data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "A decorative garden is designed with a rectangular lawn with semicircles of grass at either end. The ratio of the length of the rectangle to its width is 5:4, and the total length including the semicircles is 50 feet. Calculate the ratio of the area of the rectangle to the comb... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{\\pi}"
} | {
"index": 1600,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A decorative window is made up of a rectangle with semicircles at either end. The ratio of $AD$ to $AB$ is $3:2$. And $AB$ is 30 inches. What is the ratio of the area of the rectangle to the combined area of the semicircles?"
}
] | math | {
"style": "rule",
"ground_truth": "6:\\pi"
} | {
"index": 1601,
"solution": "1. **Set up the proportion for $AD$ and $AB$:**\n Given the ratio of $AD$ to $AB$ is $3:2$, and $AB = 30$ inches, we can write:\n \\[\n \\frac{AD}{AB} = \\frac{3}{2}\n \\]\n Substituting $AB = 30$ inches into the proportion:\n \\[\n \\frac{AD}{30} = \\frac{3}{2}\n \\]\n... |
DeepScaleR | [
{
"role": "user",
"content": "A delicious circular pie with diameter $12\\text{ cm}$ is cut into three equal-sized sector-shaped pieces. Let $l$ be the number of centimeters in the length of the longest line segment that may be drawn in one of these pieces. What is $l^2$?"
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"index": 1602,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A department dispatches 4 researchers to 3 schools to investigate the current status of the senior year review and preparation for exams, requiring at least one researcher to be sent to each school. Calculate the number of different distribution schemes."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 1603,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A department needs to arrange a duty schedule for the National Day holiday (a total of 8 days) for four people: A, B, C, and D. It is known that:\n\n- A and B each need to be on duty for 4 days.\n- A cannot be on duty on the first day, and A and B cannot be on duty on the same ... | math | {
"style": "rule",
"ground_truth": "700"
} | {
"index": 1604,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A department store offers two promotions. Promotion A says, \"Buy one pair of shoes, get the second pair for half the price.\" Promotion B says, \"Buy one pair of shoes, get $\\$10$ off the second pair.\" Jane wants to buy two pairs of shoes that cost $\\$30$ each. She can only... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 1605,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A designer has 3 fabric colors he may use for a dress: red, green, and blue. Four different patterns are available for the dress. If each dress design requires exactly one color and one pattern, how many different dress designs are possible?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 1606,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different desser... | math | {
"style": "rule",
"ground_truth": "729"
} | {
"index": 1607,
"solution": "1. **Identify the constraints and setup the problem:**\n - The chef has to prepare desserts for 7 days starting from Sunday.\n - The desserts options are cake, pie, ice cream, or pudding.\n - The same dessert cannot be served on consecutive days.\n - Cake must be served on Frid... |
DeepScaleR | [
{
"role": "user",
"content": "A diagonal from a vertex of a polygon forms a triangle with the two adjacent sides, so the number of triangles formed by the polygon is at least the number of sides of the polygon. However, the number of triangles cannot exceed the number of ways to choose 2 sides from the poly... | math | {
"style": "rule",
"ground_truth": "2023"
} | {
"index": 1608,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A diagonal of a polygon is a segment joining two nonconsecutive vertices of the polygon. How many diagonals does a regular octagon have?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 1609,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A dice is repeatedly rolled, and the upward-facing number is recorded for each roll. The rolling stops once three different numbers are recorded. If the sequence stops exactly after five rolls, calculate the total number of distinct recording sequences for these five numbers."
... | math | {
"style": "rule",
"ground_truth": "840"
} | {
"index": 1610,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A dice is thrown twice. Let $a$ be the number of dots that appear on the first throw and $b$ be the number of dots that appear on the second throw. Given the system of equations $\\begin{cases} ax+by=2 \\\\\\\\ 2x+y=3\\\\end{cases}$.\n\n(I) Find the probability that the system ... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{12}"
} | {
"index": 1611,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A die has faces numbered $1$, $2$, $3$, $3$, $4$, and $4$. Another die has faces numbered $2$, $3$, $5$, $6$, $7$, and $8$. Determine the probability that the sum of the top two numbers when both dice are rolled will be either $6$, $8$, or $10$.\nA) $\\frac{1}{36}$\nB) $\\frac{... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{36}"
} | {
"index": 1612,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A die is rolled twice continuously, resulting in numbers $a$ and $b$. What is the probability $p$, in numerical form, that the cubic equation in $x$, given by $x^{3}-(3 a+1) x^{2}+(3 a+2 b) x-2 b=0$, has three distinct real roots?"
