data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "As $n$ ranges over the positive integers, what is the maximum possible value for the greatest common divisor of $11n+3$ and $6n+1$?"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 5100,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As $n$ ranges over the positive integers, what is the sum of all possible values of the greatest common divisor of $3n+4$ and $n$?"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 5101,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As $p$ ranges over the primes greater than $5$, how many different remainders can $p^2$ leave upon division by $120$?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 5102,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As $t$ takes on all real values, the set of points $(x,y)$ defined by\n\\begin{align*}\nx &= t^2 - 2, \\\\\ny &= t^3 - 9t + 5\n\\end{align*}forms a curve that crosses itself. Compute the ordered pair $(x,y)$ where this crossing occurs."
}
] | math | {
"style": "rule",
"ground_truth": "(7,5)"
} | {
"index": 5103,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As $x$ ranges over all real numbers, find the range of\n\\[f(x) = \\sin^4 x + \\cos ^2 x.\\]Enter your answer using interval notation."
}
] | math | {
"style": "rule",
"ground_truth": "\\left[ \\frac{3}{4}, 1 \\right]"
} | {
"index": 5104,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As Emily is riding her bicycle on a long straight road, she spots Emerson skating in the same direction $1/2$ mile in front of her. After she passes him, she can see him in her rear mirror until he is $1/2$ mile behind her. Emily rides at a constant rate of $12$ miles per hour,... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 5105,
"solution": "1. **Determine the relative speed of Emily and Emerson**: Emily rides at $12$ mph and Emerson skates at $8$ mph. Since they are moving in the same direction, the relative speed of Emily with respect to Emerson is:\n \\[\n 12 \\text{ mph} - 8 \\text{ mph} = 4 \\text{ mph}\n \\]\n\... |
DeepScaleR | [
{
"role": "user",
"content": "As a prank, Tim decides to steal Nathan's fork at dinner, but so he doesn't get caught, he convinces other people to do it for him. On Monday, he convinces Joe to do it. On Tuesday, he could get either Ambie or John to do it. On Wednesday, he can't convince any of those three p... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 5106,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As a result of five measurements of the rod's length with one device (without systematic errors), the following results (in mm) were obtained: $92, 94, 103, 105, 106$. Find: a) the sample mean length of the rod; b) the sample variance and the unbiased corrected variance of the ... | math | {
"style": "rule",
"ground_truth": "42.5"
} | {
"index": 5107,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As a result of measuring the four sides and one of the diagonals of a certain quadrilateral, the following numbers were obtained: $1 ; 2 ; 2.8 ; 5 ; 7.5$. What is the length of the measured diagonal?"
}
] | math | {
"style": "rule",
"ground_truth": "2.8"
} | {
"index": 5108,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As part of his effort to take over the world, Edward starts producing his own currency. As part of an effort to stop Edward, Alex works in the mint and produces 1 counterfeit coin for every 99 real ones. Alex isn't very good at this, so none of the counterfeit coins are the rig... | math | {
"style": "rule",
"ground_truth": "\\frac{19}{28}"
} | {
"index": 5109,
"solution": "$5 \\%$ of the coins are sent to the lab, and only $.95 \\%$ of the coins are sent to the lab and counterfeit, so there is a $19 \\%$ chance that a coin sent to the lab is counterfeit and an $81 \\%$ chance that it is real. The lab could correctly detect a counterfeit coin or falsely a... |
DeepScaleR | [
{
"role": "user",
"content": "As shown below, there is a $40\\times30$ paper with a filled $10\\times5$ rectangle inside of it. We want to cut out the filled rectangle from the paper using four straight cuts. Each straight cut is a straight line that divides the paper into two pieces, and we keep the piece ... | math | {
"style": "rule",
"ground_truth": "65"
} | {
"index": 5110,
"solution": "\nWe are given a \\( 40 \\times 30 \\) rectangle (the paper) with a filled \\( 10 \\times 5 \\) rectangle inside it. The objective is to cut out the filled rectangle using four straight cuts with the aim of minimizing the total length of the cuts. Each cut divides the remaining paper i... |
DeepScaleR | [
{
"role": "user",
"content": "As shown in the diagram, $E$ is the midpoint of the leg $AB$ of trapezoid $ABCD$. $DF \\perp EC$, $DF=10$, and $EC=24$. Find the area of trapezoid $ABCD$."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 5111,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As shown in the diagram, A and B are the endpoints of the diameter of a circular track. Three miniature robots, labeled as J, Y, and B, start simultaneously and move uniformly along the circular track. Robots J and Y start from point A, while robot B starts from point B. Robot ... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 5112,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As shown in the diagram, \\( D \\), \\( E \\), and \\( F \\) are points on the sides \\( BC \\), \\( CA \\), and \\( AB \\) of \\(\\triangle ABC\\), respectively, and \\( AD \\), \\( BE \\), \\( CF \\) intersect at point \\( G \\). Given that the areas of \\(\\triangle BDG\\), ... | math | {
"style": "rule",
"ground_truth": "63"
} | {
"index": 5113,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As shown in the diagram, \\(E, F, G, H\\) are the midpoints of the sides \\(AB, BC, CD, DA\\) of the quadrilateral \\(ABCD\\). The intersection of \\(BH\\) and \\(DE\\) is \\(M\\), and the intersection of \\(BG\\) and \\(DF\\) is \\(N\\). What is \\(\\frac{S_{\\mathrm{BMND}}}{S... | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"index": 5114,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As shown in the diagram, \\(FGHI\\) is a trapezium with side \\(GF\\) parallel to \\(HI\\). The lengths of \\(FG\\) and \\(HI\\) are 50 and 20 respectively. The point \\(J\\) is on the side \\(FG\\) such that the segment \\(IJ\\) divides the trapezium into two parts of equal ar... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 5115,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As shown in the diagram, a cube with a side length of 12 cm is cut once. The cut is made along \\( IJ \\) and exits through \\( LK \\), such that \\( AI = DL = 4 \\) cm, \\( JF = KG = 3 \\) cm, and the section \\( IJKL \\) is a rectangle. The total surface area of the two resul... | math | {
"style": "rule",
"ground_truth": "1176"
} | {
"index": 5116,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As shown in the diagram, a square is divided into 4 identical rectangles, each of which has a perimeter of 20 centimeters. What is the area of this square?"
