data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "An architect is building a structure that will place vertical pillars at the vertices of regular hexagon $ABCDEF$, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of pillars at $A$, ... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 4700,
"solution": "1. **Establishing the Coordinate System and Points**:\n Let's assume the side length of the hexagon is $6$ meters for simplicity. We place the hexagon in a 3D coordinate system with $A$ at the origin, i.e., $A = (0, 0, 0)$. The coordinates of $B$ and $C$ can be calculated based on th... |
DeepScaleR | [
{
"role": "user",
"content": "An arithmetic sequence consists of $ 200$ numbers that are each at least $ 10$ and at most $ 100$. The sum of the numbers is $ 10{,}000$. Let $ L$ be the least possible value of the $ 50$th term and let $ G$ be the greatest possible value of the $ 50$th term. What is the val... | math | {
"style": "rule",
"ground_truth": "\\frac{8080}{199}"
} | {
"index": 4701,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An arithmetic sequence consists of positive terms, with the sum of the first $n$ terms denoted by $S_n$, satisfying $2S_2 = a_2(a_2 + 1)$, and given that $a_1 = 1$, find the minimum value of $\\frac{2S_n + 13}{n}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{33}{4}"
} | {
"index": 4702,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous term. For example, $2,5,8,11,14$ is an arithmetic sequence with five terms, in which the first term is $2$ and the constant added is $3$. Each row and each c... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 4703,
"solution": "\n#### Step 1: Understanding the Problem\nThe problem states that each row and column in a $5 \\times 5$ array forms an arithmetic sequence. The center square is labeled $X$. We need to determine the value of $X$.\n\n#### Step 2: Analyzing Given Information\nWe are given specific value... |
DeepScaleR | [
{
"role": "user",
"content": "An arithmetic sequence with first term $1$ has a common difference of $6$. A second sequence begins with $4$ and has a common difference of $7$. In the range of $1$ to $100$, what is the largest number common to both sequences?"
}
] | math | {
"style": "rule",
"ground_truth": "67"
} | {
"index": 4704,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An arithmetic sequence {a_n} has a sum of the first n terms as S_n, and S_6/S_3 = 4. Find the value of S_9/S_6."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{9}{4}"
} | {
"index": 4705,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An array of integers is arranged in a grid of 7 rows and 1 column with eight additional squares forming a separate column to the right. The sequence of integers in the main column of squares and in each of the two rows form three distinct arithmetic sequences. Find the value of... | math | {
"style": "rule",
"ground_truth": "\\frac{-851}{3}"
} | {
"index": 4706,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An artist arranges 1000 dots evenly around a circle, with each dot being either red or blue. A critic counts faults: each pair of adjacent red dots counts as one fault, and each pair of blue dots exactly two apart (separated by one dot) counts as another fault. What is the smal... | math | {
"style": "rule",
"ground_truth": "250"
} | {
"index": 4707,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An artist has $14$ cubes, each with an edge of $1$ meter. She stands them on the ground to form a sculpture as shown. She then paints the exposed surface of the sculpture. How many square meters does she paint?"
}
] | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 4708,
"solution": "1. **Understanding the Structure**: The sculpture is made up of 14 cubes, each with an edge of 1 meter. The cubes are arranged in a pyramid-like structure with three layers. The top layer has 1 cube, the middle layer has 4 cubes, and the bottom layer has 9 cubes.\n\n2. **Calculating Si... |
DeepScaleR | [
{
"role": "user",
"content": "An artist wants to completely cover a rectangle with identically sized squares which do not overlap and do not extend beyond the edges of the rectangle. If the rectangle is \\(60 \\frac{1}{2} \\mathrm{~cm}\\) long and \\(47 \\frac{2}{3} \\mathrm{~cm}\\) wide, what is the minimu... | math | {
"style": "rule",
"ground_truth": "858"
} | {
"index": 4709,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An assembly line produces an average of $85\\%$ first-grade products. How many products need to be selected so that with a probability of 0.997, the deviation of the frequency of first-grade products from the probability $p=0.85$ does not exceed $0.01$ in absolute value?"
}
] | math | {
"style": "rule",
"ground_truth": "11171"
} | {
"index": 4710,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An assembly line produces, on average, 85% first grade products. How many products need to be sampled so that, with a probability of 0.997, the deviation of the proportion of first grade products from 0.85 in absolute value does not exceed 0.01?"
