data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "During the New Year, Xiaoming's family bought many bottles of juice. On New Year's Eve, they drank half of the total amount minus 1 bottle. On the first day of the New Year, they drank half of the remaining amount again. On the second day of the New Year, they drank half of the... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 8100,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "During the November monthly exam at our school, approximately 1,000 science students participated, with mathematics scores distributed normally as $\\xi \\sim N(100, a^2), (a > 0)$, and a full score of 150. The statistics showed that about 60% of the students scored between 80 ... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 8101,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "During the Qingming Festival, a certain school, in order to commemorate the revolutionary martyrs, requires students to participate in the \"Qingming Sacrifice to the Martyrs\" activity by either visiting the Revolutionary Martyrs Memorial Hall or participating online. Students... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{21}"
} | {
"index": 8102,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "During the Spring Festival, a supermarket holds a promotional lottery event. When the amount spent by a customer reaches a certain threshold, they can participate in a lottery. The rules of the event are: from a box containing 3 black balls, 2 red balls, and 1 white ball (ident... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{9}"
} | {
"index": 8103,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "During the car ride home, Michael looks back at his recent math exams. A problem on Michael's calculus mid-term gets him starting thinking about a particular quadratic,\\[x^2-sx+p,\\]with roots $r_1$ and $r_2$. He notices that\\[r_1+r_2=r_1^2+r_2^2=r_1^3+r_2^3=\\cdots=r_1^{2007... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 8104,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "During the festive season when the moon is full and the country is celebrating together, a supermarket plans to reduce the selling price of grapes that cost $16$ yuan per kilogram. Through statistical analysis, it was found that when the selling price is $26$ yuan per kilogram,... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 8105,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "During the fight against the epidemic, a certain store purchased a type of disinfectant product at a cost of $8$ yuan per item. It was found during the sales process that there is a linear relationship between the daily sales quantity $y$ (items) and the selling price per item ... | math | {
"style": "rule",
"ground_truth": "525"
} | {
"index": 8106,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "During the first eleven days, 700 people responded to a survey question. Each respondent chose exactly one of the three offered options. The ratio of the frequencies of each response was \\(4: 7: 14\\). On the twelfth day, more people participated in the survey, which changed t... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"index": 8107,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "During the first year, ABC's stock price starts at $ \\$100 $ and increases $ 100\\% $. During the second year, its stock price goes down $ 25\\% $ from its price at the end of the first year. What is the price of the stock, in dollars, at the end of the second year?"
}
] | math | {
"style": "rule",
"ground_truth": "\\$150"
} | {
"index": 8108,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "During the flower exhibition in Zhengzhou Green Expo Garden, 6 volunteers are arranged to provide services in 4 exhibition areas. It is required that areas A and B each have one person, and the remaining two areas each have two people. Among them, Little Li and Little Wang cann... | math | {
"style": "rule",
"ground_truth": "156"
} | {
"index": 8109,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "During the manufacture of a steel cable, it was found that the cable has the same length as the curve given by the system of equations:\n\n$$\n\\left\\{\\begin{array}{l}\nx+y+z=8 \\\\\nx y+y z+x z=14\n\\end{array}\\right.\n$$\n\nFind the length of the cable."
}
] | math | {
"style": "rule",
"ground_truth": "4\\pi \\sqrt{\\frac{11}{3}}"
} | {
"index": 8110,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "During the military training for new freshman students, after two days of shooting practice, student A can hit the target 9 times out of 10 shots, and student B can hit the target 8 times out of 9 shots. A and B each take a shot at the same target (their shooting attempts do no... | math | {
"style": "rule",
"ground_truth": "\\frac{89}{90}"
} | {
"index": 8111,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "During the night shift, four duty personnel ate a whole barrel of pickles. If Assistant Mur ate half as much, one-tenth of the barrel would remain. If Lab Technician Trott ate half as much, one-eighth of the barrel would remain. If Intern Glupp ate half as much, one-quarter of ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{40}"
} | {
"index": 8112,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "During the past summer, 100 graduates from the city of $N$ applied to 5 different universities in our country. It turned out that during the first and second waves, each university was unable to reach exactly half of the applicants to that university. In addition, representativ... | math | {
"style": "rule",
"ground_truth": "83"
} | {
"index": 8113,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "During the preparation period of the Shanghai Expo, 5 volunteers and 2 foreign friends line up for a photo. The 2 foreign friends stand next to each other but not at either end of the line. Calculate the total number of different arrangements for the 7 individuals."
}
] | math | {
"style": "rule",
"ground_truth": "960"
} | {
"index": 8114,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "During the process of choosing trial points using the 0.618 method, if the trial interval is $[3, 6]$ and the first trial point is better than the second, then the third trial point should be at _____."