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 5117,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As shown in the diagram, circles \\( \\odot O_{1} \\) and \\( \\odot O_{2} \\) are externally tangent. The line segment \\( O_{1}O_{2} \\) intersects \\( \\odot O_{1} \\) at points \\( A \\) and \\( B \\), and intersects \\( \\odot O_{2} \\) at points \\( C \\) and \\( D \\). C... | math | {
"style": "rule",
"ground_truth": "1.2"
} | {
"index": 5118,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As shown in the diagram, four small plates \\( A, B, C, D \\) are arranged in a circular shape, with an unspecified number of candies placed on each plate. In each move, it is allowed to take all candies from 1, 3, or 4 plates, or from 2 adjacent plates. What is the maximum num... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 5119,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As shown in the diagram, in the square \\(ABCD\\), \\(AB = 2\\). Draw an arc with center \\(C\\) and radius equal to \\(CD\\), and another arc with center \\(B\\) and radius equal to \\(BA\\). The two arcs intersect at \\(E\\). What is the area of the sector \\(BAE\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"index": 5120,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As shown in the diagram, in the tetrahedron \\(A B C D\\), the face \\(A B C\\) intersects the face \\(B C D\\) at a dihedral angle of \\(60^{\\circ}\\). The projection of vertex \\(A\\) onto the plane \\(B C D\\) is \\(H\\), which is the orthocenter of \\(\\triangle B C D\\). ... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{21}}{9}"
} | {
"index": 5121,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As shown in the diagram, plane $ABDE$ is perpendicular to plane $ABC$. Triangle $ABC$ is an isosceles right triangle with $AC=BC=4$. Quadrilateral $ABDE$ is a right trapezoid with $BD \\parallel AE$, $BD \\perp AB$, $BD=2$, and $AE=4$. Points $O$ and $M$ are the midpoints of $C... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{30}}{10}"
} | {
"index": 5122,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As shown in the diagram, points \\(E\\), \\(F\\), \\(G\\), and \\(H\\) are located on the four sides of the square \\(ABCD\\) as the trisection points closer to \\(B\\), \\(C\\), \\(D\\), and \\(A\\) respectively. If the side length of the square is 6, what is the area of the s... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 5123,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As shown in the diagram, rectangle \\(ABCD\\) is inscribed in a semicircle, with \\(EF\\) as the diameter of the semicircle. Given that \\(DA = 16\\), and \\(FD = AE = 9\\), find the area of rectangle \\(ABCD\\)."
