data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Eight distinct points, $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, $P_6$, $P_7$, and $P_8$, are evenly spaced around a circle. If four points are chosen at random from these eight points to form two chords, what is the probability that the chord formed by the first two points chosen int... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 8300,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eight friends ate at a restaurant and agreed to share the bill equally. Because Judi forgot her money, each of her seven friends paid an extra \\$2.50 to cover her portion of the total bill. What was the total bill?"
}
] | math | {
"style": "rule",
"ground_truth": "$120"
} | {
"index": 8301,
"solution": "1. **Define the variables**: Let $m$ be the total bill that the 8 friends were supposed to split equally. Therefore, each friend's share would have been $\\frac{m}{8}$.\n\n2. **Adjust for Judi's share**: Since Judi did not pay, each of the remaining 7 friends paid an extra $2.50 to cov... |
DeepScaleR | [
{
"role": "user",
"content": "Eight hockey teams are competing against each other in a single round to advance to the semifinals. What is the minimum number of points that guarantees a team advances to the semifinals?"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 8302,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eight identical cubes with of size $1 \\times 1 \\times 1$ each have the numbers $1$ through $6$ written on their faces with the number $1$ written on the face opposite number $2$ , number $3$ written on the face opposite number $5$ , and number $4$ written on the... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 8303,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eight numbers \\( a_{1}, a_{2}, a_{3}, a_{4} \\) and \\( b_{1}, b_{2}, b_{3}, b_{4} \\) satisfy the equations:\n\n\\[\n\\left\\{\n\\begin{array}{l}\na_{1} b_{1} + a_{2} b_{3} = 1 \\\\\na_{1} b_{2} + a_{2} b_{4} = 0 \\\\\na_{3} b_{1} + a_{4} b_{3} = 0 \\\\\na_{3} b_{2} + a_{4} b... | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"index": 8304,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eight numbers \\( a_{1}, a_{2}, a_{3}, a_{4} \\) and \\( b_{1}, b_{2}, b_{3}, b_{4} \\) satisfy the following equations:\n\n$$\n\\left\\{\\begin{array}{c}\na_{1} b_{1}+a_{2} b_{3}=1 \\\\\na_{1} b_{2}+a_{2} b_{4}=0 \\\\\na_{3} b_{1}+a_{4} b_{3}=0 \\\\\na_{3} b_{2}+a_{4} b_{4}=1\... | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"index": 8305,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eight people are sitting around a circular table for a meeting, and the recorder is sitting between the leader and the deputy leader. Calculate the total number of different seating arrangements possible, considering arrangements that can be made identical through rotation as t... | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 8306,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{47}{256}"
} | {
"index": 8307,
"solution": "1. **Understanding the Problem:**\n We need to find the probability that no two adjacent people will stand after flipping their coins around a circular table with 8 people. Each person flips a fair coin, meaning each flip has a probability of $\\frac{1}{2}$ for heads (stand) and $\\f... |
DeepScaleR | [
{
"role": "user",
"content": "Eight points are chosen on a circle, and chords are drawn connecting every pair of points. No three chords intersect in a single point inside the circle. How many triangles with all three vertices in the interior of the circle are created?"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 8308,
"solution": "To solve this problem, we need to determine how many triangles can be formed such that all three vertices are in the interior of the circle. This is different from simply choosing any three points on the circle to form a triangle, as the problem specifies that the vertices must be in t... |
DeepScaleR | [
{
"role": "user",
"content": "Eight points are chosen on the circumference of a circle, labelled $P_{1}, P_{2}, \\ldots, P_{8}$ in clockwise order. A route is a sequence of at least two points $P_{a_{1}}, P_{a_{2}}, \\ldots, P_{a_{n}}$ such that if an ant were to visit these points in their given order, sta... | math | {
"style": "rule",
"ground_truth": "8744"
} | {
"index": 8309,
"solution": "Solution 1: How many routes are there if we are restricted to $n$ available points, and we must use all $n$ of them? The answer is $n 2^{n-2}$ : first choose the starting point, then each move after that must visit one of the two neighbors of your expanding region of visited points (do... |
DeepScaleR | [
{
"role": "user",
"content": "Eight points are spaced around at intervals of one unit around a $2 \\times 2$ square, as shown. Two of the $8$ points are chosen at random. What is the probability that the two points are one unit apart?\n[asy]\nsize((50));\ndot((5,0));\ndot((5,5));\ndot((0,5));\ndot((-5,5));\... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{7}"
} | {
"index": 8310,
"solution": "1. **Identify the total number of ways to choose two points from the eight available points.** \n Since there are 8 points, the number of ways to choose 2 points out of these 8 is given by the combination formula $\\binom{n}{k}$, where $n$ is the total number of items to choose from... |
DeepScaleR | [
{
"role": "user",
"content": "Eight positive integers are written on the faces of a cube. Each vertex is labeled with the product of the three numbers on the faces adjacent to that vertex. If the sum of the numbers on the vertices is equal to $2107$, what is the sum of the numbers written on the faces?"
