data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "An isosceles triangle with a base of $\\sqrt{2}$ has medians intersecting at a right angle. What is the area of this triangle?"
}
] | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"index": 4900,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An isosceles triangle, a square, and a regular pentagon each have a perimeter of 20 inches. What is the ratio of the side length of the triangle (assuming both equal sides for simplicity) to the side length of the square? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"index": 4901,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An isosceles, obtuse triangle has one angle with a degree measure that is 50$\\%$ larger than the measure of a right angle. What is the measure, in degrees, of one of the two smallest angles in the triangle? Express your answer as a decimal to the nearest tenth."
}
] | math | {
"style": "rule",
"ground_truth": "22.5^{\\circ}"
} | {
"index": 4902,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An iterative average of the numbers 1, 2, 3, 4, and 5 is computed the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the m... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{8}"
} | {
"index": 4903,
"solution": "To solve this problem, we need to understand how the iterative average process works and how the order of numbers affects the final result. The iterative average process can be thought of as a weighted average where numbers added later in the sequence have a greater influence on the fi... |
DeepScaleR | [
{
"role": "user",
"content": "An iterative average of the numbers 2, 3, 4, 6, and 7 is computed by arranging the numbers in some order. Find the difference between the largest and smallest possible values that can be obtained using this procedure."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{4}"
} | {
"index": 4904,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An iterative process is used to find an average of the numbers -1, 0, 5, 10, and 15. Arrange the five numbers in a certain sequence. Find the average of the first two numbers, then the average of the result with the third number, and so on until the fifth number is included. Wh... | math | {
"style": "rule",
"ground_truth": "8.875"
} | {
"index": 4905,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An o-Pod MP3 player stores and plays entire songs. Celeste has 10 songs stored on her o-Pod. The time length of each song is different. When the songs are ordered by length, the shortest song is only 30 seconds long and each subsequent song is 30 seconds longer than the previou... | math | {
"style": "rule",
"ground_truth": "\\dfrac{79}{90}"
} | {
"index": 4906,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the fin... | math | {
"style": "rule",
"ground_truth": "221"
} | {
"index": 4907,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An object in the plane moves from the origin and takes a ten-step path, where at each step the object may move one unit to the right, one unit to the left, one unit up, or one unit down. How many different points could be the final point?"
}
] | math | {
"style": "rule",
"ground_truth": "221"
} | {
"index": 4908,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An object is moving towards a converging lens with a focal length of \\( f = 10 \\ \\mathrm{cm} \\) along the line defined by the two focal points at a speed of \\( 2 \\ \\mathrm{m/s} \\). What is the relative speed between the object and its image when the object distance is \... | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"index": 4909,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An object moves $8$ cm in a straight line from $A$ to $B$, turns at an angle $\\alpha$, measured in radians and chosen at random from the interval $(0,\\pi)$, and moves $5$ cm in a straight line to $C$. What is the probability that $AC < 7$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 4910,
"solution": "1. **Setup the coordinate system and define points**:\n - Let $B = (0, 0)$, $A = (0, -8)$.\n - The possible points of $C$ create a semi-circle of radius $5$ centered at $B$.\n\n2. **Define the circles**:\n - The circle centered at $B$ with radius $5$ is described by the equation ... |
DeepScaleR | [
{
"role": "user",
"content": "An obtuse triangle with positive area has side lengths 10, 17, and \\(x\\). If \\(x\\) is an integer, what is the sum of all possible values of \\(x\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "224"
} | {
"index": 4911,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An octagon $ABCDEFGH$ is divided into eight smaller equilateral triangles, such as $\\triangle ABJ$ (where $J$ is the center of the octagon), shown in boldface in the diagram. By connecting every third vertex, we obtain a larger equilateral triangle $\\triangle ADE$, which is a... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 4912,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An octagon is inscribed in a square so that the vertices of the octagon trisect the sides of the square. The perimeter of the square is 108 centimeters. What is the number of square centimeters in the area of the octagon?"
}
] | math | {
"style": "rule",
"ground_truth": "567"
} | {
"index": 4913,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An octahedron consists of two square-based pyramids glued together along their square bases to form a polyhedron with eight faces. Imagine an ant that begins at the top vertex and walks to one of the four adjacent vertices that he randomly selects and calls vertex A. From verte... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 4914,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An octopus told me that his underwater cave is $245_{8}$ years old. How many years is this in base ten?"
}
] | math | {
"style": "rule",
"ground_truth": "165"
} | {
"index": 4915,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An octopus told me that his underwater cave is $567_{8}$ years old. How many years is this in base ten?"
}
] | math | {
"style": "rule",
"ground_truth": "375"
} | {
"index": 4916,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An odd function $f(x)$ defined on $R$ satisfies $f(x) = f(2-x)$. When $x \\in [0,1]$, $f(x) = ax^{3} + 2x + a + 1$. Find $f(2023)$."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 4917,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An odd six-digit number is called \"just cool\" if it consists of digits that are prime numbers, and no two identical digits are adjacent. How many \"just cool\" numbers exist?"