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 5124,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As shown in the diagram, square ABCD and square EFGH have their corresponding sides parallel to each other. Line CG is extended to intersect with line BD at point I. Given that BD = 10, the area of triangle BFC is 3, and the area of triangle CHD is 5, what is the length of BI?"... | math | {
"style": "rule",
"ground_truth": "15/4"
} | {
"index": 5125,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As shown in the diagram, squares \\( a \\), \\( b \\), \\( c \\), \\( d \\), and \\( e \\) are used to form a rectangle that is 30 cm long and 22 cm wide. Find the area of square \\( e \\)."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 5126,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As shown in the diagram, the area of parallelogram \\(ABCD\\) is 60. The ratio of the areas of \\(\\triangle ADE\\) and \\(\\triangle AEB\\) is 2:3. Find the area of \\(\\triangle BEF\\)."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 5127,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As shown in the diagram, there are 12 points on the circumference of a circle, dividing the circumference into 12 equal parts. How many rectangles can be formed using these equally divided points as the four vertices?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 5128,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As shown in the diagram, there is a sequence of curves \\(P_{0}, P_{1}, P_{2}, \\cdots\\). It is given that \\(P_{0}\\) is an equilateral triangle with an area of 1. Each \\(P_{k+1}\\) is obtained from \\(P_{k}\\) by performing the following operations: each side of \\(P_{k}\\)... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{5}"
} | {
"index": 5129,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As shown in the diagram, three circles intersect to create seven regions. Fill the integers $0 \\sim 6$ into the seven regions such that the sum of the four numbers within each circle is the same. What is the maximum possible value of this sum?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 5130,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As shown in the figure below, a circular park consists of an outer-ring path for walkers (white) and a ring-shaped flower garden (gray) surrounding a central circular fountain (black). The walking path is six feet wide in all places, the garden ring is eight feet wide in all pl... | math | {
"style": "rule",
"ground_truth": "38"
} | {
"index": 5131,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As shown in the figure below, a regular dodecahedron (the polyhedron consisting of $12$ congruent regular pentagonal faces) floats in space with two horizontal faces. Note that there is a ring of five slanted faces adjacent to the top face, and a ring of five slanted faces adja... | math | {
"style": "rule",
"ground_truth": "810"
} | {
"index": 5132,
"solution": "To solve this problem, we need to consider the number of ways to move from the top face to the bottom face via a sequence of adjacent faces, visiting each face at most once, and without moving from the bottom ring to the top ring. We will use casework based on the number of moves made ... |
DeepScaleR | [
{
"role": "user",
"content": "As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\\angle CDE = 110^\\circ$. Point $F$ lies on $\\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. What is the degree measure of $\\angle AFE$?"
}
] | math | {
"style": "rule",
"ground_truth": "170"
} | {
"index": 5133,
"solution": "1. **Identify the given information and setup**: We know that $ABCD$ is a square, and $E$ lies on the opposite half-plane determined by line $CD$ from point $A$. The angle $\\angle CDE = 110^\\circ$. Point $F$ lies on $\\overline{AD}$ such that $DE = DF$.\n\n2. **Extend $\\overline{AD}... |
DeepScaleR | [
{
"role": "user",
"content": "As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length 2 so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region ---- inside the hexagon but outside all of the semici... | math | {
"style": "rule",
"ground_truth": "6\\sqrt{3} - 3\\pi"
} | {
"index": 5134,
"solution": "1. **Identify the Geometry of the Problem:**\n - The problem involves a regular hexagon with side length 2.\n - Six semicircles are inscribed such that their diameters coincide with the sides of the hexagon.\n\n2. **Calculate the Area of the Regular Hexagon:**\n - The formula for... |
DeepScaleR | [
{
"role": "user",
"content": "As shown in the figure, triangle $ABC$ is divided into six smaller triangles by lines drawn from the vertices through a common interior point. The areas of four of these triangles are as indicated. Find the area of triangle $ABC$."
}
] | math | {
"style": "rule",
"ground_truth": "315"
} | {
"index": 5135,
"solution": "This problem can be done using mass points. Assign B a weight of 1 and realize that many of the triangles have the same altitude. After continuously using the formulas that state (The sum of the two weights) = (The middle weight), and (The weight $\\times$ side) = (Other weight) $\\tim... |
DeepScaleR | [
{
"role": "user",
"content": "As shown in the following figure, a heart is a shape consist of three semicircles with diameters $AB$, $BC$ and $AC$ such that $B$ is midpoint of the segment $AC$. A heart $\\omega$ is given. Call a pair $(P, P')$ bisector if $P$ and $P'$ lie on $\\omega$ and bisect its perimet... | math | {
"style": "rule",
"ground_truth": "60^\\circ"
} | {
"index": 5136,
"solution": "\nTo approach this problem, we will analyze the geometric properties and symmetrical nature of the heart shape and the properties of the cyclic quadrilateral \\(XYZT\\).\n\n1. **Understanding the Geometry of the Heart Shape:**\n - The heart shape is constructed from three semicircles... |
DeepScaleR | [
{
"role": "user",
"content": "As shown in the picture, the knight can move to any of the indicated squares of the $8 \\times 8$ chessboard in 1 move. If the knight starts from the position shown, find the number of possible landing positions after 20 consecutive moves."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 5137,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As the Chinese New Year approaches, workers from a factory begin to go home to reunite with their families starting from Monday, January 17, 2011. If the number of workers leaving the factory each day is the same, and by January 31, 121 workers remain in the factory, while the ... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 5138,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As usual, let \\( n! \\) denote the product of the integers from 1 to \\( n \\) inclusive. Determine the largest integer \\( m \\) such that \\( m! \\) divides \\( 100! + 99! + 98! \\)."