}
] | math | {
"style": "rule",
"ground_truth": "11475"
} | {
"index": 4711,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An assortment of 200 pencils is sold through a catalog for $\\$19.90$. Shipping is an additional $\\$6.95$. Including the charges for both the pencils and the shipping, what is the average cost, in cents, for each pencil? Express your answer rounded to the nearest whole number.... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 4712,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An athlete's heart beats an average of 150 times per minute while running. How many times does the athlete's heart beat during a 26-mile race if the athlete runs at a pace of 5 minutes per mile?"
}
] | math | {
"style": "rule",
"ground_truth": "19500"
} | {
"index": 4713,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An athlete's target heart rate, in beats per minute, is $80\\%$ of the theoretical maximum heart rate. The maximum heart rate is found by subtracting the athlete's age, in years, from $220$. To the nearest whole number, what is the target heart rate of an athlete who is $26$ ye... | math | {
"style": "rule",
"ground_truth": "134"
} | {
"index": 4714,
"solution": "1. **Calculate the Maximum Heart Rate**: The maximum heart rate for an athlete is calculated by subtracting the athlete's age from 220. For an athlete who is 26 years old, the calculation is:\n \\[\n \\text{Maximum Heart Rate} = 220 - 26 = 194 \\text{ beats per minute}\n \\]\n\n2... |
DeepScaleR | [
{
"role": "user",
"content": "An auditorium with $20$ rows of seats has $10$ seats in the first row. Each successive row has one more seat than the previous row. If students taking an exam are permitted to sit in any row, but not next to another student in that row, then the maximum number of students tha... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 4715,
"solution": "1. **Understanding the seating arrangement**: The first row has 10 seats, and each subsequent row has one more seat than the previous row. Therefore, the number of seats in the $i$-th row is $10 + (i-1) = 9 + i$.\n\n2. **Calculating the number of seats in each row**: The number of seat... |
DeepScaleR | [
{
"role": "user",
"content": "An author writes a book containing 60,000 words in 100 hours. For the first 20% of the time, the author writes with 50% increased productivity. How many words per hour did the author write on average?"
}
] | math | {
"style": "rule",
"ground_truth": "600"
} | {
"index": 4716,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An automobile travels $a/6$ feet in $r$ seconds. If this rate is maintained for $3$ minutes, how many yards does it travel in $3$ minutes?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10a}{r}"
} | {
"index": 4717,
"solution": "1. **Identify the rate of travel**: The automobile travels $\\frac{a}{6}$ feet in $r$ seconds. Thus, the rate of travel is:\n \\[\n \\text{Rate} = \\frac{\\frac{a}{6} \\text{ feet}}{r \\text{ seconds}}\n \\]\n\n2. **Convert the rate to yards per second**: Since there are 3 feet i... |
DeepScaleR | [
{
"role": "user",
"content": "An easel in a corner hosts three $30 \\text{ cm} \\times 40 \\text{ cm}$ shelves, with equal distances between neighboring shelves. Three spiders resided where the two walls and the middle shelf meet. One spider climbed diagonally up to the corner of the top shelf on one wall, ... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 4718,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An economist and a statistician play a game on a calculator which does only one\noperation. The calculator displays only positive integers and it is used in the following\nway: Denote by $n$ an integer that is shown on the calculator. A person types an integer,\n$m$, chosen fro... | math | {
"style": "rule",
"ground_truth": "951"
} | {
"index": 4719,
"solution": "\nTo solve this problem, we need to understand the specific condition under which the current displayed number \\( n \\) on the calculator can be transformed to another integer through the operation described, where \\( m \\) is chosen from the set \\(\\{1, 2, \\ldots, 99\\}\\).\n\nThe... |
DeepScaleR | [
{
"role": "user",
"content": "An economist-cryptographer received a cryptogram from the ruler that contained a secret decree to introduce a per-unit tax in a certain market. The cryptogram specified the amount of tax revenue that needed to be collected, emphasizing that it was not possible to collect a high... | math | {
"style": "rule",
"ground_truth": "8640"
} | {
"index": 4720,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An edge of a square can be expressed as $4x-15$ meters or as $20-3x$ meters. What is its area in square meters?"