}... | math | {
"style": "rule",
"ground_truth": "57"
} | {
"index": 8311,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eight positive integers are written on the faces of a square prism (two bases and a lateral surface consisting of four faces). Each vertex is labeled with the product of the three numbers on the faces sharing that vertex (two from the lateral faces and one from the bases). If t... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"index": 8312,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eight red boxes and eight blue boxes are randomly placed in four stacks of four boxes each. The probability that exactly one of the stacks consists of two red boxes and two blue boxes is $\\frac{m}{n}$ , where m and n are relatively prime positive integers. Find $m + n$ ."
... | math | {
"style": "rule",
"ground_truth": "843"
} | {
"index": 8313,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eight semicircles line the inside of a square with side length 2 as shown. What is the radius of the circle tangent to all of these semicircles?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}-1}{2}"
} | {
"index": 8314,
"solution": "1. **Understanding the Configuration**: \n - The square has a side length of 2.\n - Each semicircle inside the square has a diameter of 1 (since 4 semicircles fit along one side of the square, and the side length is 2).\n - Therefore, the radius of each semicircle is $\\frac{1}{2... |
DeepScaleR | [
{
"role": "user",
"content": "Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres?"
}
] | math | {
"style": "rule",
"ground_truth": "1+\\sqrt{3}"
} | {
"index": 8315,
"solution": "1. **Understanding the setup**: Each of the eight spheres is tangent to the three coordinate planes in its respective octant. This means each sphere's center is at a distance of 1 (its radius) from each of the coordinate planes. Thus, the coordinates of the centers of these spheres are... |
DeepScaleR | [
{
"role": "user",
"content": "Eight spheres of radius 100 are placed on a flat surface so that each sphere is tangent to two others and their centers are the vertices of a regular octagon. A ninth sphere is placed on the flat surface so that it is tangent to each of the other eight spheres. The radius of th... | math | {
"style": "rule",
"ground_truth": "152"
} | {
"index": 8316,
"solution": "Isolate a triangle, with base length $200$ (a side of the octagon). This triangle is obviously isoceles. Denote the other side length as $x$. Since the interior angle is $45$ degrees (due to the shape being an octagon), then we can apply Law of Cosines to this triangle. We get: \\begin... |
DeepScaleR | [
{
"role": "user",
"content": "Eight spheres of radius 2 are each tangent to the faces of an inner cube centered at the origin, with each sphere located in one of the octants, and the side length of the cube is 6. Find the radius of the smallest sphere, centered at the origin, that encloses these eight spher... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3} + 2"
} | {
"index": 8317,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eight strangers are preparing to play bridge. How many ways can they be grouped into two bridge games, meaning into unordered pairs of unordered pairs of people?"
}
] | math | {
"style": "rule",
"ground_truth": "315"
} | {
"index": 8318,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eight students from Adams school worked for $4$ days, six students from Bentley school worked for $6$ days, and seven students from Carter school worked for $10$ days. If a total amount of $\\ 1020$ was paid for the students' work, with each student receiving the same amount fo... | math | {
"style": "rule",
"ground_truth": "517.39"
} | {
"index": 8319,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eight students from a university are planning to carpool for a trip, with two students from each of the grades one, two, three, and four. How many ways are there to arrange the four students in car A, such that the last two students are from the same grade?"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 8320,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eight students from a university are preparing to carpool for a trip. There are two students from each grade level—freshmen, sophomores, juniors, and seniors—divided into two cars, Car A and Car B, with each car seating four students. The seating arrangement of the four student... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 8321,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eight students participate in a pie-eating contest. The graph shows the number of pies eaten by each participating student. Sarah ate the most pies and Tom ate the fewest. Calculate how many more pies than Tom did Sarah eat and find the average number of pies eaten by all the s... | math | {
"style": "rule",
"ground_truth": "4.5"
} | {
"index": 8322,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eight teams participated in a football tournament, and each team played exactly once against each other team. If a match was drawn then both teams received 1 point; if not then the winner of the match was awarded 3 points and the loser received no points. At the end of the tour... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 8323,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eight teams participated in a football tournament, and each team played exactly once against each other team. If a match was drawn then both teams received 1 point; if not then the winner of the match was awarded 3 points and the loser received no points. At the end of the tour... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 8324,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eight women of different heights are at a party. Each woman decides to only shake hands with women shorter than herself. How many handshakes take place?"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 8325,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eighty bricks, each measuring $3''\\times9''\\times18''$, are to be stacked one on top of another to form a tower 80 bricks tall. Each brick can be oriented so it contributes $3''$, $9''$, or $18''$ to the total height of the tower. How many different tower heights can be achie... | math | {
"style": "rule",
"ground_truth": "401"
} | {
"index": 8326,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eighty bricks, each measuring \\(3'' \\times 8'' \\times 15''\\), are to be stacked to form a column. Each brick can be oriented to contribute \\(3''\\), \\(8''\\), or \\(15''\\) to the total column height. Determine how many different total column heights can be achieved using... | math | {
"style": "rule",
"ground_truth": "961"
} | {
"index": 8327,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eighty percent of adults drink coffee and seventy percent drink tea. What is the smallest possible percent of adults who drink both coffee and tea?"