}
] | math | {
"style": "rule",
"ground_truth": "98"
} | {
"index": 5139,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "As we enter the autumn and winter seasons, the air becomes dry. A certain appliance store is preparing to purchase a batch of humidifiers. The cost price of each unit is $80$ yuan. After market research, the selling price is set at $100$ yuan per unit. The store can sell $500$ ... | math | {
"style": "rule",
"ground_truth": "12250"
} | {
"index": 5140,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Ashley, Betty, Carlos, Dick, and Elgin went shopping. Each had a whole number of dollars to spend, and together they had $56$ dollars. The absolute difference between the amounts Ashley and Betty had to spend was $19$ dollars. The absolute difference between the amounts Betty a... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 5141,
"solution": "Let's denote the amount of money each person has as follows:\n- Ashley: $A$\n- Betty: $B$\n- Carlos: $C$\n- Dick: $D$\n- Elgin: $E$\n\nFrom the problem, we have the following equations based on the absolute differences:\n1. $|A - B| = 19$\n2. $|B - C| = 7$\n3. $|C - D| = 5$\n4. $|D - E... |
DeepScaleR | [
{
"role": "user",
"content": "Assign five volunteers randomly to three different positions A, B, and C for service, with each position having at least one volunteer. \n(1) Calculate the probability that exactly two volunteers are assigned to position A. \n(2) Let the random variable $\\xi$ represent the n... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"index": 5142,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Assume $n$ is a positive integer. Considers sequences $a_0, a_1, \\ldots, a_n$ for which $a_i \\in \\{1, 2, \\ldots , n\\}$ for all $i$ and $a_n = a_0$. \r\n\r\n(a) Suppose $n$ is odd. Find the number of such sequences if $a_i - a_{i-1} \\not \\equiv i \\pmod{n}$ for all $i = 1... | math | {
"style": "rule",
"ground_truth": "(n-1)(n-2)^{n-1} - \\frac{2^{n-1} - 1}{n} - 1"
} | {
"index": 5143,
"solution": "\nLet \\( n \\) be a positive integer. Consider sequences \\( a_0, a_1, \\ldots, a_n \\) for which \\( a_i \\in \\{1, 2, \\ldots , n\\} \\) for all \\( i \\) and \\( a_n = a_0 \\).\n\n### Part (a)\nSuppose \\( n \\) is odd. We need to find the number of such sequences if \\( a_i - a_{i... |
DeepScaleR | [
{
"role": "user",
"content": "Assume $x$, $y$, $z$, and $w$ are positive integers such that $x^3 = y^2$, $z^5 = w^4$, and $z - x = 31$. Determine $w - y$."
}
] | math | {
"style": "rule",
"ground_truth": "-2351"
} | {
"index": 5144,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Assume a deck of 27 cards where each card features one of three symbols (star, circle, or square), each symbol painted in one of three colors (red, yellow, or blue), and each color applied in one of three intensities (light, medium, or dark). Each symbol-color-intensity combina... | math | {
"style": "rule",
"ground_truth": "117"
} | {
"index": 5145,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Assume an even function $f(x)$ satisfies $f(x+6) = f(x) + f(3)$ for any $x \\in \\mathbb{R}$, and $f(x) = 5x$ when $x \\in (-3, -2)$. Calculate $f(201.2)$."
}
] | math | {
"style": "rule",
"ground_truth": "-16"
} | {
"index": 5146,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Assume every 7-digit whole number is a possible telephone number except those that begin with $0$ or $1$. What fraction of telephone numbers begin with $9$ and end with $0$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{80}"
} | {
"index": 5147,
"solution": "1. **Determine the total number of valid 7-digit phone numbers ($b$):**\n - The first digit can be any digit from 2 to 9 (since 0 and 1 are not allowed), giving us 8 choices.\n - Each of the remaining six digits can be any digit from 0 to 9, giving us 10 choices for each digit.\n ... |
DeepScaleR | [
{
"role": "user",
"content": "Assume every 7-digit whole number is a possible telephone number except those that begin with a digit less than 3. What fraction of telephone numbers begin with $9$ and have $3$ as their middle digit (i.e., fourth digit)?\nA) $\\frac{1}{60}$\nB) $\\frac{1}{70}$\nC) $\\frac{1}{8... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{70}"
} | {
"index": 5148,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Assume integers \\( u \\) and \\( v \\) satisfy \\( 0 < v < u \\), and let \\( A \\) be \\((u, v)\\). Points are defined as follows: \\( B \\) is the reflection of \\( A \\) over the line \\( y = x \\), \\( C \\) is the reflection of \\( B \\) over the \\( y \\)-axis, \\( D \\)... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 5149,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Assume that $f(a+b) = f(a) + f(b) + ab$ , and that $f(75) - f(51) = 1230$ . Find $f(100)$ ."
}
] | math | {
"style": "rule",
"ground_truth": "3825"
} | {
"index": 5150,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Assume that $a$, $b$, $c$, and $d$ are positive integers such that $a^5 = b^4$, $c^3 = d^2$, and $c - a = 19$. Determine $d - b$."