}
] | math | {
"style": "rule",
"ground_truth": "5.292"
} | {
"index": 8115,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "During the regular season, Washington Redskins achieve a record of 10 wins and 6 losses. Compute the probability that their wins came in three streaks of consecutive wins, assuming that all possible arrangements of wins and losses are equally likely. (For example, the record LL... | math | {
"style": "rule",
"ground_truth": "\\frac{315}{2002}"
} | {
"index": 8116,
"solution": "Suppse the winning streaks consist of $w_{1}, w_{2}$, and $w_{3}$ wins, in chronological order, where the first winning streak is preceded by $l_{0}$ consecutive losses and the $i$ winning streak is immediately succeeded by $l_{i}$ losses. Then $w_{1}, w_{2}, w_{3}, l_{1}, l_{2}>0$ are... |
DeepScaleR | [
{
"role": "user",
"content": "During the softball season, Judy had $35$ hits. Among her hits were $1$ home run, $1$ triple and $5$ doubles. The rest of her hits were single. What percent of her hits were single?"
}
] | math | {
"style": "rule",
"ground_truth": "80\\%"
} | {
"index": 8117,
"solution": "1. **Identify the total number of hits**: Judy had a total of 35 hits during the season.\n\n2. **Determine the number of hits that were not singles**: \n - Home runs: 1\n - Triples: 1\n - Doubles: 5\n - Total non-single hits = 1 (home run) + 1 (triple) + 5 (doubles) = 7\n\n3. *... |
DeepScaleR | [
{
"role": "user",
"content": "During the weekends, Eli delivers milk in the complex plane. On Saturday, he begins at $z$ and delivers milk to houses located at $z^{3}, z^{5}, z^{7}, \\ldots, z^{2013}$, in that order; on Sunday, he begins at 1 and delivers milk to houses located at $z^{2}, z^{4}, z^{6}, \\ld... | math | {
"style": "rule",
"ground_truth": "\\frac{1005}{1006}"
} | {
"index": 8118,
"solution": "Note that the distance between two points in the complex plane, $m$ and $n$, is $|m-n|$. We have that $$\\sum_{k=1}^{1006}\\left|z^{2 k+1}-z^{2 k-1}\\right|=\\sum_{k=1}^{1006}\\left|z^{2 k}-z^{2 k-2}\\right|=\\sqrt{2012}$$ However, noting that $$|z| \\cdot \\sum_{k=1}^{1006}\\left|z^{2... |
DeepScaleR | [
{
"role": "user",
"content": "During what year after 1994 did sales increase the most number of dollars?"
}
] | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"index": 8119,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Dylan has a \\( 100 \\times 100 \\) square, and wants to cut it into pieces of area at least 1. Each cut must be a straight line (not a line segment) and must intersect the interior of the square. What is the largest number of cuts he can make?"
}
] | math | {
"style": "rule",
"ground_truth": "9999"
} | {
"index": 8120,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each block on the grid shown in the Figure is 1 unit by 1 unit. Suppose we wish to walk from $A$ to $B$ via a 7 unit path, but we have to stay on the grid -- no cutting across blocks. How many different paths can we take?[asy]size(3cm,3cm);int w=5;int h=4;int i;for (i=0; i<h;... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 8121,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each brick in the pyramid contains one number. Whenever possible, the number in each brick is the least common multiple of the numbers of the two bricks directly above it.\n\nWhat number could be in the bottom brick? Determine all possible options.\n\n(Hint: What is the least c... | math | {
"style": "rule",
"ground_truth": "2730"
} | {
"index": 8122,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each cell of a $100 \\times 100$ board is painted in either blue or white. We call a cell balanced if it has an equal number of blue and white neighboring cells. What is the maximum number of balanced cells that can be found on the board? (Cells are considered neighbors if they... | math | {
"style": "rule",
"ground_truth": "9608"
} | {
"index": 8123,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each cell of a $2 \\times 5$ grid of unit squares is to be colored white or black. Compute the number of such colorings for which no $2 \\times 2$ square is a single color."
}
] | math | {
"style": "rule",
"ground_truth": "634"
} | {
"index": 8124,
"solution": "Let $a_{n}$ denote the number of ways to color a $2 \\times n$ grid subject only to the given constraint, and $b_{n}$ denote the number of ways to color a $2 \\times n$ grid subject to the given constraint, but with the added restriction that the first column cannot be colored black-bl... |
DeepScaleR | [
{
"role": "user",
"content": "Each cell of a $3 \\times 3$ grid is labeled with a digit in the set $\\{1,2,3,4,5\\}$. Then, the maximum entry in each row and each column is recorded. Compute the number of labelings for which every digit from 1 to 5 is recorded at least once."