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 4721,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An educational center received an object with a volume of around 150 monoliths (a container designed for 150 monoliths that was almost full). Each monolith is identified either as sandy loam or loam and classified by its genesis as either marine or lake-glacial deposits. The re... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 4722,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An eight-digit integer is formed by repeating a positive four-digit integer. For example, 25,632,563 or 60,786,078 are integers of this form. What is the greatest common divisor of all eight-digit integers of this form?"
}
] | math | {
"style": "rule",
"ground_truth": "10001"
} | {
"index": 4723,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An eight-sided die (numbered 1 through 8) is rolled, and $Q$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $Q$?"
}
] | math | {
"style": "rule",
"ground_truth": "960"
} | {
"index": 4724,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An eight-sided die has its faces numbered from 1 to 8. What is the expected value of the roll of the die?"
}
] | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"index": 4725,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An eight-sided die is rolled seven times. Find the probability of rolling at least a seven at least six times."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{2048}"
} | {
"index": 4726,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An eight-sided die is rolled, and $Q$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $Q$?"
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 4727,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An eight-sided die numbered from 1 to 8 is rolled, and $P$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $P$?"
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 4728,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An eight-sided die numbered from 1 to 8 is rolled, and $Q$ is the product of the seven numbers that are visible. What is the largest number that is certain to divide $Q$?"
}
] | math | {
"style": "rule",
"ground_truth": "192"
} | {
"index": 4729,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An eight-sided die, with faces numbered from 1 to 8, is tossed three times. Given that the sum of the first two tosses equals the third, calculate the probability that at least one \"2\" is tossed."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{28}"
} | {
"index": 4730,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An elastic ball falls from a height of 128m, where each bounce reaches half of its previous height before falling again. Calculate the total distance traveled by the ball when it hits the ground for the 8th time."
}
] | math | {
"style": "rule",
"ground_truth": "382"
} | {
"index": 4731,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An electric car is charged 3 times per week for 52 weeks. The cost to charge the car each time is $0.78. What is the total cost to charge the car over these 52 weeks?"
}
] | math | {
"style": "rule",
"ground_truth": "\\$121.68"
} | {
"index": 4732,
"solution": "Since the car is charged 3 times per week for 52 weeks, it is charged \\(3 \\times 52 = 156\\) times. Since the cost per charge is $0.78, then the total cost is \\(156 \\times 0.78 = 121.68\\)."
} |
DeepScaleR | [
{
"role": "user",
"content": "An electric company installed a total of 402 poles along both sides of the road, with a distance of 20 meters between each adjacent pole. Later, all poles were replaced, and only 202 poles were installed. The distance between each adjacent pole after the replacement is $\\qquad... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 4733,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An electrician was called to repair a garland of four light bulbs connected in series, one of which has burned out. It takes 10 seconds to unscrew any bulb from the garland and 10 seconds to screw it back in. The time spent on other actions is negligible. What is the minimum ti... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 4734,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An electronic clock always displays the date as an eight-digit number. For example, January 1, 2011, is displayed as 20110101. What is the last day of 2011 that can be evenly divided by 101? The date is displayed as $\\overline{2011 \\mathrm{ABCD}}$. What is $\\overline{\\mathr... | math | {
"style": "rule",
"ground_truth": "1221"
} | {
"index": 4735,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An electronic clock displays time from 00:00:00 to 23:59:59. How much time throughout the day does the clock show a number that reads the same forward and backward?"
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"index": 4736,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An electronic flea lands on a point $k$ on the number line. In the first step, it jumps 1 unit to the left to $k_1$, in the second step it jumps 2 units to the right to $k_2$, in the third step it jumps 3 units to the left to $k_3$, in the fourth step it jumps 4 units to the ri... | math | {
"style": "rule",
"ground_truth": "-30.06"
} | {
"index": 4737,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An element is randomly chosen from among the first $15$ rows of Pascal's Triangle. What is the probability that the value of the element chosen is $1$?\n\nNote: The 1 at the top is often labelled the \"zeroth\" row of Pascal's Triangle, by convention. So to count a total of 15... | math | {
"style": "rule",
"ground_truth": "\\frac{29}{120}"
} | {
"index": 4738,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An element is randomly chosen from among the first $20$ rows of Pascal's Triangle. What is the probability that the selected element is $1$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{70}"
} | {
"index": 4739,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An element is randomly chosen from among the first $20$ rows of Pascal's Triangle. What is the probability that the value of the element chosen is $1$?\n\nNote: The 1 at the top is often labelled the \"zeroth\" row of Pascal's Triangle, by convention. So to count a total of 20 ... | math | {
"style": "rule",
"ground_truth": "\\frac{39}{210}"
} | {
"index": 4740,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An element is randomly chosen from among the first $20$ rows of Pascal’s Triangle. What is the probability that the value of the element chosen is $1$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{39}{210}"
} | {
"index": 4741,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An elevator containing 9 passengers can stop at ten different floors. The passengers exit in groups of two, three, and four people. In how many ways can this happen?"