}
] | math | {
"style": "rule",
"ground_truth": "757"
} | {
"index": 5151,
"solution": "It follows from the givens that $a$ is a perfect fourth power, $b$ is a perfect fifth power, $c$ is a perfect square and $d$ is a perfect cube. Thus, there exist integers $s$ and $t$ such that $a = t^4$, $b = t^5$, $c = s^2$ and $d = s^3$. So $s^2 - t^4 = 19$. We can factor the left-ha... |
DeepScaleR | [
{
"role": "user",
"content": "Assume that $f$ and $g$ are functions for which $f^{-1}(g(x))=5x+3$. Find $g^{-1}(f(-7))$."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 5152,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Assume that $p$, $q$, $r$, and $s$ are positive integers such that $p^3 = q^2$, $r^5 = s^4$, and $r - p = 31$. Determine $s - q$."
}
] | math | {
"style": "rule",
"ground_truth": "-2351"
} | {
"index": 5153,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Assume that $x$ is a positive real number. Which is equivalent to $\\sqrt[3]{x\\sqrt{x}}$?"
}
] | math | {
"style": "rule",
"ground_truth": "$x^{\\frac{1}{2}}$"
} | {
"index": 5154,
"solution": "1. **Rewrite the expression inside the cube root:** \n We start by expressing the term inside the cube root in terms of powers of $x$:\n \\[\n x\\sqrt{x} = x \\cdot x^{1/2} = x^{1 + 1/2} = x^{3/2}\n \\]\n\n2. **Apply the cube root to the expression:** \n Now, we take the cu... |
DeepScaleR | [
{
"role": "user",
"content": "Assume that $x_1,x_2,\\ldots,x_7$ are real numbers such that\n\\[\\begin{aligned} x_1+4x_2+9x_3+16x_4+25x_5+36x_6+49x_7 &= 1 \\\\ \n4x_1+9x_2+16x_3+25x_4+36x_5+49x_6+64x_7 &= 12 \\\\\n9x_1+16x_2+25x_3+36x_4+49x_5+64x_6+81x_7 &= 123. \\end{aligned}\\]Find the value of $16x_1+25x... | math | {
"style": "rule",
"ground_truth": "334"
} | {
"index": 5155,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Assume that $x_1,x_2,\\ldots,x_7$ are real numbers such that \\begin{align*} x_1 + 4x_2 + 9x_3 + 16x_4 + 25x_5 + 36x_6 + 49x_7 &= 1, \\\\ 4x_1 + 9x_2 + 16x_3 + 25x_4 + 36x_5 + 49x_6 + 64x_7 &= 12, \\\\ 9x_1 + 16x_2 + 25x_3 + 36x_4 + 49x_5 + 64x_6 + 81x_7 &= 123. \\end{align*} F... | math | {
"style": "rule",
"ground_truth": "334"
} | {
"index": 5156,
"solution": "We let $(x_4,x_5,x_6,x_7)=(0,0,0,0)$. Thus, we have \\begin{align*} x_1+4x_2+9x_3&=1,\\\\ 4x_1+9x_2+16x_3&=12,\\\\ 9x_1+16x_2+25x_3&=123.\\\\ \\end{align*} Grinding this out, we have $(x_1,x_2,x_3)=\\left(\\frac{797}{4},-229,\\frac{319}{4}\\right)$ which gives $\\boxed{334}$ as our fin... |
DeepScaleR | [
{
"role": "user",
"content": "Assume that $y_1, y_2, ... , y_8$ are real numbers such that\n\\[\n\\begin{aligned}\ny_1 + 4y_2 + 9y_3 + 16y_4 + 25y_5 + 36y_6 + 49y_7 + 64y_8 &= 3, \\\\\n4y_1 + 9y_2 + 16y_3 + 25y_4 + 36y_5 + 49y_6 + 64y_7 + 81y_8 &= 15, \\\\\n9y_1 + 16y_2 + 25y_3 + 36y_4 + 49y_5 + 64y_6 + 81y... | math | {
"style": "rule",
"ground_truth": "472"
} | {
"index": 5157,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Assume that a certain BASIC language limits identifiers to a maximum of three characters. The first character must be one of the 26 English letters, and the 2nd and 3rd characters can be either English letters or Arabic numerals (0, 1, 2, 3, 4, ..., 9). Find the total number of... | math | {
"style": "rule",
"ground_truth": "34658"
} | {
"index": 5158,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Assume that savings banks offer the same interest rate as the inflation rate for a year to deposit holders. The government takes away $20 \\%$ of the interest as tax. By what percentage does the real value of government interest tax revenue decrease if the inflation rate drops ... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 5159,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Assume that the probability of a certain athlete hitting the bullseye with a dart is $40\\%$. Now, the probability that the athlete hits the bullseye exactly once in two dart throws is estimated using a random simulation method: first, a random integer value between $0$ and $9$... | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"index": 5160,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Assume that the scores $X$ of 400,000 students in a math mock exam in Yunnan Province approximately follow a normal distribution $N(98,100)$. It is known that a certain student's score ranks among the top 9100 in the province. Then, the student's math score will not be lower th... | math | {
"style": "rule",
"ground_truth": "118"
} | {