}
] | math | {
"style": "rule",
"ground_truth": "2664"
} | {
"index": 8125,
"solution": "We perform casework by placing the entries from largest to smallest. - The grid must have exactly one 5 since an entry equal to 5 will be the maximum in its row and in its column. We can place this in 9 ways. - An entry equal to 4 must be in the same row or column as the 5; otherwise, ... |
DeepScaleR | [
{
"role": "user",
"content": "Each cell of an $m\\times n$ board is filled with some nonnegative integer. Two numbers in the filling are said to be adjacent if their cells share a common side. (Note that two numbers in cells that share only a corner are not adjacent). The filling is called a garden if it sa... | math | {
"style": "rule",
"ground_truth": "\\boxed{2^{mn} - 1}"
} | {
"index": 8126,
"solution": "We claim that any configuration of $0$ 's produces a distinct garden. To verify this claim, we show that, for any cell that is nonzero, the value of that cell is its distance away from the nearest zero, where distance means the shortest chain of adjacent cells connecting two cells. N... |
DeepScaleR | [
{
"role": "user",
"content": "Each corner cube is removed from this $3\\text{ cm}\\times 3\\text{ cm}\\times 3\\text{ cm}$ cube. The surface area of the remaining figure is\n[asy]\ndraw((2.7,3.99)--(0,3)--(0,0));\ndraw((3.7,3.99)--(1,3)--(1,0));\ndraw((4.7,3.99)--(2,3)--(2,0));\ndraw((5.7,3.99)--(3,3)--(3,0... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"index": 8127,
"solution": "1. **Calculate the original surface area of the cube**: \n The original cube has dimensions $3\\text{ cm} \\times 3\\text{ cm} \\times 3\\text{ cm}$. Each face of the cube is a square with an area of $3^2 = 9\\text{ cm}^2$. Since a cube has 6 faces, the total surface area of the cube... |
DeepScaleR | [
{
"role": "user",
"content": "Each corner of a rectangular prism is cut off. Two (of the eight) cuts are shown. How many edges does the new figure have?\n[asy] draw((0,0)--(3,0)--(3,3)--(0,3)--cycle); draw((3,0)--(5,2)--(5,5)--(2,5)--(0,3)); draw((3,3)--(5,5)); draw((2,0)--(3,1.8)--(4,1)--cycle,linewidth(... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 8128,
"solution": "1. **Identify the original structure**: The original figure is a rectangular prism, which has 8 vertices and 12 edges.\n\n2. **Understand the modification**: Each corner (vertex) of the prism is cut off. The cut at each vertex is such that it does not intersect any other cuts inside th... |
DeepScaleR | [
{
"role": "user",
"content": "Each day 289 students are divided into 17 groups of 17. No two students are ever in the same group more than once. What is the largest number of days that this can be done?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 8129,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each day Maria must work $8$ hours. This does not include the $45$ minutes she takes for lunch. If she begins working at $\\text{7:25 A.M.}$ and takes her lunch break at noon, then her working day will end at"
}
] | math | {
"style": "rule",
"ground_truth": "\\text{4:10 P.M.}"
} | {
"index": 8130,
"solution": "1. **Calculate the time worked before lunch:** Maria starts working at 7:25 A.M. and takes her lunch break at noon. The time interval from 7:25 A.M. to 12:00 P.M. is calculated as follows:\n \\[\n 12:00 - 7:25 = 4 \\text{ hours and } 35 \\text{ minutes}\n \\]\n\n2. **Account for ... |
DeepScaleR | [
{
"role": "user",
"content": "Each day Walter gets $3$ dollars for doing his chores or $5$ dollars for doing them exceptionally well. After $10$ days of doing his chores daily, Walter has received a total of $36$ dollars. On how many days did Walter do them exceptionally well?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 8131,
"solution": "1. Let $b$ be the number of days Walter does his chores but doesn't do them well, and let $w$ be the number of days he does his chores exceptionally well.\n\n2. We know that Walter works for $10$ days, so we have the equation:\n \\[\n b + w = 10\n \\]\n\n3. We also know that Walt... |
DeepScaleR | [
{
"role": "user",
"content": "Each day for four days, Linda traveled for one hour at a speed that resulted in her traveling one mile in an integer number of minutes. Each day after the first, her speed decreased so that the number of minutes to travel one mile increased by $5$ minutes over the preceding day... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 8132,
"solution": "1. **Understanding the problem**: Linda travels for one hour each day at a speed such that the time to travel one mile is an integer number of minutes. Each subsequent day, the time to travel one mile increases by 5 minutes. The distance she travels each day is also an integer number o... |
DeepScaleR | [
{
"role": "user",
"content": "Each day, Jenny ate $20\\%$ of the jellybeans that were in her jar at the beginning of that day. At the end of second day, 32 remained. How many jellybeans were in the jar originally?"
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 8133,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each day, John ate 30% of the chocolates that were in his box at the beginning of that day. At the end of the third day, 28 chocolates remained. How many chocolates were in the box originally?"
}
] | math | {
"style": "rule",
"ground_truth": "82"
} | {
"index": 8134,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each edge length of a rectangular solid is a prime number. If the volume of the rectangular solid is 385 cubic units, what is the total surface area, in square units, of the rectangular solid?"