}
] | math | {
"style": "rule",
"ground_truth": "50\\%"
} | {
"index": 8328,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eighty percent of dissatisfied customers leave angry reviews about a certain online store. Among satisfied customers, only fifteen percent leave positive reviews. This store has earned 60 angry reviews and 20 positive reviews. Using this data, estimate the probability that the ... | math | {
"style": "rule",
"ground_truth": "0.64"
} | {
"index": 8329,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eighty-five more than the square of a number is the same as the square of the quantity that is $17$ less than the number. What is the number?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 8330,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Either increasing the radius of a cylinder by 4 inches or the height by 10 inches results in the same volume. The original height of the cylinder is 5 inches. What is the original radius in inches?"
}
] | math | {
"style": "rule",
"ground_truth": "2 + 2\\sqrt{3}"
} | {
"index": 8331,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Either increasing the radius or the height of a cylinder by six inches will result in the same volume. The original height of the cylinder is two inches. What is the original radius in inches?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 8332,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Elbert and Yaiza each draw 10 cards from a 20-card deck with cards numbered $1,2,3, \\ldots, 20$. Then, starting with the player with the card numbered 1, the players take turns placing down the lowest-numbered card from their hand that is greater than every card previously pla... | math | {
"style": "rule",
"ground_truth": "324"
} | {
"index": 8333,
"solution": "Put each card in order and label them based on if Elbert or Yaiza got them. We will get a string of E's and Y's like EEYYYE ..., and consider the \"blocks\" of consecutive letters. It is not hard to see that only the first card of each block is played, and the number of cards played is... |
DeepScaleR | [
{
"role": "user",
"content": "Eleanor is making chocolate chip cookies for her friends. If she divides the cookies equally among $11$ of her friends, she'll have $4$ cookies left over. If she divides the cookies equally among $7$ of her friends, she'll have $1$ cookie left over. Assuming that Eleanor made f... | math | {
"style": "rule",
"ground_truth": "107"
} | {
"index": 8334,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Elective 4-4: Coordinate System and Parametric Equations\n\nGiven the parametric equations of curve \\\\(C\\\\) as \\\\(\\begin{cases}x=2\\cos \\left(\\theta\\right) \\\\ y= \\sqrt{3}\\sin \\left(\\theta\\right)\\end{cases} \\\\), in the same plane Cartesian coordinate system, ... | math | {
"style": "rule",
"ground_truth": "\\dfrac{3\\sqrt{3}}{5}"
} | {
"index": 8335,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Elective 4-4: Coordinate System and Parametric Equations\n\nIn the Cartesian coordinate system $xOy$, the curve $C$ is given by the parametric equations $\\begin{cases} & x=2\\sqrt{3}\\cos \\alpha \\\\ & y=2\\sin \\alpha \\end{cases}$, where $\\alpha$ is the parameter, $\\alpha... | math | {
"style": "rule",
"ground_truth": "6 \\sqrt{2}"
} | {
"index": 8336,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Elective 4-4: Coordinate System and Parametric Equations\n\nIn the Cartesian coordinate system $xOy$, the parametric equation of line $l_1$ is $\\begin{cases} x=2+t \\\\ y=kt \\end{cases}$ ($t$ is the parameter), and the parametric equation of line $l_2$ is $\\begin{cases} x=-2... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"index": 8337,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Elective 4-4: Coordinate System and Parametric Equations\n\nIn the Cartesian coordinate system $xOy$, the parametric equation of line $l_1$ is $\\begin{cases}x=2+t \\\\ y=kt\\end{cases}$ (where $t$ is the parameter), and the parametric equation of line $l_2$ is $\\begin{cases}x... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"index": 8338,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Elective 4-4: Coordinate System and Parametric Equations\n\nIn the Cartesian coordinate system $xOy$, with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. If the polar equation of curve $C$ is $\\rho\\cos... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{6}"
} | {
"index": 8339,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Elective 4-5: Selected Topics on Inequalities. Given the function $f(x) = |2x-1| + |2x+3|$.\n\n$(1)$ Solve the inequality $f(x) \\geqslant 6$;\n\n$(2)$ Let the minimum value of $f(x)$ be $m$, and let the positive real numbers $a, b$ satisfy $2ab + a + 2b = m$. Find the minimum ... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5} - 2"
} | {
"index": 8340,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Elena drives 45 miles in the first hour, but realizes that she will be 45 minutes late if she continues at the same speed. She increases her speed by 20 miles per hour for the rest of the journey and arrives 15 minutes early. Determine the total distance from Elena's home to th... | math | {
"style": "rule",
"ground_truth": "191.25"
} | {
"index": 8341,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Elena earns $\\$ 13.25$ per hour working at a store. How much does Elena earn in 4 hours?"