}
] | math | {
"style": "rule",
"ground_truth": "10 * 9 * 8 * 36 * 35"
} | {
"index": 4742,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ellipse $\\frac {x^{2}}{a^{2}}+ \\frac {y^{2}}{b^{2}}=1 (a>b>0)$ has its two foci and the endpoints of its minor axis all lying on the circle $x^{2}+y^{2}=1$. A line $l$ (not perpendicular to the x-axis) passing through the right focus intersects the ellipse at points A and ... | math | {
"style": "rule",
"ground_truth": "2 \\sqrt {2}"
} | {
"index": 4743,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ellipse C is defined by the equation $$\\frac {x^{2}}{a^{2}}+ \\frac {y^{2}}{b^{2}}=1$$ (a > b > 0), and its four vertices form a rhombus with side length $$\\sqrt {3}$$ and area $$2 \\sqrt {2}$$.\n(1) Find the equation of ellipse C.\n(2) A line l passes through point Q(0, -... | math | {
"style": "rule",
"ground_truth": "\\frac {4 \\sqrt {21}}{11}"
} | {
"index": 4744,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ellipse \\( \\frac{x^2}{9} + \\frac{y^2}{16} = 1 \\) has chords passing through the point \\( C = (2, 2) \\). If \\( t \\) is defined as \n\\[\nt = \\frac{1}{AC} + \\frac{1}{BC}\n\\]\nwhere \\( AC \\) and \\( BC \\) are distances from \\( A \\) and \\( B \\) (endpoints of th... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{5}}{5}"
} | {
"index": 4745,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ellipse and a hyperbola have the same foci $F\\_1(-c,0)$, $F\\_2(c,0)$. One endpoint of the ellipse's minor axis is $B$, and the line $F\\_1B$ is parallel to one of the hyperbola's asymptotes. If the eccentricities of the ellipse and hyperbola are $e\\_1$ and $e\\_2$, respec... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"index": 4746,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ellipse has a focus at coordinates $\\left(0,-\\sqrt {2}\\right)$ and is represented by the equation $2x^{2}-my^{2}=1$. Find the value of the real number $m$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\dfrac{2}{5}"
} | {
"index": 4747,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ellipse has a major axis of length 12 and a minor axis of 10. Using one focus as a center, an external circle is tangent to the ellipse. Find the radius of the circle."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{11}"
} | {
"index": 4748,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ellipse has foci $(2, 2)$ and $(2, 6)$, and it passes through the point $(14, -3).$ Given this, we can write the equation of the ellipse in standard form as \\[\\frac{(x-h)^2}{a^2} + \\frac{(y-k)^2}{b^2} = 1,\\]where $a, b, h, k$ are constants, and $a$ and $b$ are positive. ... | math | {
"style": "rule",
"ground_truth": "(8\\sqrt3, 14, 2, 4)"
} | {
"index": 4749,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ellipse has foci at $(0, 2)$ and $(3, 0)$. It has two $x$-intercepts, one of which is the origin. What is the other one? Enter your answer as an ordered pair."
}
] | math | {
"style": "rule",
"ground_truth": "\\left(\\tfrac{15}{4},0\\right)"
} | {
"index": 4750,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ellipse has foci at $(9, 20)$ and $(49, 55)$ in the $xy$-plane and is tangent to the $x$-axis. What is the length of its major axis?"
}
] | math | {
"style": "rule",
"ground_truth": "85"
} | {
"index": 4751,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ellipse has foci at $(9,20)$ and $(49,55)$ in the $xy$-plane and is tangent to the $x$-axis. What is the length of its major axis?"