"index": 5161,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Assume that the scores $X$ of 400,000 students in a math mock exam in Yunnan Province approximately follow a normal distribution $N(98,100)$. It is known that a student's score ranks among the top 9100 in the province. Then, the student's math score will not be less than ______... | math | {
"style": "rule",
"ground_truth": "118"
} | {
"index": 5162,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Assume the adjoining chart shows the $1980$ U.S. population, in millions, for each region by ethnic group. To the nearest percent, what percent of the U.S. Black population lived in the South?\n\\begin{tabular}{|c|cccc|} \\hline & NE & MW & South & West \\\\ \\hline White & 4... | math | {
"style": "rule",
"ground_truth": "56\\%"
} | {
"index": 5163,
"solution": "1. **Calculate the total Black population in the U.S.**:\n From the table, the Black population in each region is given as:\n - NE: 5 million\n - MW: 5 million\n - South: 15 million\n - West: 2 million\n\n Summing these values gives the total Black population:\n \\[\n 5... |
DeepScaleR | [
{
"role": "user",
"content": "Assume the function $f(x) = 2\\sin x \\cos^2\\left(\\frac{\\varphi}{2}\\right) + \\cos x \\sin\\varphi - \\sin x$, where $(0 < \\varphi < \\pi)$, takes its minimum value at $x = \\pi$. \n(i) Find the value of $\\varphi$ and simplify $f(x)$. \n(ii) In triangle $ABC$, $a$, $b$,... | math | {
"style": "rule",
"ground_truth": "\\frac{7\\pi}{12}"
} | {
"index": 5164,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Assume the number of passengers traveling from location A to location B per day, $X$, follows a normal distribution $N(800, 50^2)$. Let $p_0$ denote the probability that the number of passengers traveling from A to B in a day does not exceed 900. \n(1) Find the value of $p_0$.... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 5165,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Assume the quartic $x^{4}-a x^{3}+b x^{2}-a x+d=0$ has four real roots $\\frac{1}{2} \\leq x_{1}, x_{2}, x_{3}, x_{4} \\leq 2$. Find the maximum possible value of $\\frac{\\left(x_{1}+x_{2}\\right)\\left(x_{1}+x_{3}\\right) x_{4}}{\\left(x_{4}+x_{2}\\right)\\left(x_{4}+x_{3}\\r... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{4}"
} | {
"index": 5166,
"solution": "We can rewrite the expression as $$\\frac{x_{4}^{2}}{x_{1}^{2}} \\cdot \\frac{\\left(x_{1}+x_{1}\\right)\\left(x_{1}+x_{2}\\right)\\left(x_{1}+x_{3}\\right)\\left(x_{1}+x_{4}\\right)}{\\left(x_{4}+x_{1}\\right)\\left(x_{4}+x_{2}\\right)\\left(x_{4}+x_{3}\\right)\\left(x_{4}+x_{4}\\righ... |
DeepScaleR | [
{
"role": "user",
"content": "Assume we have a calendrical system in which leap years happen every four years, no matter what. In a 150-year period, what is the maximum possible number of leap years?"
}
] | math | {
"style": "rule",
"ground_truth": "38"
} | {
"index": 5167,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Assuming $a \\neq 3$, $b \\neq 4$, and $c \\neq 5$, what is the value in simplest form of the following expression?\n\\[\\frac{a-3}{5-c} \\cdot \\frac{b-4}{3-a} \\cdot \\frac{c-5}{4-b}\\]"
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 5168,
"solution": "1. **Identify the pattern in the expression**: We are given the expression:\n \\[\n \\frac{a-3}{5-c} \\cdot \\frac{b-4}{3-a} \\cdot \\frac{c-5}{4-b}\n \\]\n We notice that each fraction has a numerator and a denominator that are differences of variables and constants.\n\n2. **A... |
DeepScaleR | [
{
"role": "user",
"content": "Assuming that $x$ is a multiple of $18711$, determine the greatest common divisor of $g(x) = (4x+5)(5x+3)(6x+7)(3x+11)$ and $x$."
}
] | math | {
"style": "rule",
"ground_truth": "1155"
} | {
"index": 5169,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Assuming that the birth of a boy or a girl is equally likely, what is the probability that the three children in a family include at least one boy and one girl? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 5170,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Assuming that the clock hands move without jumps, determine how many minutes after the clock shows 8:00 will the minute hand catch up with the hour hand."
}
] | math | {
"style": "rule",
"ground_truth": "43 \\frac{7}{11}"
} | {
"index": 5171,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Assuming that the new demand is given by \\( \\frac{1}{1+ep} \\), where \\( p = 0.20 \\) and \\( e = 1.5 \\), calculate the proportionate decrease in demand."