}
] | math | {
"style": "rule",
"ground_truth": "\\$53.00"
} | {
"index": 8342,
"solution": "Elena works for 4 hours and earns $\\$ 13.25$ per hour. This means that she earns a total of $4 \\times \\$ 13.25=\\$ 53.00$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Elena intends to buy 7 binders priced at $\\textdollar 3$ each. Coincidentally, a store offers a 25% discount the next day and an additional $\\textdollar 5$ rebate for purchases over $\\textdollar 20$. Calculate the amount Elena could save by making her purchase on the day of ... | math | {
"style": "rule",
"ground_truth": "10.25"
} | {
"index": 8343,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eleven girls are standing around a circle. A ball is thrown clockwise around the circle. The first girl, Ami, starts with the ball, skips the next three girls and throws to the fifth girl, who then skips the next three girls and throws the ball to the ninth girl. If the throwin... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 8344,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eleven is the fifth prime number. What is the tenth prime number?"
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 8345,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eleven members of the Middle School Math Club each paid the same amount for a guest speaker to talk about problem solving at their math club meeting. They paid their guest speaker $\\$1A2. What is the missing digit $A$ of this $3$-digit number?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 8346,
"solution": "1. **Understanding the Problem**: The total amount paid to the guest speaker is $\\textdollar\\underline{1} \\underline{A} \\underline{2}$, where $A$ is the missing digit. Since the total amount is paid equally by 11 members, the total amount must be divisible by 11.\n\n2. **Using the ... |
DeepScaleR | [
{
"role": "user",
"content": "Eleven percent of what number is seventy-seven?"
}
] | math | {
"style": "rule",
"ground_truth": "700"
} | {
"index": 8347,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eleven positive integers from a list of fifteen positive integers are $3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23$. What is the largest possible value of the median of this list of fifteen positive integers?"
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 8348,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Eli, Joy, Paul, and Sam want to form a company; the company will have 16 shares to split among the 4 people. The following constraints are imposed: - Every person must get a positive integer number of shares, and all 16 shares must be given out. - No one person can have more sh... | math | {
"style": "rule",
"ground_truth": "315"
} | {
"index": 8349,
"solution": "We are finding the number of integer solutions to $a+b+c+d=16$ with $1 \\leq a, b, c, d \\leq 8$. We count the number of solutions to $a+b+c+d=16$ over positive integers, and subtract the number of solutions in which at least one variable is larger than 8. If at least one variable is l... |
DeepScaleR | [
{
"role": "user",
"content": "Eliminate every second number in a clockwise direction from numbers $1, 2, 3, \\cdots, 2001$ that have been placed on a circle, starting with the number 2, until only one number remains. What is the last remaining number?"
}
] | math | {
"style": "rule",
"ground_truth": "1955"
} | {
"index": 8350,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Elisa creates a sequence in a manner similar to Jacob but starts with the first term as 10. Each succeeding term depends on the outcome of flipping a fair coin: If it lands heads, the next term is obtained by doubling the previous term and subtracting 1; if it lands tails, the ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 8351,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Elisa swims laps in the pool. When she first started, she completed 10 laps in 25 minutes. Now, she can finish 12 laps in 24 minutes. By how many minutes has she improved her lap time?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 8352,
"solution": "1. **Calculate the initial lap time**: \n When Elisa started swimming, she completed 10 laps in 25 minutes. To find the time it took for one lap, we divide the total time by the number of laps:\n \\[\n \\text{Initial lap time} = \\frac{25 \\text{ minutes}}{10 \\text{ laps}} = 2.5... |
DeepScaleR | [
{
"role": "user",
"content": "Elisenda has a piece of paper in the shape of a triangle with vertices $A, B$, and $C$ such that $A B=42$. She chooses a point $D$ on segment $A C$, and she folds the paper along line $B D$ so that $A$ lands at a point $E$ on segment $B C$. Then, she folds the paper along line ... | math | {
"style": "rule",
"ground_truth": "168+48 \\sqrt{7}"
} | {
"index": 8353,
"solution": "Let $F$ be the midpoint of segment $D C$. Evidently $\\angle A D B=60^{\\circ}=\\angle B D E=\\angle E D C$. Moreover, we have $B D=D F=F C, A D=D E$, and $A B=B E$. Hence angle bisector on $B D C$ gives us that $B E=42, E C=84$, and hence angle bisector on $A B C$ gives us that if $A ... |
DeepScaleR | [
{
"role": "user",