}
] | math | {
"style": "rule",
"ground_truth": "85"
} | {
"index": 4752,
"solution": "An ellipse is defined as the set of points where the sum of the distances from the foci to the point is fixed. The length of major axis is equal to the sum of these distances $(2a)$. Thus if we find the sum of the distances, we get the answer. Let k be this fixed sum; then we get, by t... |
DeepScaleR | [
{
"role": "user",
"content": "An ellipse has foci at $F_1 = (0,2)$ and $F_2 = (3,0).$ The ellipse intersects the $x$-axis at the origin, and one other point. What is the other point of intersection?"
}
] | math | {
"style": "rule",
"ground_truth": "\\left( \\frac{15}{4}, 0 \\right)"
} | {
"index": 4753,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ellipse has its foci at $(-1, -1)$ and $(-1, -3).$ Given that it passes through the point $(4, -2),$ its equation can be written in the form \\[\\frac{(x-h)^2}{a^2} + \\frac{(y-k)^2}{b^2} = 1\\]where $a, b, h, k$ are constants, and $a$ and $b$ are positive. Find $a+k.$"
}
... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 4754,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ellipse is defined parametrically by\n\\[(x,y) = \\left( \\frac{2 (\\sin t - 1)}{2 - \\cos t}, \\frac{3 (\\cos t - 5)}{2 - \\cos t} \\right).\\]Then the equation of the ellipse can be written in the form\n\\[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,\\]where $A,$ $B,$ $C,$ $D,$ $E... | math | {
"style": "rule",
"ground_truth": "1381"
} | {
"index": 4755,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ellipse is drawn with major and minor axes of lengths 10 and 8 respectively. Using one focus as a center, a circle is drawn that is tangent to the ellipse, with no part of the circle being outside the ellipse. Compute the radius of the circle."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 4756,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ellipse satisfies the property that a light ray emitted from one focus of the ellipse, after reflecting off the ellipse, will pass through the other focus. Consider a horizontally placed elliptical billiards table that satisfies the equation $\\frac{x^2}{16} + \\frac{y^2}{9}... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 4757,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ellipse whose axes are parallel to the coordinate axes is tangent to the $x$-axis at $(4, 0)$ and tangent to the $y$-axis at $(0, 1).$ Find the distance between the foci of the ellipse."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{15}"
} | {
"index": 4758,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ellipse whose axes are parallel to the coordinate axes is tangent to the $x$-axis at $(6, 0)$ and tangent to the $y$-axis at $(0, 2)$. Find the distance between the foci of the ellipse."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{2}"
} | {
"index": 4759,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ellipse with equation\n\\[\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1\\]contains the circles $(x - 1)^2 + y^2 = 1$ and $(x + 1)^2 +y^2 = 1.$ Then the smallest possible area of the ellipse can be expressed in the form $k \\pi.$ Find $k.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{3}}{2}"
} | {
"index": 4760,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An employee arrives at the unit randomly between 7:50 and 8:30. Calculate the probability that he can clock in on time."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"index": 4761,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An employee receives an average of two requests per hour. Assuming a simple flow of requests, what is the probability of receiving four requests in four hours?"
}
] | math | {
"style": "rule",
"ground_truth": "0.0572"
} | {
"index": 4762,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An employer hired a worker for a year and promised to give him 12 rubles and a coat. However, the worker wanted to leave after 7 months. Upon settlement, he received the coat and 5 rubles. How much did the coat cost?"
