability that they would be the same color is $\\frac{5}{12}$. After Neil puts his marbles back, Jerry draws two marbles from the jar wit... | math | {
"style": "rule",
"ground_truth": "\\frac{41}{97}"
} | {
"index": 2014,
"solution": "Note that $\\frac{5}{12}=\\frac{40.97}{97 \\cdot 96}$. Of all of the original ways we could've drawn marbles, we are adding 97 ways, namely drawing the same marble twice, all of which work. Thus, the answer is $$\\frac{40 \\cdot 97+97}{97 \\cdot 96+97}=\\frac{41}{97}$$"
} |
DeepScaleR | [
{
"role": "user",
"content": "A jar contains quarters (worth $\\$0.25$ each), nickels (worth $\\$0.05$ each) and pennies (worth $\\$0.01$ each). The value of the quarters is $\\$10.00.$ The value of the nickels is $\\$10.00.$ The value of the pennies is $\\$10.00.$ If Judith randomly chooses one coin from t... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{31}"
} | {
"index": 2015,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A jar contains two red marbles, three green marbles, ten white marbles and no other marbles. Two marbles are randomly drawn from this jar without replacement. What is the probability that these two marbles drawn will both be red? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{105}"
} | {
"index": 2016,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A jar has $10$ red candies and $10$ blue candies. Terry picks two candies at random, then Mary picks two of the remaining candies at random. Given that the probability that they get the same color combination, irrespective of order, is $m/n,$ where $m$ and $n$ are relatively pr... | math | {
"style": "rule",
"ground_truth": "441"
} | {
"index": 2017,
"solution": "The probability that Terry picks two red candies is $\\frac{10 \\cdot 9}{20 \\cdot 19} = \\frac{9}{38}$, and the probability that Mary picks two red candies after Terry chooses two red candies is $\\frac{7\\cdot8}{18\\cdot17} = \\frac{28}{153}$. So the probability that they both pick t... |
DeepScaleR | [
{
"role": "user",
"content": "A jar is filled with red, orange and yellow jelly beans. The probability of randomly selecting a red jelly bean from this jar is $0.2$, and the probability of randomly selecting an orange jelly bean from this jar is $0.5$. What is the probability of randomly selecting a yello... | math | {
"style": "rule",
"ground_truth": "0.3"
} | {
"index": 2018,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A jar of peanut butter which is 3 inches in diameter and 4 inches high sells for $\\$$0.60. At the same rate, what would be the price for a jar that is 6 inches in diameter and 6 inches high?"
}
] | math | {
"style": "rule",
"ground_truth": "\\$3.60"
} | {
"index": 2019,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A journalist wants to take a photo of 5 volunteers and 2 elderly individuals they are assisting, requiring them to line up in a row. The 2 elderly individuals must stand next to each other but not at the ends. The total number of different arrangements is ____."
}
] | math | {
"style": "rule",
"ground_truth": "960"
} | {
"index": 2020,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A journey is divided into three segments: uphill, flat, and downhill. The ratio of the lengths of these segments is $1: 2: 3$, and the ratio of the times taken to walk each segment is $4: 5: 6$. Given that the person’s speed uphill is 3 km/h and the total distance is 50 km, how... | math | {
"style": "rule",
"ground_truth": "10.4166666667"
} | {
"index": 2021,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A keen archaeologist is holding a competition where participants must guess the age of a rare artifact. The age of the artifact is formed using the six digits: 2, 2, 3, 3, 7, and 9, and it must begin with an odd digit.\n\nHow many different ages can be there for the artifact?"
... | math | {
"style": "rule",
"ground_truth": "180"
} | {
"index": 2022,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A kindergarten received cards for learning to read: some are labeled \"МА\", and the rest are labeled \"НЯ\".\n\nEach child took three cards and started to form words from them. It turned out that 20 children could form the word \"МАМА\" from their cards, 30 children could form... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 2023,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A king summoned two wise men. He gave the first one 100 blank cards and instructed him to write a positive number on each (the numbers do not have to be different), without showing them to the second wise man. Then, the first wise man can communicate several distinct numbers to... | math | {
"style": "rule",
"ground_truth": "101"
} | {
"index": 2024,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A king traversed a $9 \\times 9$ chessboard, visiting each square exactly once. The king's route is not a closed loop and may intersect itself. What is the maximum possible length of such a route if the length of a move diagonally is $\\sqrt{2}$ and the length of a move vertica... | math | {
"style": "rule",
"ground_truth": "16 + 64 \\sqrt{2}"
} | {
"index": 2025,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A kite has sides $15$ units and $20$ units meeting at a right angle, and its diagonals are $24$ units and $x$ units. Find the area of the kite."