}
] | math | {
"style": "rule",
"ground_truth": "334"
} | {
"index": 8135,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each edge of a cube is colored either red or black. Every face of the cube has at least one black edge. The smallest number possible of black edges is"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 8136,
"solution": "1. **Understanding the Problem**: We need to color the edges of a cube using two colors, red and black, such that every face of the cube has at least one black edge. We aim to find the minimum number of black edges required.\n\n2. **Analyzing the Cube's Geometry**: A cube has 6 faces, ... |
DeepScaleR | [
{
"role": "user",
"content": "Each edge of a regular tetrahedron is divided into three equal parts. Through each point of division, two planes are drawn, each parallel to one of the two faces of the tetrahedron that do not pass through this point. Into how many parts do the constructed planes divide the tet... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 8137,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each edge of a regular tetrahedron is divided into three equal parts. Through each resulting division point, two planes are drawn, parallel to the two faces of the tetrahedron that do not pass through that point. Into how many parts do the constructed planes divide the tetrahed... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 8138,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each edge of a regular tetrahedron is given a stripe. The choice of which edge to stripe is made at random. What is the probability that there is at least one triangle face with all its edges striped?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1695}{4096}"
} | {
"index": 8139,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of the opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling t... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{16}"
} | {
"index": 8140,
"solution": "\nTo solve this problem, we need to determine the probability that a continuous stripe encircles the cube. We will analyze the problem using the approach described in Solution 1, which is both clear and concise.\n\n1. **Total Possible Stripe Combinations**:\n Each face of the cube ha... |
DeepScaleR | [
{
"role": "user",
"content": "Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of the opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{16}"
} | {
"index": 8141,
"solution": "\n1. **Total Possible Stripe Combinations**:\n Each face of the cube has two possible orientations for the stripe. Since there are six faces, the total number of stripe combinations is calculated as:\n \\[\n 2^6 = 64\n \\]\n\n2. **Counting Favorable Outcomes**:\n To have a co... |
DeepScaleR | [
{
"role": "user",
"content": "Each face of a cube is painted either red or blue, each with probability 1/2. The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{16}"
} | {
"index": 8142,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each face of a die is arranged so that the sum of the numbers on opposite faces is 7. In the arrangement shown with three dice, only seven faces are visible. What is the sum of the numbers on the faces that are not visible in the given image?"
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"index": 8143,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each face of a fair six-sided die is marked with one of the numbers $1, 2, \\cdots, 6$. When two such identical dice are rolled, the sum of the numbers on the top faces of these dice is the score for that roll. What is the probability that the product of the scores from three s... | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"index": 8144,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each face of a rectangular prism is painted with a single narrow stripe from one vertex to the diagonally opposite vertex. The choice of the vertex pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling th... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{16}"
} | {
"index": 8145,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each face of a regular tetrahedron is labeled with one of the numbers $1, 2, 3, 4$. Four identical regular tetrahedrons are simultaneously rolled onto a table. What is the probability that the product of the four numbers on the faces touching the table is divisible by 4?"
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{13}{16}$"
} | {
"index": 8146,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each face of a regular tetrahedron is labeled with one of the numbers 1, 2, 3, 4. Four identical regular tetrahedrons are simultaneously rolled onto a table. Calculate the probability that the product of the four numbers on the faces touching the table is divisible by 4."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{16}"
} | {
"index": 8147,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each face of a regular tetrahedron is painted either red, white, or blue. Two colorings are considered indistinguishable if two congruent tetrahedra with those colorings can be rotated so that their appearances are identical. How many distinguishable colorings are possible?"
... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 8148,
"solution": "1. **Identify the Problem Context**: We are given a regular tetrahedron with each face painted either red, white, or blue. We need to find the number of distinguishable colorings, considering that colorings are indistinguishable if one tetrahedron can be rotated to look like another.\n... |
DeepScaleR | [
{
"role": "user",
"content": "Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths $\\sqrt{21}$ and $\\sqrt{31}$. The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive inte... | math | {
"style": "rule",
"ground_truth": "125"
} | {
"index": 8149,
"solution": "Let us inscribe a tetrahedron $ACB'D'$ in given parallelepiped so that its edges coincide with the diagonals of the faces of the parallelepiped. Note that the three edges outgoing from the vertex $B'$ have the same length $b$, and the three edges at the base have a different length $a.... |
DeepScaleR | [
{
"role": "user",
"content": "Each half of this figure is composed of 3 red triangles, 5 blue triangles and 8 white triangles. When the upper half is folded down over the centerline, 2 pairs of red triangles coincide, as do 3 pairs of blue triangles. There are 2 red-white pairs. How many white pairs coincid... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 8150,
"solution": "1. **Identify the total number of triangles in each half**: Each half of the figure contains 3 red triangles, 5 blue triangles, and 8 white triangles.\n\n2. **Analyze the red triangles**: When the upper half is folded down, 2 pairs of red triangles coincide. This means 4 red triangles ... |
DeepScaleR | [
{
"role": "user",
"content": "Each integer 1 through 9 is written on a separate slip of paper and all nine slips are put into a hat. Jack picks one of these slips at random and puts it back. Then Jill picks a slip at random. Which digit is most likely to be the units digit of the sum of Jack's integer and J... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 8151,
"solution": "1. **Understanding the Problem**: Each integer from $1$ to $9$ is placed on a slip of paper and put into a hat. Jack and Jill each draw a slip (with replacement), and we want to find the most likely units digit of the sum of the two integers they draw.\n\n2. **Setting Up the Problem**:... |
DeepScaleR | [
{
"role": "user",
"content": "Each interior angle of a regular polygon measures $140^\\circ$. How many sides does the polygon have?"