}
] | math | {
"style": "rule",
"ground_truth": "729"
} | {
"index": 4918,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An old clock's minute and hour hands overlap every 66 minutes of standard time. Calculate how much the old clock's 24 hours differ from the standard 24 hours."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 4919,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An omino is a 1-by-1 square or a 1-by-2 horizontal rectangle. An omino tiling of a region of the plane is a way of covering it (and only it) by ominoes. How many omino tilings are there of a 2-by-10 horizontal rectangle?"
}
] | math | {
"style": "rule",
"ground_truth": "7921"
} | {
"index": 4920,
"solution": "There are exactly as many omino tilings of a 1-by-$n$ rectangle as there are domino tilings of a 2-by-$n$ rectangle. Since the rows don't interact at all, the number of omino tilings of an $m$-by-$n$ rectangle is the number of omino tilings of a 1-by-$n$ rectangle raised to the $m$ th ... |
DeepScaleR | [
{
"role": "user",
"content": "An open box is constructed by starting with a rectangular sheet of metal 10 in. by 14 in. and cutting a square of side $x$ inches from each corner. The resulting projections are folded up and the seams welded. The volume of the resulting box is:"
}
] | math | {
"style": "rule",
"ground_truth": "140x - 48x^2 + 4x^3"
} | {
"index": 4921,
"solution": "To find the volume of the box formed by cutting squares from each corner of a rectangular sheet and folding the sides up, we follow these steps:\n\n1. **Identify the dimensions of the original rectangle:**\n The original metal sheet is a rectangle with dimensions $10$ inches by $14$ ... |
DeepScaleR | [
{
"role": "user",
"content": "An ordered list of four numbers is called a quadruple. A quadruple $(p, q, r, s)$ of integers with $p, q, r, s \\geq 0$ is chosen at random such that $2 p+q+r+s=4$. What is the probability that $p+q+r+s=3$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{11}"
} | {
"index": 4922,
"solution": "First, we count the number of quadruples $(p, q, r, s)$ of non-negative integer solutions to the equation $2 p+q+r+s=4$. Then, we determine which of these satisfies $p+q+r+s=3$. This will allow us to calculate the desired probability. Since each of $p, q, r$, and $s$ is a non-negative ... |
DeepScaleR | [
{
"role": "user",
"content": "An ordered pair $(n,p)$ is *juicy* if $n^{2} \\equiv 1 \\pmod{p^{2}}$ and $n \\equiv -1 \\pmod{p}$ for positive integer $n$ and odd prime $p$ . How many juicy pairs exist such that $n,p \\leq 200$ ?\n\n\nProposed by Harry Chen (Extile)"
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 4923,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ordered pair $(a, b)$ of positive integers is called spicy if $\\operatorname{gcd}(a+b, ab+1)=1$. Compute the probability that both $(99, n)$ and $(101, n)$ are spicy when $n$ is chosen from $\\{1,2, \\ldots, 2024\\}$ uniformly at random."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{96}{595}"
} | {
"index": 4924,
"solution": "We claim that $(a, b)$ is spicy if and only if both $\\operatorname{gcd}(a+1, b-1)=1$ and $\\operatorname{gcd}(a-1, b+1)=1$. To prove the claim, we note that $$\\operatorname{gcd}(a+b, ab+1)=\\operatorname{gcd}(a+b, b(-b)+1)=\\operatorname{gcd}(a+b, b^{2}-1)$$ Hence, we have $$\\begin{... |
DeepScaleR | [
{
"role": "user",
"content": "An ordered pair $(a, c)$ of integers, each of which has an absolute value less than or equal to 6, is chosen at random. What is the probability that the equation $ax^2 - 3ax + c = 0$ will not have distinct real roots both greater than 2?\nA) $\\frac{157}{169}$ B) $\\frac{167}{... | math | {
"style": "rule",
"ground_truth": "\\frac{167}{169}"
} | {
"index": 4925,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ordered pair $(m,n)$ of non-negative integers is called \"simple\" if the addition $m+n$ in base $10$ requires no carrying. Find the number of simple ordered pairs of non-negative integers that sum to $1492$."