}
] | math | {
"style": "rule",
"ground_truth": "216"
} | {
"index": 2026,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A kite is inscribed in a circle with center $O$ and radius $60$ . The diagonals of the kite meet at a point $P$ , and $OP$ is an integer. The minimum possible area of the kite can be expressed in the form $a\\sqrt{b}$ , where $a$ and $b$ are positive integers and $b... | math | {
"style": "rule",
"ground_truth": "239"
} | {
"index": 2027,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A kite-shaped field is planted uniformly with wheat. The sides of the kite are 120 m and 80 m, with angles between the unequal sides being \\(120^\\circ\\) and the other two angles being \\(60^\\circ\\) each. At harvest, the wheat at any point in the field is brought to the nea... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 2028,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A kitten bites off a quarter of a sausage from one end, then a puppy bites off a third of the remaining piece from the opposite end, then the kitten again bites off a quarter from its end, and the puppy bites off a third from its end, and so on. You need to tie a string around ... | math | {
"style": "rule",
"ground_truth": "1:1"
} | {
"index": 2029,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A knight begins on the lower-left square of a standard chessboard. How many squares could the knight end up at after exactly 2009 legal knight's moves?"
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 2030,
"solution": "The knight goes from a black square to a white square on every move, or vice versa, so after 2009 moves he must be on a square whose color is opposite of what he started on. So he can only land on half the squares after 2009 moves. Note that he can access any of the 32 squares (there a... |
DeepScaleR | [
{
"role": "user",
"content": "A labor employment service center has 7 volunteers ready to provide labor employment guidance on the streets over the weekend, with 6 people to be arranged for duty on Saturday and Sunday. If 3 people are arranged for each day, there are a total of _______ different arrangement... | math | {
"style": "rule",
"ground_truth": "140"
} | {
"index": 2031,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A laboratory has $10$ experimental mice, among which $3$ have been infected with a certain virus, and the remaining $7$ are healthy. Random medical examinations are conducted one by one until all $3$ infected mice are identified. The number of different scenarios where the last... | math | {
"style": "rule",
"ground_truth": "1512"
} | {
"index": 2032,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A ladder is leaning against a house with its lower end 15 feet from the house. When the lower end is pulled 9 feet farther from the house, the upper end slides 13 feet down. How long is the ladder (in feet)?"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 2033,
"solution": "Of course the house makes a right angle with the ground, so we can use the Pythagorean theorem. Let $x$ be the length of the ladder and $y$ be the original height at which it touched the house. Then we are given $x^{2}=15^{2}+y^{2}=24^{2}+(y-13)^{2}$. Isolating $y$ in the second equati... |
DeepScaleR | [
{
"role": "user",
"content": "A lady made 3 round doilies with radii of 2, 3, and 10 inches, respectively. She placed them on a round table so that each doily touches the two others and the edge of the table. What is the radius of the table?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 2034,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A lame king is a chess piece that can move from a cell to any cell that shares at least one vertex with it, except for the cells in the same column as the current cell. A lame king is placed in the top-left cell of a $7 \\times 7$ grid. Compute the maximum number of cells it ca... | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 2035,
"solution": "Color the columns all-black and all-white, alternating by column. Each move the lame king takes will switch the color it's on. Assuming the king starts on a black cell, there are 28 black and 21 white cells, so it can visit at most $22+21=43$ cells in total, which is easily achievable.... |
DeepScaleR | [
{
"role": "user",
"content": "A laptop is originally priced at $800. The store offers a $15\\%$ discount, followed by another $10\\%$ discount on the discounted price. Tom also has a special membership card giving an additional $5\\%$ discount on the second discounted price. What single percent discount wou... | math | {
"style": "rule",
"ground_truth": "27.325\\%"
} | {
"index": 2036,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A laptop is originally priced at $\\$1200$. It is on sale for $15\\%$ off. John applies two additional coupons: one gives $10\\%$ off the discounted price, and another gives $5\\%$ off the subsequent price. What single percent discount would give the same final price as these t... | math | {