}
] | math | {
"style": "rule",
"ground_truth": "4.8"
} | {
"index": 4763,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An empty $2020 \\times 2020 \\times 2020$ cube is given, and a $2020 \\times 2020$ grid of square unit cells is drawn on each of its six faces. A [i]beam[/i] is a $1 \\times 1 \\times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following co... | math | {
"style": "rule",
"ground_truth": "3030"
} | {
"index": 4764,
"solution": "\nTo address this problem, we need to determine the smallest number of beams that can be placed inside a \\(2020 \\times 2020 \\times 2020\\) cube such that they satisfy the given conditions: they must be \\(1 \\times 1 \\times 2020\\) and can only touch the faces of the cube or each o... |
DeepScaleR | [
{
"role": "user",
"content": "An empty $2020 \\times 2020 \\times 2020$ cube is given, and a $2020 \\times 2020$ grid of square unit cells is drawn on each of its six faces. A beam is a $1 \\times 1 \\times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions... | math | {
"style": "rule",
"ground_truth": "\\[\n3030\n\\]"
} | {
"index": 4765,
"solution": "Take one vertex of the cube as origin and establish 3D coordinates along the cube's edges.\nDefine a beam as $x-dir$ if its long edge is parallel to x-axis. Similarly for $y-dir$ and $z-dir$ .\nDefine a beam's location as (direction, ( $1 \\times 1$ face's location in 2D coordinate).\n... |
DeepScaleR | [
{
"role": "user",
"content": "An engineer invested $\\$10,\\!000$ in a six-month savings certificate that paid a simple annual interest rate of $12\\%$. After six months, she invested the total value of her investment in another six-month certificate. After six more months, the investment was worth $\\$11,\... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 4766,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An engineer invested $\\$15,\\!000$ in a nine-month savings certificate that paid a simple annual interest rate of $9\\%$. After nine months, she invested the total value of her investment in another nine-month certificate. After another nine months, the investment was worth $\... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 4767,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An engraver makes plates with letters. He engraves the same letters in the same amount of time, but different letters may take different times. On two plates, \"ДОМ МОДЫ\" (DOM MODY) and \"ВХОД\" (VKHOD), together he spent 50 minutes, and one plate \"В ДЫМОХОД\" (V DYMOHOD) too... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 4768,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An enterprise has four employees participating in a vocational skills assessment. Each employee can draw any one of the four available assessment projects to participate in. Calculate the probability that exactly one project is not selected."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{16}"
} | {
"index": 4769,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An entire floor is tiled with blue and white tiles. The floor has a repeated tiling pattern that forms every $8 \\times 8$ square. Each of the four corners of this square features an asymmetrical arrangement of tiles where the bottom left $4 \\times 4$ segment within each $8 \\... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 4770,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An entrepreneur invested $\\$20,\\!000$ in a nine-month term deposit that paid a simple annual interest rate of $8\\%$. After the term ended, she reinvested all the proceeds into another nine-month term deposit. At the end of the second term, her total investment had grown to $... | math | {
"style": "rule",
"ground_truth": "7.840"
} | {
"index": 4771,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An entrepreneur invested \\$12,000 in a three-month savings certificate that paid a simple annual interest rate of $8\\%$. After three months, she invested the total value of her investment in another three-month certificate. After three more months, the investment was worth \\... | math | {
"style": "rule",
"ground_truth": "24\\%"
} | {
"index": 4772,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An entrepreneur took out a discounted loan of 12 million HUF with a fixed annual interest rate of 8%. What will be the debt after 10 years if they can repay 1.2 million HUF annually?"
}
] | math | {
"style": "rule",
"ground_truth": "8523225"
} | {
"index": 4773,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An entry in a grid is called a saddle point if it is the largest number in its row and the smallest number in its column. Suppose that each cell in a $3 \\times 3$ grid is filled with a real number, each chosen independently and uniformly at random from the interval $[0,1]$. Co... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{10}"
} | {
"index": 4774,
"solution": "With probability 1, all entries of the matrix are unique. If this is the case, we claim there can only be one saddle point. To see this, suppose $A_{i j}$ and $A_{k l}$ are both saddle points. They cannot be in the same row, since they cannot both be the greatest number in the same row... |
DeepScaleR | [
{
"role": "user",
"content": "An environmental agency decides to expand its monitoring teams due to new regulations requiring more extensive testing. They estimate needing 120 new employees to monitor water pollution and 105 new employees to monitor air pollution. Additionally, they need 65 new employees ca... | math | {
"style": "rule",
"ground_truth": "160"
} | {
"index": 4775,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An environmental agency needs to hire a number of new employees so that 85 of the new employees will be able to monitor water pollution, 73 of the new employees will be able to monitor air pollution, and exactly 27 of the new employees will be able to monitor both. (These 27 ar... | math | {
"style": "rule",
"ground_truth": "131"
} | {
"index": 4776,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An equiangular hexagon has side lengths $1,1, a, 1,1, a$ in that order. Given that there exists a circle that intersects the hexagon at 12 distinct points, we have $M<a<N$ for some real numbers $M$ and $N$. Determine the minimum possible value of the ratio $\\frac{N}{M}$."