}
] | math | {
"style": "rule",
"ground_truth": "300"
} | {
"index": 4926,
"solution": "Since no carrying over is allowed, the range of possible values of any digit of $m$ is from $0$ to the respective digit in $1492$ (the values of $n$ are then fixed). Thus, the number of ordered pairs will be $(1 + 1)(4 + 1)(9 + 1)(2 + 1) = 2\\cdot 5\\cdot 10\\cdot 3 = \\boxed{300}$.\n ... |
DeepScaleR | [
{
"role": "user",
"content": "An ordinary $6$-sided die has a number on each face from $1$ to $6$ (each number appears on one face). How many ways can I paint two faces of a die red, so that the numbers on the red faces don't add up to $7$?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 4927,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An ordinary clock in a factory is running slow so that the minute hand passes the hour hand at the usual dial position (12 o'clock, etc.) but only every 69 minutes. At time and one-half for overtime, the extra pay to which a $4.00 per hour worker should be entitled after workin... | math | {
"style": "rule",
"ground_truth": "$2.60"
} | {
"index": 4928,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An oreo shop now sells $5$ different flavors of oreos, $3$ different flavors of milk, and $2$ different flavors of cookies. Alpha and Gamma decide to purchase some items. Since Alpha is picky, he will order no more than two different items in total, avoiding replicas. To be equ... | math | {
"style": "rule",
"ground_truth": "2100"
} | {
"index": 4929,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An oreo shop sells $5$ different flavors of oreos and $3$ different flavors of milk. Alpha and Beta decide to purhcase some oreos. Since Alpha is picky, he will not order more than 1 of the same flavor. To be just as weird, Beta will only order oreos, but she will be willing to... | math | {
"style": "rule",
"ground_truth": "351"
} | {
"index": 4930,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An organization has $30$ employees, $20$ of whom have a brand A computer while the other $10$ have a brand B computer. For security, the computers can only be connected to each other and only by cables. The cables can only connect a brand A computer to a brand B computer. Emplo... | math | {
"style": "rule",
"ground_truth": "191"
} | {
"index": 4931,
"solution": "To solve this problem, we need to determine the maximum number of cables that can be used such that every employee can communicate with each other, under the constraint that cables can only connect a brand A computer to a brand B computer.\n\n#### Step 1: Understand the problem constra... |
DeepScaleR | [
{
"role": "user",
"content": "An organization has a structure where there is one president, two vice-presidents (VP1 and VP2), and each vice-president supervises two managers. If the organization currently has 12 members, in how many different ways can the leadership (president, vice-presidents, and manager... | math | {
"style": "rule",
"ground_truth": "554400"
} | {
"index": 4932,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An organization initially consists of 10 leaders and a certain number of regular members. Each year, all leaders are replaced, and each regular member recruits two new members. After this, 10 new leaders are elected from outside the organization. Initially, there are 30 people ... | math | {
"style": "rule",
"ground_truth": "1180990"
} | {
"index": 4933,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An organization starts with 20 people, consisting of 7 leaders and 13 regular members. Each year, all leaders are replaced. Every regular member recruits one new person to join as a regular member, and 5% of the regular members decide to leave the organization voluntarily. Afte... | math | {
"style": "rule",
"ground_truth": "172"
} | {
"index": 4934,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An six-digit integer is formed by repeating a positive three-digit integer. For example, 123,123 or 456,456 are integers of this form. What is the greatest common divisor of all six-digit integers of this form?"
}
] | math | {
"style": "rule",
"ground_truth": "1001"
} | {
"index": 4935,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An uncrossed belt is fitted without slack around two circular pulleys with radii of $14$ inches and $4$ inches. \nIf the distance between the points of contact of the belt with the pulleys is $24$ inches, then the distance \nbetween the centers of the pulleys in inches is"
}
... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 4936,
"solution": "1. **Identify the setup**: We have two pulleys with radii $14$ inches and $4$ inches respectively. The distance between the points of contact of the belt with the pulleys is $24$ inches. We denote the centers of the smaller and larger pulleys as $A$ and $B$ respectively.\n\n2. **Draw p... |
DeepScaleR | [
{
"role": "user",
"content": "An underground line has $26$ stops, including the first and the final one, and all the stops are numbered from $1$ to $26$ according to their order. Inside the train, for each pair $(x,y)$ with $1\\leq x < y \\leq 26$ there is exactly one passenger that goes from the ... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 4937,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An unfair coin has the property that when flipped four times, it has the same probability of turning up 2 heads and 2 tails (in any order) as 3 heads and 1 tail (in any order). What is the probability of getting a head in any one flip?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 4938,
"solution": "Let $p$ be the probability of getting a head in one flip. There are 6 ways to get 2 heads and 2 tails, each with probability $p^{2}(1-p)^{2}$, and 4 ways to get 3 heads and 1 tail, each with probability $p^{3}(1-p)$. We are given that $6 p^{2}(1-p)^{2}=4 p^{3}(1-p)$. Clearly $p$ is not... |
DeepScaleR | [
{
"role": "user",
"content": "An unfair coin lands on heads with probability $\\frac34$ and tails with probability $\\frac14$. A heads flip gains $\\$3$, but a tails flip loses $\\$8$. What is the expected worth of a coin flip? Express your answer as a decimal rounded to the nearest hundredth."