"content": "Elizabetta wants to write the integers 1 to 9 in the regions of the shape shown, with one integer in each region. She wants the product of the integers in any two regions that have a common edge to be not more than 15. In how many ways can she do this?"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 8354,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Ella adds up all the odd integers from 1 to 499, inclusive. Mike adds up all the integers from 1 to 500, inclusive. What is Ella's sum divided by Mike's sum?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{500}{1001}"
} | {
"index": 8355,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Ella walks to her university library, averaging 80 steps per minute, with each of her steps covering 80 cm. It takes her 20 minutes to get to the library. Her friend Tia, going to the same library by the same route, averages 120 steps per minute, but her steps are only 70 cm lo... | math | {
"style": "rule",
"ground_truth": "15.24"
} | {
"index": 8356,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Ellen wants to color some of the cells of a $4 \\times 4$ grid. She wants to do this so that each colored cell shares at least one side with an uncolored cell and each uncolored cell shares at least one side with a colored cell. What is the largest number of cells she can color... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 8357,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Ellie and Sam run on the same circular track but in opposite directions, with Ellie running counterclockwise and Sam running clockwise. Ellie completes a lap every 120 seconds, while Sam completes a lap every 75 seconds. They both start from the same starting line simultaneousl... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{12}"
} | {
"index": 8358,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Ellie's drawer of hair clips contains 4 red clips, 5 blue clips, and 7 green clips. Each morning, she randomly chooses one hair clip to wear for the day. She returns this clip to the drawer each evening. One morning, Kyne removes $k$ hair clips before Ellie can make her daily s... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 8359,
"solution": "Before Kyne removes hair clips, Ellie has 4 red clips and $4+5+7=16$ clips in total, so the probability that she randomly chooses a red clip is $\frac{4}{16}$ which equals $\frac{1}{4}$. After Kyne removes the clips, the probability that Ellie chooses a red clip is $2 \times \frac{1}{4... |
DeepScaleR | [
{
"role": "user",
"content": "Ellina has twelve blocks, two each of red ($\\textbf{R}$), blue ($\\textbf{B}$), yellow ($\\textbf{Y}$), green ($\\textbf{G}$), orange ($\\textbf{O}$), and purple ($\\textbf{P}$). Call an arrangement of blocks $\\textit{even}$ if there is an even number of blocks between each p... | math | {
"style": "rule",
"ground_truth": "247"
} | {
"index": 8360,
"solution": "We can simply use constructive counting. First, let us place the red balls; choose the first slot in $12$ ways, and the second in $6$ ways, because the number is cut in half due to the condition in the problem. This gives $12 \\cdot 6$ ways to place the blue balls. Similarly, there are... |
DeepScaleR | [
{
"role": "user",
"content": "Elliot and Emily run a 12 km race. They start at the same point, run 6 km up a hill, and return to the starting point by the same route. Elliot has a 8 minute head start and runs at the rate of 12 km/hr uphill and 18 km/hr downhill. Emily runs 14 km/hr uphill and 20 km/hr downh... | math | {
"style": "rule",
"ground_truth": "\\frac{169}{48}"
} | {
"index": 8361,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Elmer the emu takes $44$ equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in $12$ equal leaps. The telephone poles are evenly spaced, and the $41$st pole along this road is exactly one mile ($5280$ feet) fr... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 8362,
"solution": "1. **Calculate the number of gaps between the poles**: \n There are 41 poles, so there are \\(41 - 1 = 40\\) gaps between the poles.\n\n2. **Calculate the total number of strides and leaps**:\n - Elmer takes 44 strides per gap, so for 40 gaps, he takes \\(44 \\times 40 = 1760\\) st... |
DeepScaleR | [
{
"role": "user",
"content": "Elmer's new car gives $50\\%$ percent better fuel efficiency, measured in kilometers per liter, than his old car. However, his new car uses diesel fuel, which is $20\\%$ more expensive per liter than the gasoline his old car used. By what percent will Elmer save money if he use... | math | {
"style": "rule",
"ground_truth": "20\\%"
} | {
"index": 8363,
"solution": "1. **Define the fuel efficiency and cost per liter:**\n - Let the fuel efficiency of Elmer's old car be $x$ kilometers per liter.\n - The new car has $50\\%$ better fuel efficiency, so it runs at $\\frac{3}{2}x$ kilometers per liter.\n\n2. **Convert the new car's efficiency to the ... |
DeepScaleR | [
{
"role": "user",
"content": "Elmo bakes cookies at a rate of one per 5 minutes. Big Bird bakes cookies at a rate of one per 6 minutes. Cookie Monster *consumes* cookies at a rate of one per 4 minutes. Together Elmo, Big Bird, Cookie Monster, and Oscar the Grouch produce cookies at a net rate of one per 8 m... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 8364,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Elmo makes $N$ sandwiches for a fundraiser. For each sandwich he uses $B$ globs of peanut butter at $4$ cents per glob and $J$ blobs of jam at $5$ cents per blob. The cost of the peanut butter and jam to make all the sandwiches is $\\$2.53$. Assume that $B$, $J$, and $N$ are ... | math | {
"style": "rule",
"ground_truth": "\\$1.65"