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 8152,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each letter represents a non-zero digit. What is the value of $t?$ \\begin{align*}\nc + o &= u \\\\\nu + n &= t \\\\\nt + c &= s \\\\\no + n + s &= 12\n\\end{align*}"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 8153,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each morning of her five-day workweek, Jane bought either a $50$-cent muffin or a $75$-cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy?"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 8154,
"solution": "Let $b$ be the number of bagels Jane bought, and $m$ be the number of muffins. Since Jane buys these items over a five-day workweek, we have the equation:\n\\[ b + m = 5 \\]\n\nThe cost of a bagel is $75$ cents and the cost of a muffin is $50$ cents. The total cost for the week, in cen... |
DeepScaleR | [
{
"role": "user",
"content": "Each number in the list $1,2,3,\\ldots,10$ is either colored red or blue. Numbers are colored independently, and both colors are equally probable. The expected value of the number of positive integers expressible as a sum of a red integer and a blue integer can be written as ... | math | {
"style": "rule",
"ground_truth": "455"
} | {
"index": 8155,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of $1000$ elves has a hat, red on the inside and blue on the outside or vise versa. An elf with a hat that is red outside can only lie, and an elf with a hat that is blue outside can only tell the truth. One day every elf tells every other elf, “Your hat is red on the ou... | math | {
"style": "rule",
"ground_truth": "998"
} | {
"index": 8156,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of $2010$ boxes in a line contains a single red marble, and for $1 \\le k \\le 2010$, the box in the $k\\text{th}$ position also contains $k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 8157,
"solution": "1. **Understanding the problem setup**: Each of the 2010 boxes contains 1 red marble and an increasing number of white marbles as we move from the first to the last box. Isabella draws marbles one by one from each box starting from the first box and stops when she draws a red marble.\n... |
DeepScaleR | [
{
"role": "user",
"content": "Each of $2011$ boxes in a line contains two red marbles, and for $1 \\le k \\le 2011$, the box in the $k\\text{th}$ position also contains $k+1$ white marbles. Liam begins at the first box and successively draws a single marble at random from each box, in order. He stops when h... | math | {
"style": "rule",
"ground_truth": "62"
} | {
"index": 8158,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of $6$ balls is randomly and independently painted either black or white with equal probability. What is the probability that every ball is different in color from more than half of the other $5$ balls?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{16}"
} | {
"index": 8159,
"solution": "1. **Understanding the Problem**: We need to find the probability that each of the 6 balls is painted a color such that it is different from the color of more than half of the other 5 balls. This means each ball must be different from at least 3 of the other balls.\n\n2. **Condition fo... |
DeepScaleR | [
{
"role": "user",
"content": "Each of $a, b$ and $c$ is equal to a number from the list $3^{1}, 3^{2}, 3^{3}, 3^{4}, 3^{5}, 3^{6}, 3^{7}, 3^{8}$. There are $N$ triples $(a, b, c)$ with $a \\leq b \\leq c$ for which each of $\\frac{ab}{c}, \\frac{ac}{b}$ and $\\frac{bc}{a}$ is equal to an integer. What is th... | math | {
"style": "rule",
"ground_truth": "86"
} | {
"index": 8160,
"solution": "We write $a=3^{r}, b=3^{s}$ and $c=3^{t}$ where each of $r, s, t$ is between 1 and 8, inclusive. Since $a \\leq b \\leq c$, then $r \\leq s \\leq t$. Next, we note that $\\frac{ab}{c}=\\frac{3^{r} 3^{s}}{3^{t}}=3^{r+s-t}$, $\\frac{ac}{b}=\\frac{3^{r} 3^{t}}{3^{s}}=3^{r+t-s}$, and $\\fr... |
DeepScaleR | [
{
"role": "user",
"content": "Each of $a_1,$ $a_2,$ $\\dots,$ $a_{100}$ is equal to $1$ or $-1.$ Find the minimum positive value of\n\\[\\sum_{1 \\le i < j \\le 100} a_i a_j.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 8161,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of $b_1, b_2, \\dots, b_{150}$ is equal to $2$ or $-2$. Find the minimum positive value of\n\\[\\sum_{1 \\le i < j \\le 150} b_i b_j.\\]"
}
] | math | {
"style": "rule",
"ground_truth": "38"
} | {
"index": 8162,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of 100 students sends messages to 50 different students. What is the least number of pairs of students who send messages to each other?"