}
] | math | {
"style": "rule",
"ground_truth": "\\$0.25"
} | {
"index": 4939,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An unpainted cone has radius \\( 3 \\mathrm{~cm} \\) and slant height \\( 5 \\mathrm{~cm} \\). The cone is placed in a container of paint. With the cone's circular base resting flat on the bottom of the container, the depth of the paint in the container is \\( 2 \\mathrm{~cm} \... | math | {
"style": "rule",
"ground_truth": "59"
} | {
"index": 4940,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An up-right path between two lattice points $P$ and $Q$ is a path from $P$ to $Q$ that takes steps of 1 unit either up or to the right. A lattice point $(x, y)$ with $0 \\leq x, y \\leq 5$ is chosen uniformly at random. Compute the expected number of up-right paths from $(0,0)$... | math | {
"style": "rule",
"ground_truth": "175"
} | {
"index": 4941,
"solution": "For a lattice point $(x, y)$, let $F(x, y)$ denote the number of up-right paths from $(0,0)$ to $(5,5)$ that don't pass through $(x, y)$, and let $$S=\\sum_{0 \\leq x \\leq 5} \\sum_{0 \\leq y \\leq 5} F(x, y)$$ Our answer is $\\frac{S}{36}$, as there are 36 lattice points $(x, y)$ wit... |
DeepScaleR | [
{
"role": "user",
"content": "An up-right path between two lattice points $P$ and $Q$ is a path from $P$ to $Q$ that takes steps of length 1 unit either up or to the right. How many up-right paths from $(0,0)$ to $(7,7)$, when drawn in the plane with the line $y=x-2.021$, enclose exactly one bounded region ... | math | {
"style": "rule",
"ground_truth": "637"
} | {
"index": 4942,
"solution": "We will make use of a sort of bijection which is typically used to prove the closed form for the Catalan numbers. We will count these paths with complementary counting. Since both the starting and ending points are above the line $x-2.021$, any path which traverses below this line (and... |
DeepScaleR | [
{
"role": "user",
"content": "An up-right path from $(a, b) \\in \\mathbb{R}^{2}$ to $(c, d) \\in \\mathbb{R}^{2}$ is a finite sequence $(x_{1}, y_{1}), \\ldots,(x_{k}, y_{k})$ of points in $\\mathbb{R}^{2}$ such that $(a, b)=(x_{1}, y_{1}),(c, d)=(x_{k}, y_{k})$, and for each $1 \\leq i<k$ we have that eit... | math | {
"style": "rule",
"ground_truth": "0.2937156494680644"
} | {
"index": 4943,
"solution": "Note that any up-right path must pass through exactly one point of the form $(n,-n)$ (i.e. a point on the upper-left to lower-right diagonal), and the number of such paths is $\\binom{800}{400-n}^{2}$ because there are $\\binom{800}{400-n}$ up-right paths from $(-400,-400)$ to $(n,-n)$... |
DeepScaleR | [
{
"role": "user",
"content": "An up-right path from $(a, b) \\in \\mathbb{R}^{2}$ to $(c, d) \\in \\mathbb{R}^{2}$ is a finite sequence $\\left(x_{1}, y_{1}\\right), \\ldots,\\left(x_{k}, y_{k}\\right)$ of points in $\\mathbb{R}^{2}$ such that $(a, b)=\\left(x_{1}, y_{1}\\right),(c, d)=\\left(x_{k}, y_{k}\\... | math | {
"style": "rule",
"ground_truth": "1750"
} | {
"index": 4944,
"solution": "The number of up-right paths from $(0,0)$ to $(4,4)$ is $\\binom{8}{4}$ because any such upright path is identical to a sequence of 4 U's and 4 R's, where $U$ corresponds to a step upwards and R corresponds to a step rightwards. Therefore, the total number of pairs of (possibly interse... |
DeepScaleR | [
{
"role": "user",
"content": "An urn contains $4$ green balls and $6$ blue balls. A second urn contains $16$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is $0.58$. Find $N$."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 4945,
"solution": "First, we find the probability both are green, then the probability both are blue, and add the two probabilities. The sum should be equal to $0.58$.\n The probability both are green is $\\frac{4}{10}\\cdot\\frac{16}{16+N}$, and the probability both are blue is $\\frac{6}{10}\\cdot\\fra... |
DeepScaleR | [
{
"role": "user",
"content": "An urn contains $k$ balls labeled with $k$, for all $k = 1, 2, \\ldots, 2016$. What is the minimum number of balls we must draw, without replacement and without looking at the balls, to ensure that we have 12 balls with the same number?"
}
] | math | {
"style": "rule",
"ground_truth": "22122"
} | {
"index": 4946,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An urn contains 101 balls, exactly 3 of which are red. The balls are drawn one by one without replacement. On which draw is it most likely to pull the second red ball?"
}
] | math | {
"style": "rule",
"ground_truth": "51"
} | {
"index": 4947,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An urn contains marbles of four colors: red, white, blue, and green. When four marbles are drawn without replacement, the following events are equally likely:\n(a) the selection of four red marbles;\n(b) the selection of one white and three red marbles;\n(c) the selection of on... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 4948,
"solution": "1. **Define Variables and Equations**:\n Let $r$, $w$, $b$, and $g$ represent the number of red, white, blue, and green marbles, respectively, and let $n$ be the total number of marbles. Thus, we have:\n \\[ r + w + b + g = n \\]\n\n2. **Calculate the Probabilities**:\n - The num... |
DeepScaleR | [
{
"role": "user",
"content": "An urn contains nine balls, numbered from 1 to 9. José and Maria each simultaneously draw one ball from the urn. They form a two-digit number with the number on José's ball as the tens digit and the number on Maria's ball as the units digit. What is the probability that this nu... | math | {
"style": "rule",
"ground_truth": "4/9"
} | {
"index": 4949,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"index": 4950,
"solution": "To solve this problem, we need to calculate the probability that after four operations, the urn contains exactly three red balls and three blue balls. We start with one red ball and one blue ball in the urn.\n\n#### Step 1: Understanding the possible sequences\nTo have three balls of e... |
DeepScaleR | [
{
"role": "user",
"content": "An urn initially contains two red balls and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation five times: he draws a ball from the urn at random and then takes a ball of the same color from the box and adds those two matching ... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{21}"
} | {
"index": 4951,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An urn initially contains two red balls and one blue ball. George undertakes the operation of randomly drawing a ball and then adding two more balls of the same color from a box into the urn. This operation is done three times. After these operations, the urn has a total of nin... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{10}"
} | {
"index": 4952,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "An urn is filled with coins and beads, all of which are either silver or gold. Twenty percent of the objects in the urn are beads. Forty percent of the coins in the urn are silver. What percent of objects in the urn are gold coins?"