}
] | math | {
"style": "rule",
"ground_truth": "0.23077"
} | {
"index": 5172,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "At $2:15$ o'clock, the hour and minute hands of a clock form an angle of:"
}
] | math | {
"style": "rule",
"ground_truth": "22\\frac {1}{2}^{\\circ}"
} | {
"index": 5173,
"solution": "To find the angle between the hour and minute hands of a clock at a given time, we can use the formula:\n\\[\n\\text{Angle} = \\left|\\frac{60h - 11m}{2}\\right|\n\\]\nwhere $h$ is the hour and $m$ is the minute.\n\n1. **Identify the values of $h$ and $m$:**\n At $2:15$, $h = 2$ and ... |
DeepScaleR | [
{
"role": "user",
"content": "At 11:00 a.m. how many degrees are in the smaller angle formed by the minute hand and the hour hand of the clock?"
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 5174,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "At 12 o'clock, the angle between the hour hand and the minute hand is 0 degrees. After that, at what time do the hour hand and the minute hand form a 90-degree angle for the 6th time? (12-hour format)"
}
] | math | {
"style": "rule",
"ground_truth": "3:00"
} | {
"index": 5175,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "At 17:00, the speed of a racing car was 30 km/h. Every subsequent 5 minutes, the speed increased by 6 km/h. Determine the distance traveled by the car from 17:00 to 20:00 on the same day."
}
] | math | {
"style": "rule",
"ground_truth": "425.5"
} | {
"index": 5176,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "At 1:00 PM, two identical recreational boats set off in opposite directions from a pier on a river. At the same time, a raft also departed from the pier. An hour later, one of the boats turned around and started moving back. The other boat did the same at 3:00 PM. What is the s... | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"index": 5177,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "At 30 palm trees on different parts of an uninhabited island, a sign is attached.\n\n- On 15 of them it says: \"Exactly under 15 signs a treasure is buried.\"\n- On 8 of them it says: \"Exactly under 8 signs a treasure is buried.\"\n- On 4 of them it says: \"Exactly under 4 sig... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 5178,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "At 7:00 AM, Xiaoming leaves his house and heads to school at a speed of 52 meters per minute. When he reaches the school, the hour and minute hands on his watch are positioned symmetrically around the number 7 on the clock. It's known that Xiaoming walked for less than an hour.... | math | {
"style": "rule",
"ground_truth": "1680"
} | {
"index": 5179,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "At 7:00, five sheep, designated as A, B, C, D, and E, have distances to Wolf Castle forming an arithmetic sequence with a common difference of 20 meters. At 8:00, these same five sheep have distances to Wolf Castle forming another arithmetic sequence, but with a common differen... | math | {
"style": "rule",
"ground_truth": "140"
} | {
"index": 5180,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "At 7:10 in the morning, Xiao Ming's mother wakes him up and asks him to get up. However, Xiao Ming sees the time in the mirror and thinks that it is not yet time to get up. He tells his mother, \"It's still early!\" Xiao Ming mistakenly believes that the time is $\\qquad$ hours... | math | {
"style": "rule",
"ground_truth": "4:50"
} | {
"index": 5181,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "At 8:00 AM, Xiao Cheng and Xiao Chen set off from locations A and B respectively, heading towards each other. They meet on the way at 9:40 AM. Xiao Cheng says: \"If I had walked 10 km more per hour, we would have met 10 minutes earlier.\" Xiao Chen says: \"If I had set off half... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 5182,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "At 9:00, a pedestrian set off on a journey. An hour later, a cyclist set off from the same starting point. At 10:30, the cyclist caught up with the pedestrian and continued ahead, but after some time, the bicycle broke down. After repairing the bike, the cyclist resumed the jou... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 5183,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "At Academic Academy, to pass an algebra test you must score at least $80\\%$. If there are 35 problems on the test, what is the greatest number you can miss and still pass?"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 5184,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "At Alpine School, there are 15 soccer players. Every player is enrolled in either physics or mathematics class, but not necessarily both. If 9 players are taking physics and 4 players are registered for both physics and mathematics, how many are taking mathematics?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 5185,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "At Archimedes Academy, there are three teachers, Mrs. Algebra, Mr. Calculus, and Ms. Statistics, teaching students preparing for the AMC 8 contest. Mrs. Algebra has 13 students, Mr. Calculus has 10 students, and Ms. Statistics has 12 students. If 3 students are enrolled in both... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 5186,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "At Barker High School, a total of 36 students are on either the baseball team, the hockey team, or both. If there are 25 students on the baseball team and 19 students on the hockey team, how many students play both sports?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 5187,
"solution": "The two teams include a total of $25+19=44$ players. There are exactly 36 students who are on at least one team. Thus, there are $44-36=8$ students who are counted twice. Therefore, there are 8 students who play both baseball and hockey."