"style": "rule",
"ground_truth": "27.325\\%"
} | {
"index": 2037,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A large $5 \\times 5 \\times 5$ cube is formed using 125 small $1 \\times 1 \\times 1$ cubes. There are three central columns, each passing through the small cube at the very centre of the large cube: one from top to bottom, one from front to back, and one from left to right. A... | math | {
"style": "rule",
"ground_truth": "192"
} | {
"index": 2038,
"solution": "The original $5 \\times 5 \\times 5$ cube has 6 faces, each of which is $5 \\times 5$. When the three central columns of cubes is removed, one of the '1 \\times 1$ squares' on each face is removed. This means that the surface area of each face is reduced by 1 to $5 \\times 5 - 1 = 24$.... |
DeepScaleR | [
{
"role": "user",
"content": "A large batch of tires contains $1.5\\%$ defects. What should be the sample size for the probability of finding at least one defective tire in the sample to be more than $0.92 ?$"
}
] | math | {
"style": "rule",
"ground_truth": "168"
} | {
"index": 2039,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A large candle is $119$ centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes $10$ seconds to burn down the first centimeter from the top, $20$ seconds to burn down the second ... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 2040,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A large candle is $119$ centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes $10$ seconds to burn down the first centimeter from the top, $20$ seconds to burn down the second ... | math | {
"style": "rule",
"ground_truth": "350"
} | {
"index": 2041,
"solution": "We find that $T=10(1+2+\\cdots +119)$. From Gauss's formula, we find that the value of $T$ is $10(7140)=71400$. The value of $\\frac{T}{2}$ is therefore $35700$. We find that $35700$ is $10(3570)=10\\cdot \\frac{k(k+1)}{2}$, so $3570=\\frac{k(k+1)}{2}$. As a result, $7140=k(k+1)$, whic... |
DeepScaleR | [
{
"role": "user",
"content": "A large chest contains 10 smaller chests. In each of the smaller chests, either 10 even smaller chests are placed or nothing is placed. In each of those smaller chests, either 10 smaller chests are placed or none, and so on. After this, there are exactly 2006 chests with conten... | math | {
"style": "rule",
"ground_truth": "18054"
} | {
"index": 2042,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A large cube is formed by stacking 27 unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. The number of unit cubes that the plane intersects is"
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 2043,
"solution": "1. **Understanding the Problem:**\n - We have a large cube formed by stacking 27 smaller unit cubes (3x3x3).\n - A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal.\n - We need to find the number of unit cubes that the plane inte... |
DeepScaleR | [
{
"role": "user",
"content": "A large cube is made up of 27 small cubes. A plane is perpendicular to one of the diagonals of this large cube and bisects the diagonal. How many small cubes are intersected by this plane?"
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 2044,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A large equilateral triangle with side length $12$ is to be filled completely with non-overlapping equilateral triangles, each of side length $2$. Calculate the number of small triangles needed to fill the large triangle."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 2045,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A large number \\( y \\) is defined by \\( 2^33^54^45^76^57^38^69^{10} \\). Determine the smallest positive integer that, when multiplied with \\( y \\), results in a product that is a perfect square."
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 2046,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A large rectangle consists of three identical squares and three identical small rectangles. The perimeter of a square is 24, and the perimeter of a small rectangle is 16. What is the perimeter of the large rectangle?\n\nThe perimeter of a shape is the sum of its side lengths."
... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"index": 2047,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A large rectangle consists of three identical squares and three identical small rectangles. The perimeter of each square is 24, and the perimeter of each small rectangle is 16. What is the perimeter of the large rectangle?"
}
] | math | {
"style": "rule",
"ground_truth": "52"
} | {
"index": 2048,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A large rectangle has side lengths of $(x+7)$ and $(x+5)$. In the large rectangle, there is a rectangular hole with side lengths of $(2x-3)$ and $(x-2)$. What is the area of the large rectangle (not including the area of the hole)? Express your answer as a polynomial in $x$."