}
] | math | {
"style": "rule",
"ground_truth": "48\\%"
} | {
"index": 4953,
"solution": "1. **Identify the percentage of beads and coins:**\n Given that 20% of the objects in the urn are beads, the remaining objects must be coins. Therefore, the percentage of coins in the urn is:\n \\[\n 100\\% - 20\\% = 80\\%\n \\]\n\n2. **Determine the composition of the coins:**... |
DeepScaleR | [
{
"role": "user",
"content": "Ana and Banana are rolling a standard six-sided die. Ana rolls the die twice, obtaining $a_{1}$ and $a_{2}$, then Banana rolls the die twice, obtaining $b_{1}$ and $b_{2}$. After Ana's two rolls but before Banana's two rolls, they compute the probability $p$ that $a_{1} b_{1}+a... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"index": 4954,
"solution": "If either $a_{1}$ or $a_{2}$ is relatively prime to 6, then $p=\\frac{1}{6}$. If one of them is a multiple of 2 but not 6, while the other is a multiple of 3 but not 6, we also have $p=\\frac{1}{6}$. In other words, $p=\\frac{1}{6}$ if $\\operatorname{gcd}(a_{1}, a_{2})$ is coprime to ... |
DeepScaleR | [
{
"role": "user",
"content": "Ana and Bonita were born on the same date in different years, $n$ years apart. Last year Ana was $5$ times as old as Bonita. This year Ana's age is the square of Bonita's age. What is $n?$"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 4955,
"solution": "\nLet's denote Ana's age this year as $A$ and Bonita's age this year as $B$. Given that Ana and Bonita were born $n$ years apart, we have:\n\\[ A = B + n \\]\n\nFrom the problem, we know:\n1. Last year, Ana was $5$ times as old as Bonita:\n \\[ A - 1 = 5(B - 1) \\]\n2. This year, Ana... |
DeepScaleR | [
{
"role": "user",
"content": "Ana and Luíza train every day for the Big Race that will take place at the end of the year at school, each running at the same speed. The training starts at point $A$ and ends at point $B$, which are $3000 \\mathrm{~m}$ apart. They start at the same time, but when Luíza finishe... | math | {
"style": "rule",
"ground_truth": "125"
} | {
"index": 4956,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Ana has an iron material of mass $20.2$ kg. She asks Bilyana to make $n$ weights to be used in a classical weighning scale with two plates. Bilyana agrees under the condition that each of the $n$ weights is at least $10$ g. Determine the smallest possible value of $n$ ... | math | {
"style": "rule",
"ground_truth": "2020"
} | {
"index": 4957,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Ana's monthly salary was $2000$ in May. In June she received a 20% raise. In July she received a 20% pay cut. After the two changes in June and July, Ana's monthly salary was"
}
] | math | {
"style": "rule",
"ground_truth": "1920"
} | {
"index": 4958,
"solution": "1. **Calculate the salary after the raise in June:**\n Ana's initial salary in May is $2000. A 20\\% raise means her salary is increased by $2000 \\times 20\\% = $400. Therefore, her new salary in June becomes:\n \\[\n 2000 + 400 = 2400\n \\]\n Alternatively, this can be calc... |
DeepScaleR | [
{
"role": "user",
"content": "Ana, Bob, and Cao bike at constant rates of $8.6$ meters per second, $6.2$ meters per second, and $5$ meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking alon... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"index": 4959,
"solution": "Let P, Q, and R be the east-west distance of the field, the north-south distance, and the distance from the southeast corner to point D, respectively.\n Ana's distance to point D = $P + Q + (P - R) = 2P + Q - R$\n Bob's distance to point D = $Q + R$\n Cao's distance to point D = $\\sqr... |
DeepScaleR | [
{
"role": "user",
"content": "Analogical reasoning is an important method of reasoning. Based on the similarity of two things in some characteristics, conclusions can be drawn that they may be similar in other characteristics. Reading perception: In addition and subtraction of fractions with different denom... | math | {
"style": "rule",
"ground_truth": "-\\frac{1011}{2023}"
} | {
"index": 4960,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Analogous to the exponentiation of rational numbers, we define the division operation of several identical rational numbers (all not equal to $0$) as \"division exponentiation,\" denoted as $a^{ⓝ}$, read as \"$a$ circle $n$ times.\" For example, $2\\div 2\\div 2$ is denoted as ... | math | {
"style": "rule",
"ground_truth": "-73"
} | {
"index": 4961,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Analyzing the intersection of $y = x^3 - 6x + 2$ and $y = m$ where $-10 < m < 10$, define $L(m)$ as the smallest $x$ coordinate of their intersection points. Calculate the function $r = \\frac{L(-m) - L(m)}{m}$ as $m$ approaches zero. Find the limit of this function."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"index": 4962,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Anastasia is taking a walk in the plane, starting from $(1,0)$. Each second, if she is at $(x, y)$, she moves to one of the points $(x-1, y),(x+1, y),(x, y-1)$, and $(x, y+1)$, each with $\\frac{1}{4}$ probability. She stops as soon as she hits a point of the form $(k, k)$. Wha... | math | {
"style": "rule",
"ground_truth": "\\frac{3-\\sqrt{3}}{3}"
} | {
"index": 4963,
"solution": "The key idea is to consider $(a+b, a-b)$, where $(a, b)$ is where Anastasia walks on. Then, the first and second coordinates are independent random walks starting at 1, and we want to find the probability that the first is divisible by 3 when the second reaches 0 for the first time. Le... |
DeepScaleR | [
{
"role": "user",
"content": "Anca and Bruce drove along a highway. Bruce drove at 50 km/h and Anca at 60 km/h, but stopped to rest. How long did Anca stop?"