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"index": 8163,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of 8 balls is randomly and independently painted either black or white with equal probability. Calculate the probability that every ball is different in color from at least half of the other 7 balls."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{35}{128}"
} | {
"index": 8164,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of 8 balls is randomly and independently painted either black or white with equal probability. Calculate the probability that every ball is different in color from more than half of the other 7 balls."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{35}{128}"
} | {
"index": 8165,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of Natalie's blueberry bushes yields eight containers of blueberries. If she can trade five containers of blueberries for two zucchinis, how many bushes does Natalie need to pick in order to have forty-eight zucchinis?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 8166,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of \\( A \\), \\( B \\), \\( C \\), and \\( D \\) is a positive two-digit integer. These integers satisfy each of the equations\n\n\\[\n\\begin{aligned}\nB &= 3C \\\\\nD &= 2B - C \\\\\nA &= B + D\n\\end{aligned}\n\\]\n\nWhat is the largest possible value of \\( A + B + C ... | math | {
"style": "rule",
"ground_truth": "204"
} | {
"index": 8167,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of a group of $50$ girls is blonde or brunette and is blue eyed of brown eyed. If $14$ are blue-eyed blondes, $31$ are brunettes, and $18$ are brown-eyed, then the number of brown-eyed brunettes is"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 8168,
"solution": "1. **Identify the total number of girls and their characteristics:**\n - Total number of girls = $50$\n - Characteristics: blonde or brunette, blue-eyed or brown-eyed\n\n2. **Calculate the number of blondes:**\n - Given that there are $31$ brunettes, the number of blondes is:\n ... |
DeepScaleR | [
{
"role": "user",
"content": "Each of five, standard, six-sided dice is rolled once. What is the probability that there is at least one pair but not a three-of-a-kind (that is, there are two dice showing the same value, but no three dice show the same value)?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{25}{36}"
} | {
"index": 8169,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of given $100$ numbers was increased by $1$. Then each number was increased by $1$ once more. Given that the fi\frst time the sum of the squares of the numbers was not changed \ffind how this sum was changed the second time."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 8170,
"solution": "\nLet the original 100 numbers be \\( a_1, a_2, \\ldots, a_{100} \\). Initially, the sum of their squares is \n\n\\[\nS = \\sum_{i=1}^{100} a_i^2.\n\\]\n\nWhen each number is increased by 1 for the first time, the new numbers are \\( a_1 + 1, a_2 + 1, \\ldots, a_{100} + 1 \\). The new ... |
DeepScaleR | [
{
"role": "user",
"content": "Each of six, standard, six-sided dice is rolled once. What is the probability that there is exactly one pair and one triplet (three dice showing the same value), and the remaining dice show different values?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{25}{162}"
} | {
"index": 8171,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the $100$ students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are $42$ students who cannot sing, $65$ students who cannot dance, and $29$ students who cannot act. How many... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"index": 8172,
"solution": "1. **Calculate the number of students who can sing, dance, and act**:\n - Students who can sing: $100 - 42 = 58$\n - Students who can dance: $100 - 65 = 35$\n - Students who can act: $100 - 29 = 71$\n\n2. **Sum the number of students who can sing, dance, or act**:\n - Total (ig... |
DeepScaleR | [
{
"role": "user",
"content": "Each of the $12$ edges of a cube is labeled $0$ or $1$. Two labelings are considered different even if one can be obtained from the other by a sequence of one or more rotations and/or reflections. For how many such labelings is the sum of the labels on the edges of each of the ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 8173,
"solution": "To solve this problem, we need to ensure that each face of the cube has exactly two edges labeled with $1$ and two edges labeled with $0$. We will analyze the possible configurations that satisfy this condition.\n\n#### Step 1: Understanding the Cube's Faces and Edges\nA cube has 6 fac... |
DeepScaleR | [
{
"role": "user",
"content": "Each of the $20$ balls is tossed independently and at random into one of the $5$ bins. Let $p$ be the probability that some bin ends up with $3$ balls, another with $5$ balls, and the other three with $4$ balls each. Let $q$ be the probability that every bin ends up with $4$ ba... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 8174,
"solution": "1. **Define the sets and probabilities**: Let $A$ be the set of all configurations where the balls are distributed as $3{-}5{-}4{-}4{-}4$ among the bins, and let $B$ be the set of all configurations where the balls are distributed as $4{-}4{-}4{-}4{-}4$. Define $p = \\frac{|A|}{N}$ and... |