} | {
"index": 8365,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Elon Musk's Starlink project belongs to his company SpaceX. He plans to use tens of thousands of satellites to provide internet services to every corner of the Earth. A domestic company also plans to increase its investment in the development of space satellite networks to deve... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 8366,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Emilia wishes to create a basic solution with $7 \\%$ hydroxide $(\\mathrm{OH})$ ions. She has three solutions of different bases available: $10 \\%$ rubidium hydroxide $(\\mathrm{Rb}(\\mathrm{OH})$ ), $8 \\%$ cesium hydroxide $(\\mathrm{Cs}(\\mathrm{OH})$ ), and $5 \\%$ franci... | math | {
"style": "rule",
"ground_truth": "1 \\%"
} | {
"index": 8367,
"solution": "Suppose that Emilia uses $R$ liters of $\\mathrm{Rb}(\\mathrm{OH}), C$ liters of $\\mathrm{Cs}(\\mathrm{OH})$, and $F$ liters of $\\mathrm{Fr}(\\mathrm{OH})$, then we have $$\\frac{10 \\% \\cdot R+8 \\% \\cdot C+5 \\% \\cdot F}{R+C+F}=7 \\% \\text { and } \\frac{5 \\% \\cdot F}{R+C+F} ... |
DeepScaleR | [
{
"role": "user",
"content": "Emilia writes down the numbers $5, x$, and 9. Valentin calculates the mean (average) of each pair of these numbers and obtains 7, 10, and 12. What is the value of $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 8368,
"solution": "Since the average of 5 and 9 is $\frac{5+9}{2}=7$, then the averages of 5 and $x$ and of $x$ and 9 must be 10 and 12. In other words, $\frac{5+x}{2}$ and $\frac{x+9}{2}$ are equal to 10 and 12 in some order. Adding these, we obtain $\frac{5+x}{2}+\frac{x+9}{2}=10+12$ or $\frac{14+2x}{2... |
DeepScaleR | [
{
"role": "user",
"content": "Emily and John each solved three-quarters of the homework problems individually and the remaining one-quarter together. Emily correctly answered 70% of the problems she solved alone, achieving an overall accuracy of 76% on her homework. John had an 85% success rate with the pro... | math | {
"style": "rule",
"ground_truth": "87.25\\%"
} | {
"index": 8369,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Emily cycles at a constant rate of 15 miles per hour, and Leo runs at a constant rate of 10 miles per hour. If Emily overtakes Leo when he is 0.75 miles ahead of her, and she can view him in her mirror until he is 0.6 miles behind her, calculate the time in minutes it takes for... | math | {
"style": "rule",
"ground_truth": "16.2"
} | {
"index": 8370,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Emily has 8 blue marbles and 7 red marbles. She randomly selects a marble, notes its color, and returns it to the bag. She repeats this process 6 times. What is the probability that she selects exactly three blue marbles?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3512320}{11390625}"
} | {
"index": 8371,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Emily ordered her playing cards by suit in the order $$A,2,3,4,5,6,7,8,9,10,J,Q,K,A,2,3,\\cdots.$$What is the $42$nd card?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 8372,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Emily paid for a $\\$2$ sandwich using 50 coins consisting of pennies, nickels, and dimes, and received no change. How many dimes did Emily use?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 8373,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Emily sees a ship traveling at a constant speed along a straight section of a river. She walks parallel to the riverbank at a uniform rate faster than the ship. She counts $210$ equal steps walking from the back of the ship to the front. Walking in the opposite direction, she c... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 8374,
"solution": "1. **Define Variables:**\n Let $L$ be the length of the ship, $E$ be the length of Emily's step, and $S$ be the length of the ship's step (the distance the ship travels while Emily takes one step).\n\n2. **Set Up Equations:**\n - When Emily walks from the back of the ship to the fr... |
DeepScaleR | [
{
"role": "user",
"content": "Emily's broken clock runs backwards at five times the speed of a regular clock. How many times will it display the correct time in the next 24 hours? Note that it is an analog clock that only displays the numerical time, not AM or PM. The clock updates continuously."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 8375,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Emily's quiz scores so far are: 92, 95, 87, 89 and 100. What score does she need to get on the sixth quiz to make the arithmetic mean of the six scores equal 93?"
}
] | math | {
"style": "rule",
"ground_truth": "95"
} | {
"index": 8376,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Emma had just been given some coins by her parents. On the way to school she lost exactly half of them, and then by retracing her steps she found exactly four-fifths of the coins she had lost. What fraction of the coins that she received from her parents were still missing af... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{10}"
} | {
"index": 8377,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Emma has $100$ fair coins. She flips all the coins once. Any coin that lands on tails is tossed again, and the process continues up to four times for coins that sequentially land on tails. Calculate the expected number of coins that finally show heads."