}
] | math | {
"style": "rule",
"ground_truth": "40 \\text{ minutes}"
} | {
"index": 4964,
"solution": "Bruce drove 200 km in 4 hours. Anca drove the same distance in \\( 3 \\frac{1}{3} \\) hours. The difference is \\( \\frac{2}{3} \\) hours, or 40 minutes."
} |
DeepScaleR | [
{
"role": "user",
"content": "Anca and Bruce left Mathville at the same time. They drove along a straight highway towards Staton. Bruce drove at $50 \\mathrm{~km} / \\mathrm{h}$. Anca drove at $60 \\mathrm{~km} / \\mathrm{h}$, but stopped along the way to rest. They both arrived at Staton at the same time. ... | math | {
"style": "rule",
"ground_truth": "40 \\text{ minutes}"
} | {
"index": 4965,
"solution": "Since Bruce drove 200 km at a speed of $50 \\mathrm{~km} / \\mathrm{h}$, this took him $\\frac{200}{50}=4$ hours. Anca drove the same 200 km at a speed of $60 \\mathrm{~km} / \\mathrm{h}$ with a stop somewhere along the way. Since Anca drove 200 km at a speed of $60 \\mathrm{~km} / \\m... |
DeepScaleR | [
{
"role": "user",
"content": "Ancient astronaut theorist Nutter B. Butter claims that the Caloprians from planet Calop, 30 light years away and at rest with respect to the Earth, wiped out the dinosaurs. The iridium layer in the crust, he claims, indicates spaceships with the fuel necessary to travel at 30%... | math | {
"style": "rule",
"ground_truth": "111"
} | {
"index": 4966,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Anders is solving a math problem, and he encounters the expression $\\sqrt{15!}$. He attempts to simplify this radical by expressing it as $a \\sqrt{b}$ where $a$ and $b$ are positive integers. The sum of all possible distinct values of $ab$ can be expressed in the form $q \\cd... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 4967,
"solution": "Note that $15!=2^{11} \\cdot 3^{6} \\cdot 5^{3} \\cdot 7^{2} \\cdot 11^{1} \\cdot 13^{1}$. The possible $a$ are thus precisely the factors of $2^{5} \\cdot 3^{3} \\cdot 5^{1} \\cdot 7^{1}=$ 30240. Since $\\frac{ab}{15!}=\\frac{ab}{a^{2}b}=\\frac{1}{a}$, we have $$q =\\frac{1}{15!} \\su... |
DeepScaleR | [
{
"role": "user",
"content": "Andrea and Lauren are $20$ kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between them decreasing at a rate of $1$ kilometer per minute. After $5$ minutes, Andrea stops biking because of a flat tire and waits... | math | {
"style": "rule",
"ground_truth": "65"
} | {
"index": 4968,
"solution": "1. **Define Variables:**\n Let $v_A$ be Andrea's speed and $v_L$ be Lauren's speed, both in kilometers per hour.\n\n2. **Set Up Equations:**\n Given that Andrea travels at three times the speed of Lauren, we have:\n \\[ v_A = 3v_L \\]\n Also, the rate at which the distance betw... |
DeepScaleR | [
{
"role": "user",
"content": "Andrea flips a fair coin repeatedly, continuing until she either flips two heads in a row (the sequence $H H$ ) or flips tails followed by heads (the sequence $T H$ ). What is the probability that she will stop after flipping $H H$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "1/4"
} | {
"index": 4969,
"solution": "The only way that Andrea can ever flip $H H$ is if she never flips $T$, in which case she must flip two heads immediately at the beginning. This happens with probability $\\frac{1}{4}$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Andrea notices that the 40-foot tree next to her is casting a 10-foot shadow. How tall, in inches, is Andrea if she is casting a 15-inch shadow at the same time?"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 4970,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Andreas, Boyu, Callista, and Diane each randomly choose an integer from 1 to 9, inclusive. Each of their choices is independent of the other integers chosen and the same integer can be chosen by more than one person. The probability that the sum of their four integers is even i... | math | {
"style": "rule",
"ground_truth": "78"
} | {
"index": 4971,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Andrew and John are both Beatles fans. Their respective collections share nine of the same albums. Andrew has seventeen albums in his collection. Six albums are in John's collection, but not Andrew's. How many albums are in either Andrew's or John's collection, but not both... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 4972,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Andrew flips a fair coin $5$ times, and counts the number of heads that appear. Beth flips a fair coin $6$ times and also counts the number of heads that appear. Compute the probability Andrew counts at least as many heads as Beth."