}
] | math | {
"style": "rule",
"ground_truth": "3/4"
} | {
"index": 1613,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A die with faces showing the numbers $0,1,2,3,4,5$ is rolled until the total sum of the rolled numbers exceeds 12. What is the most likely value of this sum?"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 1614,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A digit is inserted between the digits of a two-digit number to form a three-digit number. Some two-digit numbers, when a certain digit is inserted in between, become three-digit numbers that are $k$ times the original two-digit number (where $k$ is a positive integer). What is... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 1615,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A digit is written to the right of the units digit of $757$. If the resulting four-digit number is divisible by $3$, how many possibilities are there for the digit that was written?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 1616,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A digit was crossed out from a six-digit number, resulting in a five-digit number. When this five-digit number was subtracted from the original six-digit number, the result was 654321. Find the original six-digit number."
}
] | math | {
"style": "rule",
"ground_truth": "727023"
} | {
"index": 1617,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A digital 12-hour clock has a malfunction such that every time it should display a \"2\", it instead shows a \"5\". For example, when it is 2:27 PM, the clock incorrectly shows 5:57 PM. What fraction of the day will the clock show the correct time?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{8}"
} | {
"index": 1618,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A digital clock displays time in a 24-hour format (from 00:00 to 23:59). Find the largest possible sum of the digits in this time display."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 1619,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A digital clock shows the time $4:56$. How many minutes will pass until the clock next shows a time in which all of the digits are consecutive and are in increasing order?"
}
] | math | {
"style": "rule",
"ground_truth": "458"
} | {
"index": 1620,
"solution": "We would like to find the first time after $4:56$ where the digits are consecutive digits in increasing order. It would make sense to try 5:67, but this is not a valid time. Similarly, the time cannot start with $6,7,8$ or 9. No time starting with 10 or 11 starts with consecutive incre... |
DeepScaleR | [
{
"role": "user",
"content": "A digital clock shows the time 4:56. How many minutes will pass until the clock next shows a time in which all of the digits are consecutive and are in increasing order?"
}
] | math | {
"style": "rule",
"ground_truth": "458"
} | {
"index": 1621,
"solution": "We would like to find the first time after 4:56 where the digits are consecutive digits in increasing order. It would make sense to try 5:67, but this is not a valid time. Similarly, the time cannot start with 6, 7, 8, or 9. No time starting with 10 or 11 starts with consecutive increa... |
DeepScaleR | [
{
"role": "user",
"content": "A digital music library contains 12 days of music and takes up 16,000 megabytes of disk space. On average, how many megabytes of disk space does one hour of music in this library take up? Express your answer to the nearest whole number."
}
] | math | {
"style": "rule",
"ground_truth": "56"
} | {
"index": 1622,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A digital watch displays hours and minutes in a 24-hour format. Calculate the largest possible sum of the digits in the display."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 1623,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A digital watch displays hours and minutes in a 24-hour format. Calculate the largest possible sum of the digits in this display."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 1624,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A digital watch displays hours and minutes in a 24-hour format. Find the largest possible sum of the digits in the display."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 1625,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display?"