DeepScaleR | [
{
"role": "user",
"content": "Each of the $2001$ students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between $80$ percent and $85$ percent of the school population, and the number who study French is between $30$ percent and $40$ percent. Let $m$ ... | math | {
"style": "rule",
"ground_truth": "298"
} | {
"index": 8175,
"solution": "Let $S$ be the percent of people who study Spanish, $F$ be the number of people who study French, and let $S \\cup F$ be the number of students who study both. Then $\\left\\lceil 80\\% \\cdot 2001 \\right\\rceil = 1601 \\le S \\le \\left\\lfloor 85\\% \\cdot 2001 \\right\\rfloor = 170... |
DeepScaleR | [
{
"role": "user",
"content": "Each of the $25$ balls is tossed independently and at random into one of $5$ bins. Let $r$ be the probability that one bin ends up with $6$ balls, another with $7$ balls, and the other three with $4$ balls each. Let $s$ be the probability that one bin ends up with $5$ balls and... | math | {
"style": "rule",
"ground_truth": "\\frac{10 \\cdot \\binom{25}{6}\\binom{19}{7}\\binom{12}{4}\\binom{8}{4}\\binom{4}{4}}{\\binom{25}{5}\\binom{20}{5}\\binom{15}{5}\\binom{10}{5}\\binom{5}{5}}"
} | {
"index": 8176,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the $2500$ students at a university studies either Physics or Chemistry, and some study both. The number who study Physics is between $70\\%$ and $75\\%$ of the university population, and the number who study Chemistry is between $40\\%$ and $45\\%$. Let $m$ be the smal... | math | {
"style": "rule",
"ground_truth": "250"
} | {
"index": 8177,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the $39$ students in the eighth grade at Lincoln Middle School has one dog or one cat or both a dog and a cat. Twenty students have a dog and $26$ students have a cat. How many students have both a dog and a cat?"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 8178,
"solution": "1. **Define the Sets:**\n Let $D$ be the set of students who have a dog and $C$ be the set of students who have a cat. We know:\n - $|D| = 20$ (students with a dog)\n - $|C| = 26$ (students with a cat)\n - Total number of students, $|D \\cup C| = 39$.\n\n2. **Use the Principle ... |
DeepScaleR | [
{
"role": "user",
"content": "Each of the $5$ sides and the $5$ diagonals of a regular pentagon are randomly and independently colored red or blue with equal probability. What is the probability that there will be a triangle whose vertices are among the vertices of the pentagon such that all of its sides ha... | math | {
"style": "rule",
"ground_truth": "\\frac{253}{256}"
} | {
"index": 8179,
"solution": "To solve this problem, we need to calculate the probability that there exists a triangle (a $K_3$ subgraph) in a complete graph $K_5$ (a pentagon with all diagonals drawn) such that all its edges are colored the same color (either all red or all blue).\n\n#### Step 1: Understanding the... |
DeepScaleR | [
{
"role": "user",
"content": "Each of the $6$ sides and the $9$ diagonals of a regular hexagon are randomly and independently colored red, blue, or green with equal probability. What is the probability that there will be a triangle whose vertices are among the vertices of the hexagon such that all of its si... | math | {
"style": "rule",
"ground_truth": "\\frac{872}{1000}"
} | {
"index": 8180,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the 33 warriors either always lies or always tells the truth. It is known that each warrior has exactly one favorite weapon: a sword, a spear, an axe, or a bow.\n\nOne day, Uncle Chernomor asked each warrior four questions:\n- Is your favorite weapon a sword?\n- Is your... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 8181,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the boxes in the figure below is a square. How many different squares can be traced using the lines in the figure?\n\n[asy]\nunitsize(0.2inch);\ndraw((0,1)--(1,1)--(1,4)--(0,4)--(0,1));\ndraw((1,1)--(1,0)--(2,0)--(2,5)--(1,5)--(1,4));\ndraw((2,0)--(3,0)--(3,5)--(2,5));\... | math | {
"style": "rule",
"ground_truth": "38"
} | {
"index": 8182,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the cells of a $7 \\times 7$ grid is painted with a color chosen randomly and independently from a set of $N$ fixed colors. Call an edge hidden if it is shared by two adjacent cells in the grid that are painted the same color. Determine the least $N$ such that the... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 8183,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the digits 3, 5, 6, 7, and 8 is placed one to a box in the diagram. If the two-digit number is subtracted from the three-digit number, what is the smallest difference?"
}
] | math | {
"style": "rule",
"ground_truth": "269"
} | {
"index": 8184,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the eight letters in \"GEOMETRY\" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word \"ANGLE\"? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"index": 8185,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the eight vertices of a rectangular prism is truncated, and each face of the prism is divided by a diagonal cut into two triangles. Calculate the total number of edges of the altered shape."