}
] | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"index": 4973,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Andrew's grandfather's age is twelve times Andrew's age. If Andrew's grandfather was 55 years old when Andrew was born, how many years old is Andrew now?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 4974,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Andrey drove a car to the airport in a neighboring city. After one hour of driving at a speed of 60 km/h, he realized that if he did not change his speed, he would be 20 minutes late. He then sharply increased his speed, and as a result, covered the remaining distance at an ave... | math | {
"style": "rule",
"ground_truth": "180"
} | {
"index": 4975,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Andrey found the product of all the numbers from 1 to 11 inclusive and wrote the result on the board. During a break, someone accidentally erased three digits, leaving the number $399 * 68 * *$. Help restore the missing digits without recalculating the product."
}
] | math | {
"style": "rule",
"ground_truth": "39916800"
} | {
"index": 4976,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Andy and Bethany have a rectangular array of numbers with $40$ rows and $75$ columns. Andy adds the numbers in each row. The average of his $40$ sums is $A$. Bethany adds the numbers in each column. The average of her $75$ sums is $B$. What is the value of $\\frac{A}{B}$?"
}
... | math | {
"style": "rule",
"ground_truth": "\\frac{15}{8}"
} | {
"index": 4977,
"solution": "Let's denote the rectangular array of numbers as $a_{ij}$ where $i$ ranges from $1$ to $40$ (rows) and $j$ ranges from $1$ to $75$ (columns).\n\n1. **Summing Rows and Columns:**\n - Andy sums each row, so for row $i$, the sum is $S_i = \\sum_{j=1}^{75} a_{ij}$.\n - Bethany sums eac... |
DeepScaleR | [
{
"role": "user",
"content": "Andy is attempting to solve the quadratic equation $$64x^2 - 96x - 48 = 0$$ by completing the square. He aims to rewrite the equation in the form $$(ax + b)^2 = c,$$ where \\(a\\), \\(b\\), and \\(c\\) are integers and \\(a > 0\\). Determine the value of \\(a + b + c\\)."
}
] | math | {
"style": "rule",
"ground_truth": "86"
} | {
"index": 4978,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Andy the Ant lives on a coordinate plane and is currently at $(-20, 20)$ facing east (that is, in the positive $x$-direction). Andy moves $1$ unit and then turns $90^{\\circ}$ left. From there, Andy moves $2$ units (north) and then turns $90^{\\circ}$ left. He then moves $3$ un... | math | {
"style": "rule",
"ground_truth": "$(-1030, -990)$"
} | {
"index": 4979,
"solution": "To solve this problem, we need to understand the pattern of Andy's movements and how his position changes with each move. Andy starts at $(-20, 20)$ and moves in a spiral pattern, increasing the distance he moves by $1$ unit after each turn, and always turning left.\n\n1. **Initial Mov... |
DeepScaleR | [
{
"role": "user",
"content": "Andy, Beth, Charlie, and Daniel take a test with thirty questions. Andy and Beth together get the same number of questions wrong as Charlie and Daniel together. Andy and Daniel together get four more questions wrong than Beth and Charlie do together. If Charlie gets five questi... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 4980,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Ang, Ben, and Jasmin each have $5$ blocks, colored red, blue, yellow, white, and green; and there are $5$ empty boxes. Each of the people randomly and independently of the other two people places one of their blocks into each box. The probability that at least one box receives ... | math | {
"style": "rule",
"ground_truth": "471"
} | {
"index": 4981,
"solution": "1. **Understanding the Problem:**\n - Ang, Ben, and Jasmin each have 5 blocks of different colors and there are 5 boxes.\n - They independently place one block in each box.\n - We need to find the probability that at least one box ends up with 3 blocks of the same color.\n\n2. **... |
DeepScaleR | [
{
"role": "user",
"content": "Angela and Barry share a piece of land. The ratio of the area of Angela's portion to the area of Barry's portion is $3: 2$. They each grow corn and peas on their piece of land. The entire piece of land is covered by corn and peas in the ratio $7: 3$. On Angela's portion of the ... | math | {
"style": "rule",
"ground_truth": "$11: 9$"
} | {
"index": 4982,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Angela has $a$ marbles, Brian has twice as many marbles as Angela, Caden has three times as many marbles as Brian, and Daryl has five times the number of marbles Caden has. If in total Angela, Brian, Caden and Daryl have 78 marbles, what is the value of $a?$"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 4983,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Angela has deposited $\\$8,\\!000$ into an account that pays $6\\%$ interest compounded annually.\n\nBob has deposited $\\$10,\\!000$ into an account that pays $7\\%$ simple annual interest.\n\nIn $20$ years Angela and Bob compare their respective balances. To the nearest dolla... | math | {
"style": "rule",
"ground_truth": "\\$1,\\!657"
} | {
"index": 4984,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Angelica wants to choose a three-digit code for her suitcase lock. To make it easier to remember, Angelica wants all the digits in her code to be in non-decreasing order. How many different possible codes does Angelica have to choose from?"