}
... | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{3}+3}{2}"
} | {
"index": 4777,
"solution": "We claim that the greatest possible value of $M$ is $\\sqrt{3}-1$, whereas the least possible value of $N$ is 3 . To begin, note that the condition requires the circle to intersect each side of the hexagon at two points on its interior. This implies that the center must be inside the h... |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral hexagon with side length 1 has interior angles $90^{\\circ}, 120^{\\circ}, 150^{\\circ}, 90^{\\circ}, 120^{\\circ}, 150^{\\circ}$ in that order. Find its area."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3+\\sqrt{3}}{2}"
} | {
"index": 4778,
"solution": "The area of this hexagon is the area of a $\\frac{3}{2} \\times\\left(1+\\frac{\\sqrt{3}}{2}\\right)$ rectangle (with the $90^{\\circ}$ angles of the hexagon at opposite vertices) minus the area of an equilateral triangle with side length 1. Then this is $$\\frac{6+3 \\sqrt{3}}{4}-\\fr... |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral pentagon $AMNPQ$ is inscribed in triangle $ABC$ such that $M\\in\\overline{AB}$ , $Q\\in\\overline{AC}$ , and $N,P\\in\\overline{BC}$ .\n \nSuppose that $ABC$ is an equilateral triangle of side length $2$ , and that $AMNPQ$ has a line of symmetry per... | math | {
"style": "rule",
"ground_truth": "5073"
} | {
"index": 4779,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle $ABC$ is divided by nine lines parallel to $BC$ into ten bands that are equally wide. We colour the bands alternately red and blue, with the smallest band coloured red. The difference between the total area in red and the total area in blue is $20$ ... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 4780,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle $ABC$ shares a side with a square $BCDE$ . If the resulting pentagon has a perimeter of $20$ , what is the area of the pentagon? (The triangle and square do not overlap)."
}
] | math | {
"style": "rule",
"ground_truth": "16 + 4\\sqrt{3}"
} | {
"index": 4781,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle $ABC$ has an area of $27\\sqrt{3}$. The rays trisecting $\\angle BAC$ intersect side $BC$ at points $D$ and $E$. Find the area of $\\triangle ADE$."
}
] | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3}"
} | {
"index": 4782,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle $ABC$ is inscribed in the ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ such that vertex $B$ is at $(0,b)$, and side $\\overline{AC}$ is parallel to the $x$-axis. The foci $F_1$ and $F_2$ of the ellipse lie on sides $\\overline{BC}$ and $\\overline{A... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{5}"
} | {
"index": 4783,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle $ABC$ shares a common side $BC$ with a square $BCDE,$ as pictured. What is the number of degrees in $\\angle DAE$ (not pictured)? [asy]\npair pA, pB, pC, pD, pE;\npA = (0, 0);\npB = pA + dir(300);\npC = pA + dir(240);\npD = pC + dir(270);\npE = pB + dir(... | math | {
"style": "rule",
"ground_truth": "30^\\circ."
} | {
"index": 4784,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle $PQR$ is inscribed in the ellipse $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1,$ so that $Q$ is at $(0,b),$ and $\\overline{PR}$ is parallel to the $x$-axis, as shown below. Also, foci $F_1$ and $F_2$ lie on sides $\\overline{QR}$ and $\\overline{PQ},$ resp... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{5}"
} | {
"index": 4785,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle ABC has a side length of 4. A right isosceles triangle DBE, where $DB=EB=1$ and angle $D\\hat{B}E = 90^\\circ$, is cut from triangle ABC. Calculate the perimeter of the remaining quadrilateral."