... | math | {
"style": "rule",
"ground_truth": "-x^2+19x+29"
} | {
"index": 2049,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A large rectangle is partitioned into four rectangles by two segments parallel to its sides. The areas of three of the resulting rectangles are shown. What is the area of the fourth rectangle?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 2050,
"solution": "Let's denote the large rectangle as $ABCD$, with $AB$ and $CD$ being the length and $BC$ and $AD$ being the width. Suppose two segments parallel to $AB$ and $BC$ partition $ABCD$ into four smaller rectangles: $PQRS$, $PQTU$, $UVWX$, and $RSWX$. Given the areas of three of these rectang... |
DeepScaleR | [
{
"role": "user",
"content": "A large rectangle is tiled by some $1\\times1$ tiles. In the center there is a small rectangle tiled by some white tiles. The small rectangle is surrounded by a red border which is fi\fve tiles wide. That red border is surrounded by a white border which is fi\fve tiles wide. ... | math | {
"style": "rule",
"ground_truth": "350"
} | {
"index": 2051,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A large rectangle measures 15 units by 20 units. One-quarter of this rectangle is shaded. If half of this quarter rectangle is shaded, what fraction of the large rectangle is shaded?\nA) $\\frac{1}{24}$\nB) $\\frac{1}{12}$\nC) $\\frac{1}{10}$\nD) $\\frac{1}{8}$\nE) $\\frac{1}{6... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"index": 2052,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A large rectangular garden contains two flower beds in the shape of congruent isosceles right triangles and a trapezoidal playground. The parallel sides of the trapezoid measure $30$ meters and $46$ meters. Determine the fraction of the garden occupied by the flower beds.\nA) $... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{23}"
} | {
"index": 2053,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A large shopping mall designed a lottery activity to reward its customers. In the lottery box, there are $8$ small balls of the same size, with $4$ red and $4$ black. The lottery method is as follows: each customer draws twice, picking two balls at a time from the lottery box e... | math | {
"style": "rule",
"ground_truth": "\\frac{6}{7}"
} | {
"index": 2054,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A large sphere has a volume of $288\\pi$ cubic units. A smaller sphere has a volume which is $12.5\\%$ of the volume of the larger sphere. What is the ratio of the radius of the smaller sphere to the radius of the larger sphere? Express your answer as a common fraction."
}
... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 2055,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A large square has each side divided into four equal parts. A square is inscribed such that its vertices touch these division points, as illustrated below. Determine the ratio of the area of the inscribed square to the large square.\n[asy]\ndraw((0,0)--(4,0)--(4,4)--(0,4)--cycl... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{8}"
} | {
"index": 2056,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A large square region is paved with $n^2$ square black tiles, where each tile measures $t$ inches on each side. Surrounding each tile is a white border that is $w$ inches wide. When $n=30$, it's given that the black tiles cover $81\\%$ of the area of the large square region. Fi... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"index": 2057,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A large supermarket has recorded the number of customers during eight holidays (unit: hundreds of people) as $29$, $30$, $38$, $25$, $37$, $40$, $42$, $32$. What is the $75$th percentile of this data set?"
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"index": 2058,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A large supermarket purchased a popular disinfectant laundry detergent. Due to the rise in raw material prices, the cost price per bottle of detergent this year increased by $4$ compared to last year. The quantity of detergent purchased for $1440$ yuan this year is the same as ... | math | {
"style": "rule",
"ground_truth": "8100"
} | {
"index": 2059,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A large urn contains $100$ balls, of which $36 \\%$ are red and the rest are blue. How many of the blue balls must be removed so that the percentage of red balls in the urn will be $72 \\%$? (No red balls are to be removed.)"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 2060,
"solution": "1. **Calculate the initial number of red and blue balls:**\n Given that $36\\%$ of the balls are red, and there are $100$ balls in total, the number of red balls is:\n \\[\n 0.36 \\times 100 = 36 \\text{ red balls}\n \\]\n The rest of the balls are blue, so the number of blue... |
DeepScaleR | [
{
"role": "user",
"content": "A larger equilateral triangle ABC with side length 5 has a triangular corner DEF removed from one corner, where DEF is an isosceles triangle with DE = EF = 2, and DF = 2\\sqrt{2}. Calculate the perimeter of the remaining quadrilateral."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 2061,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A larger grid is considered for the next challenge, where each segment must still only be traversed in a rightward or downward direction. Starting from point $A$, located at the top-left of a 3x3 grid, to point $B$ at the bottom-right. How many different routes can be taken?"
... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 2062,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A larger square contains two non-overlapping shapes: a circle with diameter $2$ and a rectangle with side lengths $2$ and $4$. Find the smallest possible side length of the larger square such that these shapes can fit without overlapping, and then, find the area of the square $... | math | {
"style": "rule",
"ground_truth": "\\sqrt{8 - \\pi}"
} | {
"index": 2063,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A larger square is constructed, and another square is formed inside it by connecting the midpoints of each side of the larger square. If the area of the larger square is 100, what is the area of the smaller square formed inside?"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 2064,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A laser is placed at the point $(3,5)$. The laser beam travels in a straight line. Larry wants the beam to hit and bounce off the $y$-axis, then hit and bounce off the $x$-axis, then hit the point $(7,5)$. What is the total distance the beam will travel along this path?"