}
] | math | {
"style": "rule",
"ground_truth": "93.75"
} | {
"index": 8378,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Emma writes down the smallest positive multiple of 30 that is a perfect square, the smallest positive multiple of 30 that is a perfect cube, and all the multiples of 30 between them. How many integers are in Emma's list?"
}
] | math | {
"style": "rule",
"ground_truth": "871"
} | {
"index": 8379,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Emma's telephone number is $548-1983$ and her apartment number contains different digits. The sum of the digits in her four-digit apartment number is the same as the sum of the digits in her phone number. What is the lowest possible value for Emma’s apartment number?"
}
] | math | {
"style": "rule",
"ground_truth": "9876"
} | {
"index": 8380,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Employees from department X are 30, while the employees from department Y are 20. Since employees from the same department do not interact, the number of employees from department X that will shake hands with the employees from department Y equals 30, and the number of employee... | math | {
"style": "rule",
"ground_truth": "600"
} | {
"index": 8381,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Encrypt integers by the following method: the digit of each number becomes the units digit of its product with 7, then replace each digit _a_ with $10 - _a_$. If a number is encrypted by the above method and becomes 473392, then the original number is ______."
}
] | math | {
"style": "rule",
"ground_truth": "891134"
} | {
"index": 8382,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Encrypt integers using the following method: each digit of the number becomes the units digit of its product with 7, then replace each digit $a$ with $10-a$. If a number is encrypted using the above method and the result is 473392, then the original number is ______."
}
] | math | {
"style": "rule",
"ground_truth": "891134"
} | {
"index": 8383,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Engineer Sergei received a research object with a volume of approximately 200 monoliths (a container designed for 200 monoliths, which was almost completely filled). Each monolith has a specific designation (either \"sand loam\" or \"clay loam\") and genesis (either \"marine\" ... | math | {
"style": "rule",
"ground_truth": "77"
} | {
"index": 8384,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Engineers use the formula $L=\\frac{25T^4}{H^2}$ to find the crushing load for square pillars. If $T = 4$ and $H = 8$, what is the value of $L$?"
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 8385,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Enlarge each edge of a graph by four times its original size. This is equivalent to enlarging the graph by a scale of \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "4:1"
} | {
"index": 8386,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Enter all the solutions to\n\\[ \\sqrt{4x-3}+\\frac{10}{\\sqrt{4x-3}}=7,\\]separated by commas."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac 74,7"
} | {
"index": 8387,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Entrepreneurs Vasiliy Petrovich and Petr Gennadievich opened a clothing factory \"ViP.\" Vasiliy Petrovich invested 200 thousand rubles, while Petr Gennadievich invested 350 thousand rubles. The factory was successful, and after a year, Anastasia Alekseevna approached them with... | math | {
"style": "rule",
"ground_truth": "1000000"
} | {
"index": 8388,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Equilateral $\\triangle A B C$ has side length 6. Let $\\omega$ be the circle through $A$ and $B$ such that $C A$ and $C B$ are both tangent to $\\omega$. A point $D$ on $\\omega$ satisfies $C D=4$. Let $E$ be the intersection of line $C D$ with segment $A B$. What is the lengt... | math | {
"style": "rule",
"ground_truth": "\\frac{20}{13}"
} | {
"index": 8389,
"solution": "Let $F$ be the second intersection of line $C D$ with $\\omega$. By power of a point, we have $C F=9$, so $D F=5$. This means that $\\frac{[A D B]}{[A F B]}=\\frac{D E}{E F}=\\frac{D E}{5-D E}$. Now, note that triangle $C A D$ is similar to triangle $C F A$, so $\\frac{F A}{A D}=\\frac... |
DeepScaleR | [
{
"role": "user",
"content": "Equilateral $\\triangle ABC$ has side length $1$, and squares $ABDE$, $BCHI$, $CAFG$ lie outside the triangle. What is the area of hexagon $DEFGHI$?"