} |
DeepScaleR | [
{
"role": "user",
"content": "At Beaumont High School, there are 12 players on the baseball team. All 12 players are taking at least one of biology or chemistry. If 7 players are taking biology and 2 players are taking both sciences, how many players are taking chemistry?"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 5188,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "At Beaumont High School, there are 20 players on the basketball team. All 20 players are taking at least one of biology or chemistry. (Biology and chemistry are two different science courses at the school.) If there are 8 players taking biology and 4 players are taking both ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 5189,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "At CMU, markers come in two colors: blue and orange. Zachary fills a hat randomly with three markers such that each color is chosen with equal probability, then Chase shuffles an additional orange marker into the hat. If Zachary chooses one of the markers in the hat at random a... | math | {
"style": "rule",
"ground_truth": "39"
} | {
"index": 5190,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "At Central Middle School the $108$ students who take the AMC 8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of $15$ cookies, lists these ite... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 5191,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "At Central Middle School, the $150$ students who participate in the Math Club plan to gather for a game night. Due to an overlapping school event, attendance is expected to drop by $40\\%$. Walter and Gretel are preparing cookies using a new recipe that makes $18$ cookies per b... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 5192,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "At Clover View Junior High, one half of the students go home on the school bus. One fourth go home by automobile. One tenth go home on their bicycles. The rest walk home. What fractional part of the students walk home?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{20}"
} | {
"index": 5193,
"solution": "To find the fractional part of the students who walk home, we need to subtract the fractions of students who use other means of transportation from the whole.\n\n1. **Identify the fractions for each mode of transportation:**\n - School bus: $\\frac{1}{2}$\n - Automobile: $\\frac{1}... |
DeepScaleR | [
{
"role": "user",
"content": "At Easter-Egg Academy, each student has two eyes, each of which can be eggshell, cream, or cornsilk. It is known that $30 \\%$ of the students have at least one eggshell eye, $40 \\%$ of the students have at least one cream eye, and $50 \\%$ of the students have at least one co... | math | {
"style": "rule",
"ground_truth": "80 \\%"
} | {
"index": 5194,
"solution": "For the purposes of this solution, we abbreviate \"eggshell\" by \"egg\", and \"cornsilk\" by \"corn\". We know that there are only six combinations of eye color possible: egg-cream, egg-corn, egg-egg, cream-corn, cream-cream, corn-corn. If we let the proportions for each of these be r... |
DeepScaleR | [
{
"role": "user",
"content": "At Euclid High School, the number of students taking the AMC 10 was $60$ in 2002, $66$ in 2003, $70$ in 2004, $76$ in 2005, $78$ in 2006, and is $85$ in 2007. Between what two consecutive years was there the largest percentage increase?"
}
] | math | {
"style": "rule",
"ground_truth": "2002 and 2003"
} | {
"index": 5195,
"solution": "To find the largest percentage increase in the number of students taking the AMC 10 between consecutive years, we calculate the percentage increase for each interval:\n\n1. **From 2002 to 2003:**\n \\[\n \\frac{66 - 60}{60} \\times 100\\% = \\frac{6}{60} \\times 100\\% = 10\\%\n ... |
DeepScaleR | [
{
"role": "user",
"content": "At Euclid Middle School the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are $11$ students in Mrs. Germain's class, $8$ students in Mr. Newton's class, and $9$ students in Mrs. Young's class taking the AMC $8$ this year. How many mathematics students... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 5196,
"solution": "1. **Identify the number of students in each class:**\n - Mrs. Germain's class: 11 students\n - Mr. Newton's class: 8 students\n - Mrs. Young's class: 9 students\n\n2. **Assumption of no overlap in students:**\n Since the problem does not mention any students being in more than... |
DeepScaleR | [
{
"role": "user",
"content": "At Euler Middle School, $198$ students voted on two issues in a school referendum with the following results: $149$ voted in favor of the first issue and $119$ voted in favor of the second issue. If there were exactly $29$ students who voted against both issues, how many studen... | math | {
"style": "rule",
"ground_truth": "99"
} | {
"index": 5197,
"solution": "1. **Define the Sets:**\n Let $A$ be the set of students who voted in favor of the first issue, and $B$ be the set of students who voted in favor of the second issue. We are given:\n - $|A| = 149$\n - $|B| = 119$\n - Total students, $|U| = 198$\n - Students against both issue... |
DeepScaleR | [
{
"role": "user",
"content": "At Frank's Fruit Market, 3 bananas cost as much as 2 apples, and 6 apples cost as much as 4 oranges. How many oranges cost as much as 18 bananas?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 5198,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "At Frank's Fruit Market, 6 bananas cost as much as 4 apples, and 5 apples cost as much as 3 oranges. In addition, 4 oranges cost as much as 7 pears. How many pears cost as much as 36 bananas?"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 5199,
"solution": ""
} |
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