}
] | math | {
"style": "rule",
"ground_truth": "10\\sqrt{2}"
} | {
"index": 2065,
"solution": "1. **Identify Points and Reflections**:\n Let $A = (3,5)$ and $D = (7,5)$. We need to find points $B$ on the $y$-axis and $C$ on the $x$-axis where the laser beam bounces off.\n\n2. **Reflection Properties**:\n When a beam hits and bounces off a coordinate axis, the angle of incide... |
DeepScaleR | [
{
"role": "user",
"content": "A lateral face of a regular triangular pyramid $SABC$ is inclined to the base plane $ABC$ at an angle $\\alpha = \\operatorname{arctg} \\frac{3}{4}$. Points $M, N, K$ are midpoints of the sides of the base $ABC$. The triangle $MNK$ serves as the lower base of a rectangular pris... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 2066,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A lattice point in an $xy$-coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y = mx + 2$ passes through no lattice point with $0 < x \\leq 100$ for all $m$ such that $\\frac{1}{2} < m < a$. What is the maximum possible value of $a$?"
}... | math | {
"style": "rule",
"ground_truth": "\\frac{50}{99}"
} | {
"index": 2067,
"solution": "1. **Understanding the Problem**: We need to find the maximum value of $a$ such that the line $y = mx + 2$ does not pass through any lattice points for $0 < x \\leq 100$ and $\\frac{1}{2} < m < a$.\n\n2. **Impact of the Constant Term**: The constant term \"+2\" shifts the line vertical... |
DeepScaleR | [
{
"role": "user",
"content": "A lattice point in an $xy$-coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y = mx + 5$ passes through no lattice point with $0 < x \\leq 150$ for all $m$ such that $0 < m < b$. What is the maximum possible value of $b$?\n**A)** $\\frac... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{151}"
} | {
"index": 2068,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A lattice point in an $xy$-coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y = mx +2$ passes through no lattice point with $0 < x \\le 100$ for all $m$ such that $\\frac{1}{2} < m < a$. What is the maximum possible value of $a$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{50}{99}"
} | {
"index": 2069,
"solution": "1. **Understanding the Problem**: We need to find the maximum value of $a$ such that the line $y = mx + 2$ does not pass through any lattice points for $0 < x \\leq 100$ and $\\frac{1}{2} < m < a$. A lattice point $(x, y)$ is where both $x$ and $y$ are integers.\n\n2. **Condition for L... |
DeepScaleR | [
{
"role": "user",
"content": "A lattice point in the $x,y$-plane is a point both of whose coordinates are integers (not necessarily positive). How many lattice points lie on the graph of the equation $x^2-y^2=47$?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 2070,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A lattice point in the plane is a point of the form $(n, m)$, where $n$ and $m$ are integers. Consider a set $S$ of lattice points. We construct the transform of $S$, denoted by $S^{\\prime}$, by the following rule: the pair $(n, m)$ is in $S^{\\prime}$ if and only if any of $(... | math | {
"style": "rule",
"ground_truth": "421"
} | {
"index": 2071,
"solution": "Transforming it $k \\geq 1$ times yields the 'diamond' of points $(n, m)$ such that $|n|+|m| \\leq k$. The diamond contains $(k+1)^{2}+k^{2}$ lattice points (this can be seen by rotating the plane 45 degrees and noticing the lattice points in the transforms form two squares, one of whi... |
DeepScaleR | [
{
"role": "user",
"content": "A lattice point is a point in the plane with integer coordinates. How many lattice points are on the line segment whose endpoints are $(3,17)$ and $(48,281)$? (Include both endpoints of the segment in your count.)"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 2072,
"solution": "1. **Identify the coordinates of the endpoints and calculate the differences in coordinates:**\n - The endpoints of the line segment are $(3,17)$ and $(48,281)$.\n - The difference in the $x$-coordinates is $48 - 3 = 45$.\n - The difference in the $y$-coordinates is $281 - 17 = 2... |
DeepScaleR | [
{
"role": "user",
"content": "A lattice point is a point whose coordinates are both integers. How many lattice points are on the boundary or inside the region bounded by $y=|x|$ and $y=-x^2+6$?"