}
] | math | {
"style": "rule",
"ground_truth": "3+\\sqrt3"
} | {
"index": 8390,
"solution": "1. **Identify the Configuration**: We start by noting that $\\triangle ABC$ is equilateral with side length $1$. Squares $ABDE$, $BCHI$, and $CAFG$ are constructed outside the triangle. We need to find the area of hexagon $DEFGHI$.\n\n2. **Understanding the Hexagon**: Each side of the ... |
DeepScaleR | [
{
"role": "user",
"content": "Equilateral $\\triangle ABC$ has side length $2$, $M$ is the midpoint of $\\overline{AC}$, and $C$ is the midpoint of $\\overline{BD}$. What is the area of $\\triangle CDM$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"index": 8391,
"solution": "\n1. **Understanding the Problem:**\n - We have an equilateral triangle $\\triangle ABC$ with side length $2$.\n - $M$ is the midpoint of $\\overline{AC}$, and $C$ is the midpoint of $\\overline{BD}$.\n - We need to find the area of $\\triangle CDM$.\n\n2. **Identifying Key Prope... |
DeepScaleR | [
{
"role": "user",
"content": "Equilateral $\\triangle ABC$ has side length $2$, and shapes $ABDE$, $BCHT$, $CAFG$ are formed outside the triangle such that $ABDE$ and $CAFG$ are squares, and $BCHT$ is an equilateral triangle. What is the area of the geometric shape formed by $DEFGHT$?\nA) $3\\sqrt{3} - 1$\n... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3} - 2"
} | {
"index": 8392,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Equilateral $\\triangle ABC$ has side length $600$. Points $P$ and $Q$ lie outside the plane of $\\triangle ABC$ and are on opposite sides of the plane. Furthermore, $PA=PB=PC$, and $QA=QB=QC$, and the planes of $\\triangle PAB$ and $\\triangle QAB$ form a $120^{\\circ}$ dihedr... | math | {
"style": "rule",
"ground_truth": "450"
} | {
"index": 8393,
"solution": "Draw a good diagram. Draw $CH$ as an altitude of the triangle. Scale everything down by a factor of $100\\sqrt{3}$, so that $AB=2\\sqrt{3}$. Finally, call the center of the triangle U. Draw a cross-section of the triangle via line $CH$, which of course includes $P, Q$. From there, we c... |
DeepScaleR | [
{
"role": "user",
"content": "Equilateral $\\triangle ABC$ has side length $\\sqrt{111}$. There are four distinct triangles $AD_1E_1$, $AD_1E_2$, $AD_2E_3$, and $AD_2E_4$, each congruent to $\\triangle ABC$, with $BD_1 = BD_2 = \\sqrt{11}$. Find $\\sum_{k=1}^4(CE_k)^2$."
}
] | math | {
"style": "rule",
"ground_truth": "677"
} | {
"index": 8394,
"solution": "This method uses complex numbers with $A$ as the origin. Let $A=0$, $B=\\sqrt{111}$, $C = \\sqrt{111}\\theta$, where $\\theta = e^{i \\pi/3} = \\frac{1}{2} + \\frac{\\sqrt{3}}{2}i$.\n Also, let $x$ be $D_1$ or $D_2$. Then\n$|x|=\\sqrt{111}, |x-\\sqrt{111}|=\\sqrt{11}$\n Therefore, $11 ... |
DeepScaleR | [
{
"role": "user",
"content": "Equilateral $\\triangle ABC$ is inscribed in a circle of radius $2$. Extend $\\overline{AB}$ through $B$ to point $D$ so that $AD=13,$ and extend $\\overline{AC}$ through $C$ to point $E$ so that $AE = 11.$ Through $D,$ draw a line $l_1$ parallel to $\\overline{AE},$ and throug... | math | {
"style": "rule",
"ground_truth": "865"
} | {
"index": 8395,
"solution": "Note that $AB=2\\sqrt3$, $DF=11$, and $EF=13$. If we take a homothety of the parallelogram with respect to $A$, such that $F$ maps to $G$, we see that $\\frac{[ABG]}{[ACG]}=\\frac{11}{13}$. Since $\\angle AGB=\\angle AGC=60^{\\circ}$, from the sine area formula we have $\\frac{BG}{CG}=... |
DeepScaleR | [
{
"role": "user",
"content": "Equilateral $\\triangle DEF$ has side length $300$. Points $R$ and $S$ lie outside the plane of $\\triangle DEF$ and are on opposite sides of the plane. Furthermore, $RA=RB=RC$, and $SA=SB=SC$, and the planes containing $\\triangle RDE$ and $\\triangle SDE$ form a $150^{\\circ}... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"index": 8396,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Equilateral triangle $ABC$ and square $BCDE$ are coplanar, as shown. What is the number of degrees in the measure of angle $CAD$?\n\n[asy]\nsize(70);\ndraw((0,0)--(20,0)--(20,20)--(0,20)--cycle);\ndraw((0,20)--(10,37.3)--(20,20));\ndraw((10,37.3)--(20,0));\nlabel(\"$A$\",(10,37... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 8397,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Equilateral triangle $ABC$ has a side length of $12$. There are three distinct triangles $AD_1E_1$, $AD_2E_2$, and $AD_3E_3$, each congruent to triangle $ABC$, with $BD_1 = BD_2 = BD_3 = 6$. Find $\\sum_{k=1}^3(CE_k)^2$."
}
] | math | {
"style": "rule",
"ground_truth": "432"
} | {
"index": 8398,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Equilateral triangle $ABC$ has a side length of $\\sqrt{144}$. There are four distinct triangles $AD_1E_1$, $AD_1E_2$, $AD_2E_3$, and $AD_2E_4$, each congruent to triangle $ABC$, with $BD_1 = BD_2 = \\sqrt{12}$. Additionally, $BD_1$ and $BD_2$ are placed such that $\\angle ABD_... | math | {
"style": "rule",
"ground_truth": "576"
} | {
"index": 8399,
"solution": ""
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.