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 8186,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the equations \\( a x^{2} - b x + c = 0 \\) and \\( c x^{2} - a x + b = 0 \\) has two distinct real roots. The sum of the roots of the first equation is non-negative, and the product of the roots of the first equation is 9 times the sum of the roots of the second equati... | math | {
"style": "rule",
"ground_truth": "-3"
} | {
"index": 8187,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the first $150$ positive integers is painted on a different marble, and the $150$ marbles are placed in a bag. If $n$ marbles are chosen (without replacement) from the bag, what is the smallest value of $n$ such that we are guaranteed to choose three marbles wit... | math | {
"style": "rule",
"ground_truth": "101"
} | {
"index": 8188,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the first eight prime numbers is placed in a bowl. Two primes are drawn without replacement. What is the probability, expressed as a common fraction, that the sum of the two numbers drawn is a prime number?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac17"
} | {
"index": 8189,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the five numbers 1, 4, 7, 10, and 13 is placed in one of the five squares so that the sum of the three numbers in the horizontal row equals the sum of the three numbers in the vertical column. The largest possible value for the horizontal or vertical sum is\n[asy] draw(... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 8190,
"solution": "\nLet's denote the five squares as follows, where $a$, $b$, $c$, $d$, and $e$ represent the numbers 1, 4, 7, 10, and 13 placed in the squares:\n\n\\[\n\\begin{array}{|c|c|}\n\\hline\na & b \\\\\n\\hline\nc & d \\\\\n\\hline\n\\end{array}\n\\quad\n\\begin{array}{|c|}\n\\hline\ne \\\\\n\... |
DeepScaleR | [
{
"role": "user",
"content": "Each of the following 15 cards has a letter on one side and a positive integer on the other side. What is the minimum number of cards that need to be turned over to check if the following statement is true? 'If a card has a lower case letter on one side, then it has an odd inte... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 8191,
"solution": "Each card fits into exactly one of the following categories: (A) lower case letter on one side, even integer on the other side (B) lower case letter on one side, odd integer on the other side (C) upper case letter on one side, even integer on the other side (D) upper case letter on one... |
DeepScaleR | [
{
"role": "user",
"content": "Each of the four digits of the integer 2024 is even. How many integers between 1000 and 9999, inclusive, have the property that all four of their digits are even?"
}
] | math | {
"style": "rule",
"ground_truth": "500"
} | {
"index": 8192,
"solution": "The integers between 1000 and 9999, inclusive, are all four-digit positive integers of the form $abcd$. We want each of $a, b, c$, and $d$ to be even. There are 4 choices for $a$, namely $2, 4, 6, 8$. ($a$ cannot equal 0.) There are 5 choices for each of $b, c$ and $d$, namely $0, 2, 4... |
DeepScaleR | [
{
"role": "user",
"content": "Each of the integers $1,2, \\ldots, 729$ is written in its base-3 representation without leading zeroes. The numbers are then joined together in that order to form a continuous string of digits: $12101112202122 \\ldots \\ldots$ How many times in this string does the substring 0... | math | {
"style": "rule",
"ground_truth": "148"
} | {
"index": 8193,
"solution": "Ignore $729=3^{6}=1000000_{3}$ since it will not contribute to a 012 substring. Break into cases on how 012 appears: (i) when an individual integer contains the string 012 ; (ii) when 01 are the last two digits of an integer and 2 is the first digit of the next integer; and (iii) when ... |
DeepScaleR | [
{
"role": "user",
"content": "Each of the integers 226 and 318 has digits whose product is 24. How many three-digit positive integers have digits whose product is 24?"
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 8194,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the integers 334 and 419 has digits whose product is 36. How many 3-digit positive integers have digits whose product is 36?"
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 8195,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the integers \\(1, 2, 3, \\ldots, 9\\) is assigned to a vertex of a regular 9-sided polygon (every vertex receives exactly one integer from \\(\\{1, 2, \\ldots, 9\\}\\), and no two vertices receive the same integer) so that the sum of the integers assigned to any three ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 8196,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the lateral edges of a pyramid is equal to 269/32. The base of the pyramid is a triangle with sides 13, 14, 15. Find the volume of the pyramid."
}
] | math | {
"style": "rule",
"ground_truth": "483/8"
} | {
"index": 8197,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Each of the letters $W$, $X$, $Y$, and $Z$ represents a different integer in the set $\\{ 1,2,3,4\\}$, but not necessarily in that order. If $\\frac{W}{X} - \\frac{Y}{Z}=1$, then the sum of $W$ and $Y$ is"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 8198,
"solution": "1. **Identify the possible values for fractions:** Given that $\\text{W}, \\text{X}, \\text{Y}, \\text{Z}$ are different integers from the set $\\{1, 2, 3, 4\\}$, we need to find pairs $(\\text{W}, \\text{X})$ and $(\\text{Y}, \\text{Z})$ such that $\\frac{\\text{W}}{\\text{X}} - \\fra... |
DeepScaleR | [
{
"role": "user",
"content": "Each of the letters in \"GEOMETRY\" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word \"RHYME\"? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 8199,
"solution": ""
} |
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