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 2073,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A lattice point is a point whose coordinates are integers. How many lattice points are on the boundary or inside the region bounded by \\( y = |x| \\) and \\( y = -x^2 + 8 \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 2074,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A leak formed in the hold of a ship. A pump was immediately switched on to remove the water, but it couldn't keep up, and after 10 minutes, the water level rose by 20 cm. Then, a second pump of equal power was turned on, and after 5 minutes, the water level dropped by 10 cm. Th... | math | {
"style": "rule",
"ground_truth": "1.25"
} | {
"index": 2075,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A lemming sits at a corner of a square with side length $10$ meters. The lemming runs $6.2$ meters along a diagonal toward the opposite corner. It stops, makes a $90^{\\circ}$ right turn and runs $2$ more meters. A scientist measures the shortest distance between the lemming an... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 2076,
"solution": "1. **Understanding the problem**: A lemming starts at a corner of a square with side length $10$ meters. It moves $6.2$ meters along a diagonal towards the opposite corner, then makes a $90^{\\circ}$ right turn and runs $2$ meters. We need to find the average of the shortest distances ... |
DeepScaleR | [
{
"role": "user",
"content": "A lemming starts at a corner of a rectangular area measuring 8 meters by 15 meters. It dashes diagonally across the rectangle towards the opposite corner for 11.3 meters. Then the lemming makes a $90^{\\circ}$ right turn and sprints upwards for 3 meters. Calculate the average o... | math | {
"style": "rule",
"ground_truth": "5.75"
} | {
"index": 2077,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A library has six identical copies of a certain book. At any given time, some of these copies are at the library and some are checked out. How many different ways are there for some of the books to be in the library and the rest to be checked out if at least one book is in the ... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 2078,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A license plate in a certain state consists of 4 digits, not necessarily distinct, and 2 letters, also not necessarily distinct. These six characters may appear in any order, except that the two letters must appear next to each other. How many distinct license plates are possi... | math | {
"style": "rule",
"ground_truth": "33,\\!800,\\!000"
} | {
"index": 2079,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A license plate in a certain state consists of 5 digits, not necessarily distinct, and 3 letters, with the condition that at least one of the letters must be a vowel (A, E, I, O, U). These letters do not need to be next to each other but must be in a sequence. How many distinct... | math | {
"style": "rule",
"ground_truth": "4,989,000,000"
} | {
"index": 2080,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A lieutenant is training recruits in marching drills. Upon arriving at the parade ground, he sees that all the recruits are arranged in several rows, with each row having the same number of soldiers, and that the number of soldiers in each row is 5 more than the number of rows.... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 2081,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A light flashes green every 3 seconds. Determine the number of times the light has flashed green after 671 seconds."
}
] | math | {
"style": "rule",
"ground_truth": "154"
} | {
"index": 2082,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A light flashes in one of three different colors: red, green, and blue. Every $3$ seconds, it flashes green. Every $5$ seconds, it flashes red. Every $7$ seconds, it flashes blue. If it is supposed to flash in two colors at once, it flashes the more infrequent color. How ... | math | {
"style": "rule",
"ground_truth": "154"
} | {
"index": 2083,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A light pulse starts at a corner of a reflective square. It bounces around inside the square, reflecting off of the square's perimeter $n$ times before ending in a different corner. The path of the light pulse, when traced, divides the square into exactly 2021 regions. Compute ... | math | {
"style": "rule",
"ground_truth": "129"
} | {
"index": 2084,
"solution": "The main claim is that if the light pulse reflects vertically (on the left/right edges) $a$ times and horizontally $b$ times, then $\\operatorname{gcd}(a+1, b+1)=1$, and the number of regions is $\\frac{(a+2)(b+2)}{2}$. This claim can be conjectured by looking at small values of $a$ an... |
DeepScaleR | [
{
"role": "user",
"content": "A lighthouse emits a yellow signal every 15 seconds and a red signal every 28 seconds. The yellow signal is first seen 2 seconds after midnight, and the red signal is first seen 8 seconds after midnight. At what time will both signals be seen together for the first time?"