}
] | math | {
"style": "rule",
"ground_truth": "220"
} | {
"index": 4985,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Angie's class has 2 girls for every 3 boys. If there are 20 students in the class, how many girls are in Angie's class?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 4986,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Angie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side. What is the probability that Angie and Carlos are seated opposite each other?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 4987,
"solution": "1. **Fix Angie's Position:** Assume Angie is seated at one side of the square table. This does not affect the generality of the problem due to the symmetry of the seating arrangement.\n\n2. **Count Total Arrangements:** With Angie's position fixed, there are 3 remaining seats for Bridg... |
DeepScaleR | [
{
"role": "user",
"content": "Angle $ABC$ of $\\triangle ABC$ is a right angle. The sides of $\\triangle ABC$ are the diameters of semicircles as shown. The area of the semicircle on $\\overline{AB}$ equals $8\\pi$, and the arc of the semicircle on $\\overline{AC}$ has length $8.5\\pi$. What is the radius o... | math | {
"style": "rule",
"ground_truth": "7.5"
} | {
"index": 4988,
"solution": "1. **Identify the given information and the relationship between the elements:**\n - $\\triangle ABC$ is a right triangle with $\\angle ABC = 90^\\circ$.\n - The sides of $\\triangle ABC$ are the diameters of semicircles.\n - The area of the semicircle on $\\overline{AB}$ is $8\\... |
DeepScaleR | [
{
"role": "user",
"content": "Angle $EAB$ is a right angle, and $BE=9$ units. What is the number of square units in the sum of the areas of the two squares $ABCD$ and $AEFG$?\n\n[asy]\ndraw((0,0)--(1,1)--(0,2)--(-1,1)--cycle);\ndraw((0,2)--(2,4)--(0,6)--(-2,4)--cycle);\ndraw((1,1)--(2,4));\ndraw((-1,1)--(-2... | math | {
"style": "rule",
"ground_truth": "81"
} | {
"index": 4989,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Angle ABC is a right angle. The diagram shows four quadrilaterals, where three are squares on each side of triangle ABC, and one square is on the hypotenuse. The sum of the areas of all four squares is 500 square centimeters. What is the number of square centimeters in the area... | math | {
"style": "rule",
"ground_truth": "\\frac{500}{3}"
} | {
"index": 4990,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Angle PQR is a right angle. The three quadrilaterals shown are squares. The sum of the areas of the three squares is 338 square centimeters. What is the number of square centimeters in the area of the largest square?\n\n[asy]\ndraw((0,0)--(12,0)--(0,5)--cycle);\ndot((0,0));\ndo... | math | {
"style": "rule",
"ground_truth": "169"
} | {
"index": 4991,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Angle bisectors $AA', BB'$ and $CC'$ are drawn in triangle $ABC$ with angle $\\angle B= 120^o$ . Find $\\angle A'B'C'$ ."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 4992,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Angles $A$ and $B$ are supplementary. If the measure of angle $A$ is $8$ times angle $B$, what is the measure of angle A?"
}
] | math | {
"style": "rule",
"ground_truth": "160"
} | {
"index": 4993,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Angles $C$ and $D$ are supplementary. If the measure of angle $C$ is $12$ times angle $D$, what is the measure of angle $C$?"
}
] | math | {
"style": "rule",
"ground_truth": "166.15"
} | {
"index": 4994,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Angry reviews about the work of an online store are left by $80\\%$ of dissatisfied customers (those who were poorly served in the store). Of the satisfied customers, only $15\\%$ leave a positive review. A certain online store earned 60 angry and 20 positive reviews. Using thi... | math | {
"style": "rule",
"ground_truth": "0.64"
} | {
"index": 4995,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Anh read a book. On the first day she read $n$ pages in $t$ minutes, where $n$ and $t$ are positive integers. On the second day Anh read $n + 1$ pages in $t + 1$ minutes. Each day thereafter Anh read one more page than she read on the previous day, and it took her one more minu... | math | {
"style": "rule",
"ground_truth": "53"
} | {
"index": 4996,
"solution": "We could see that both $374$ and $319$ are divisible by $11$ in the outset, and that $34$ and $29$, the quotients, are relatively prime. Both are the $average$ number of minutes across the $11$ days, so we need to subtract $\\left \\lfloor{\\frac{11}{2}}\\right \\rfloor=5$ from each to... |
DeepScaleR | [
{
"role": "user",
"content": "Anh traveled 75 miles on the interstate and 15 miles on a mountain pass. The speed on the interstate was four times the speed on the mountain pass. If Anh spent 45 minutes driving on the mountain pass, determine the total time of his journey in minutes."
}
] | math | {
"style": "rule",
"ground_truth": "101.25"
} | {
"index": 4997,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Anita attends a baseball game in Atlanta and estimates that there are 50,000 fans in attendance. Bob attends a baseball game in Boston and estimates that there are 60,000 fans in attendance. A league official who knows the actual numbers attending the two games note that:\ni. T... | math | {
"style": "rule",
"ground_truth": "22000"
} | {
"index": 4998,
"solution": "To find the largest possible difference between the numbers attending the two games, we need to consider the maximum and minimum possible attendances based on the given estimates and the conditions provided.\n\n1. **Estimate for Atlanta:**\n - Anita estimates 50,000 fans.\n - The a... |
DeepScaleR | [
{
"role": "user",
"content": "Anička received a rectangular cake for her birthday. She cut the cake with two straight cuts. The first cut was made such that it intersected both longer sides of the rectangle at one-third of their length. The second cut was made such that it intersected both shorter sides of ... | math | {
"style": "rule",
"ground_truth": "2/15"
} | {
"index": 4999,
"solution": ""
} |
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