}
] | math | {
"style": "rule",
"ground_truth": "10 + \\sqrt{2}"
} | {
"index": 4786,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle \\( ABC \\) is inscribed in the ellipse \\( \\frac{x^2}{p^2} + \\frac{y^2}{q^2} = 1 \\), such that vertex \\( B \\) is at \\( (0, q) \\), and \\( \\overline{AC} \\) is parallel to the \\( x \\)-axis. The foci \\( F_1 \\) and \\( F_2 \\) of the ellipse li... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{5}"
} | {
"index": 4787,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle and a circle intersect so that each side of the triangle contains a chord of the circle equal in length to the radius of the circle. What is the ratio of the area of the triangle to the area of the circle? Express your answer as a common fraction in term... | math | {
"style": "rule",
"ground_truth": "\\frac{9\\sqrt{3}}{4\\pi}"
} | {
"index": 4788,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle and a regular hexagon have equal perimeters. If the triangle's area is 4, what is the area of the hexagon?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 4789,
"solution": "1. **Assign Variables:**\n Let the side length of the equilateral triangle be $s$ and the side length of the regular hexagon be $t$.\n\n2. **Equal Perimeters:**\n Since the triangle and the hexagon have equal perimeters, and the triangle has 3 sides while the hexagon has 6 sides, w... |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle and a square are inscribed in a circle as shown. $ABC$ is isosceles. The triangle and square share a common vertex. What is the number of degrees in the measure of the angle indicated by the question mark?\n\n[asy]\n\nimport markers; defaultpen(linewidth... | math | {
"style": "rule",
"ground_truth": "75^\\circ"
} | {
"index": 4790,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle and a square both have perimeters of 48 inches. What is the ratio of the length of the side of the triangle to the length of the side of the square? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac43"
} | {
"index": 4791,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle and a square have the same perimeter of 12 inches. What is the ratio of the side length of the triangle to the side length of the square? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 4792,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle has an area of $64\\sqrt{3}$ $\\text{cm}^2$. If each side of the triangle is decreased by 4 cm, by how many square centimeters is the area decreased?"
}
] | math | {
"style": "rule",
"ground_truth": "28\\sqrt{3}"
} | {
"index": 4793,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle has sides 8 units long. An equilateral triangle with sides 4 units long is cut off at the top, leaving an isosceles trapezoid. What is the ratio of the area of the smaller triangle to the area of the trapezoid? Express your answer as a common fraction."
... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 4794,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle has sides of length 2 units. A second equilateral triangle is formed having sides that are $150\\%$ of the length of the sides of the first triangle. A third equilateral triangle is formed having sides that are $150\\%$ of the length of the sides of the ... | math | {
"style": "rule",
"ground_truth": "237.5\\%"
} | {
"index": 4795,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle has two vertices at $(0,5)$ and $(8,5)$. If the third vertex is in the first quadrant, what is the y-coordinate? Express your answer in simplest radical form. [asy]\ndraw((-1,0)--(11,0),Arrows);\ndraw((0,-1)--(0,12),Arrows);\nfor(int i=0;i<11;++i)\n{draw... | math | {
"style": "rule",
"ground_truth": "5+4\\sqrt{3}"
} | {
"index": 4796,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle is drawn with a side of length $a$. A new equilateral triangle is formed by joining the midpoints of the sides of the first one. Then a third equilateral triangle is formed by joining the midpoints of the sides of the second; and so on forever. The limit... | math | {
"style": "rule",
"ground_truth": "6a"
} | {
"index": 4797,
"solution": "1. **Identify the sequence of perimeters**: \n - The perimeter of the first equilateral triangle is $3a$ since each side is $a$ and there are three sides.\n - Each subsequent triangle is formed by joining the midpoints of the sides of the previous triangle, thus each side of the ne... |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle is inscribed in a circle. A smaller equilateral triangle has one vertex coinciding with a vertex of the larger triangle and another vertex on the midpoint of a side of the larger triangle. What percent of the area of the larger triangle is the area of th... | math | {
"style": "rule",
"ground_truth": "25\\%"
} | {
"index": 4798,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An equilateral triangle is inscribed in the ellipse whose equation is $x^2+4y^2=4$. One vertex of the triangle is $(0,1)$, one altitude is contained in the y-axis, and the square of the length of each side is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integ... | math | {
"style": "rule",
"ground_truth": "937"
} | {
"index": 4799,
"solution": "Solving for $y$ in terms of $x$ gives $y=\\sqrt{4-x^2}/2$, so the two other points of the triangle are $(x,\\sqrt{4-x^2}/2)$ and $(-x,\\sqrt{4-x^2}/2)$, which are a distance of $2x$ apart. Thus $2x$ equals the distance between $(x,\\sqrt{4-x^2}/2)$ and $(0,1)$, so by the distance formu... |
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