}
] | math | {
"style": "rule",
"ground_truth": "92"
} | {
"index": 2085,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line $l$ does not pass through the origin $O$ and intersects an ellipse $\\frac{x^{2}}{2}+y^{2}=1$ at points $A$ and $B$. $M$ is the midpoint of segment $AB$. Determine the product of the slopes of line $AB$ and line $OM$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"index": 2086,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line $l$ in the coordinate plane has the equation $2x - 3y + 30 = 0$. This line is rotated $90^\\circ$ counterclockwise about the point $(15,10)$ to obtain line $k'$. Find the $x$-coordinate of the $x$-intercept of $k'$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{65}{3}"
} | {
"index": 2087,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line $l$ is drawn through the fixed point $M(4,0)$, intersecting the parabola $y^{2}=4x$ at points $A$ and $B$. $F$ is the focus of the parabola. Calculate the minimum area of the triangle $\\triangle AFB$."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 2088,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line $l$ is tangent to the circle $x^{2}+y^{2}=1$ and the sum of its intercepts on the two coordinate axes is equal to $\\sqrt{3}$. Find the area of the triangle formed by line $l$ and the two coordinate axes."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 2089,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line $l$ passes through two points $P(-1,2)$ and $Q(2,-2)$, and intersects the hyperbola $(y-2)^{2}-x^{2}=1$ at two points $A$ and $B$. \n$(1)$ Write the parametric equation of $l$ as required by the question; \n$(2)$ Find the distance between the midpoint $M$ of $AB$ and p... | math | {
"style": "rule",
"ground_truth": "5 \\sqrt {65}"
} | {
"index": 2090,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line $l$ passing through the focus of the parabola $y=4x^2$ intersects the parabola at points $A(x_1, y_1)$ and $B(x_2, y_2)$. If $y_1+y_2=2$, then the length of segment $AB$ equals \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{8}"
} | {
"index": 2091,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line $l$ with an inclination angle of $45^\\circ$ passes through the focus $F$ of the parabola $y^2=8x$ and intersects the parabola at points $A$ and $B$. Find the value of $|\\overrightarrow{FA}| \\cdot |\\overrightarrow{FB}|$."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 2092,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line $x=k$ intersects the graph of $y=\\log_5 x$ and the graph of $y=\\log_5 (x + 4)$. The distance between the points of intersection is $0.5$. Given that $k = a + \\sqrt{b}$, where $a$ and $b$ are integers, what is $a+b$?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 2093,
"solution": "1. **Identify the points of intersection**: The line $x=k$ intersects the graphs of $y=\\log_5 x$ and $y=\\log_5 (x+4)$. The points of intersection are $(k, \\log_5 k)$ and $(k, \\log_5 (k+4))$ respectively.\n\n2. **Set up the equation for the distance between the points**: The distanc... |
DeepScaleR | [
{
"role": "user",
"content": "A line $y = -2$ intersects the graph of $y = 5x^2 + 2x - 6$ at points $C$ and $D$. Find the distance between points $C$ and $D$, expressed in the form $\\frac{\\sqrt{p}}{q}$ where $p$ and $q$ are positive coprime integers. What is $p - q$?"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 2094,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line \\( l \\) passes through the focus \\( F \\) of the parabola \\( y^2 = 4x \\) and intersects the parabola at points \\( A \\) and \\( B \\). Point \\( M \\) is given as \\( (4,0) \\). Extending \\( AM \\) and \\( BM \\) intersects the parabola again at points \\( C \\) a... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 2095,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line \\( l \\), parallel to the diagonal \\( AC_1 \\) of a unit cube \\( ABCDA_1B_1C_1D_1 \\), is equidistant from the lines \\( BD \\), \\( A_1D_1 \\), and \\( CB_1 \\). Find the distances from the line \\( l \\) to these lines."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{6}"
} | {
"index": 2096,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line \\( y = -\\frac{2}{3}x + 6 \\) crosses the \\( x \\)-axis at \\( P \\) and the \\( y \\)-axis at \\( Q \\). Point \\( T(r,s) \\) is on the line segment \\( PQ \\). If the area of \\( \\triangle POQ \\) is four times the area of \\( \\triangle TOP \\), what is the value o... | math | {
"style": "rule",
"ground_truth": "8.25"
} | {
"index": 2097,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line connects points $(2,1)$ and $(7,3)$ on a square that has vertices at $(2,1)$, $(7,1)$, $(7,6)$, and $(2,6)$. What fraction of the area of the square is above this line?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"index": 2098,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A line containing the points $(2, 16)$ and $(-8, -4)$ intersects the $y$-axis at what point?"
}
] | math | {
"style": "rule",
"ground_truth": "(0, 12)"
} | {
"index": 2099,
"solution": ""
} |
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