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 1626,
"solution": "1. **Understanding the Display Format**: The digital watch displays time in a 12-hour format with AM and PM, showing hours and minutes. The hours can range from 01 to 12, and the minutes from 00 to 59.\n\n2. **Maximizing the Hour Digits**: \n - The hours are displayed as either 01, 0... |
DeepScaleR | [
{
"role": "user",
"content": "A digital watch displays time in a 24-hour format showing only hours and minutes. Find the largest possible sum of the digits in the display."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 1627,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A digital watch displays time in a 24-hour format, showing hours and minutes. Calculate the largest possible sum of the digits in this display."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 1628,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A digital watch now displays time in a 24-hour format, showing hours and minutes. Find the largest possible sum of the digits when it displays time in this format, where the hour ranges from 00 to 23 and the minutes range from 00 to 59."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 1629,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A dilation of the plane—that is, a size transformation with a positive scale factor—sends the circle of radius $2$ centered at $A(2,2)$ to the circle of radius $3$ centered at $A’(5,6)$. What distance does the origin $O(0,0)$, move under this transformation?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{13}"
} | {
"index": 1630,
"solution": "1. **Identify the centers and radii of the original and dilated circles:**\n - Original circle: Center $A(2,2)$, Radius $2$\n - Dilated circle: Center $A'(5,6)$, Radius $3$\n\n2. **Determine the dilation factor:**\n - The dilation factor $k$ can be calculated by the ratio of the ... |
DeepScaleR | [
{
"role": "user",
"content": "A disk with radius $1$ is externally tangent to a disk with radius $5$. Let $A$ be the point where the disks are tangent, $C$ be the center of the smaller disk, and $E$ be the center of the larger disk. While the larger disk remains fixed, the smaller disk is allowed to roll al... | math | {
"style": "rule",
"ground_truth": "58"
} | {
"index": 1631,
"solution": "First, we determine how far the small circle goes. For the small circle to rotate completely around the circumference, it must rotate $5$ times (the circumference of the small circle is $2\\pi$ while the larger one has a circumference of $10\\pi$) plus the extra rotation the circle get... |
DeepScaleR | [
{
"role": "user",
"content": "A district in a city is laid out in an $11 \\times 11$ grid. Every day, a sprinkler truck departs from the bottom-left corner $A(0,0)$ and travels along the streets to reach the top-right corner $B(10,10)$. At each intersection, the driver randomly chooses a direction, as long ... | math | {
"style": "rule",
"ground_truth": "1 - \\frac{\\binom{18}{9}}{\\binom{20}{10}}"
} | {
"index": 1632,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A divisor of a number is a proper divisor if it is not equal to the number. What is the sum of the proper divisors of $432$?"
}
] | math | {
"style": "rule",
"ground_truth": "808"
} | {
"index": 1633,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A doctor told Mikael to take a pill every 75 minutes. He took his first pill at 11:05. At what time did he take his fourth pill?"
}
] | math | {
"style": "rule",
"ground_truth": "14:50"
} | {
"index": 1634,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A dodecahedron consists of two pentagonal-based pyramids glued together along their pentagonal bases, forming a polyhedron with 12 faces. Consider an ant at the top vertex of one of the pyramids, selecting randomly one of the five adjacent vertices, designated as vertex A. From... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"index": 1635,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A dog and a cat simultaneously grab a sausage from different ends with their teeth. If the dog bites off its piece and runs away, the cat will get 300 grams more than the dog. If the cat bites off its piece and runs away, the dog will get 500 grams more than the cat. How much s... | math | {
"style": "rule",
"ground_truth": "400"
} | {
"index": 1636,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A domino is a 1-by-2 or 2-by-1 rectangle. A domino tiling of a region of the plane is a way of covering it (and only it) completely by nonoverlapping dominoes. For instance, there is one domino tiling of a 2-by-1 rectangle and there are 2 tilings of a 2-by-2 rectangle (one cons... | math | {
"style": "rule",
"ground_truth": "89"
} | {
"index": 1637,
"solution": "The number of tilings of a 2-by-$n$, rectangle is the $n$th Fibonacci number $F_{n}$, where $F_{0}=F_{1}=1$ and $F_{n}=F_{n-1}+F_{n-1}$ for $n \\geq 2$. (This is not hard to show by induction.) The answer is 89."
} |
DeepScaleR | [
{
"role": "user",
"content": "A domino is a rectangular tile composed of two squares. An integer is represented on both squares, and each integer 0-9 is paired with every integer 0-9 exactly once to form a complete set. A $\\textit{double}$ is a domino that has the same integer on both of its squares. What ... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{11}"
} | {
"index": 1638,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A dormitory is installing a shower room for 100 students. How many shower heads are economical if the boiler preheating takes 3 minutes per shower head, and it also needs to be heated during the shower? Each group is allocated 12 minutes for showering."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 1639,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A dormitory of a certain high school senior class has 8 people. In a health check, the weights of 7 people were measured to be 60, 55, 60, 55, 65, 50, 50 (in kilograms), respectively. One person was not measured due to some reasons, and it is known that the weight of this stude... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 1640,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A dot is marked at each vertex of a triangle $A B C$. Then, 2,3 , and 7 more dots are marked on the sides $A B, B C$, and $C A$, respectively. How many triangles have their vertices at these dots?"
}
] | math | {
"style": "rule",
"ground_truth": "357"
} | {
"index": 1641,
"solution": "Altogether there are $3+2+3+7=15$ dots, and thus $\\binom{15}{3}=455$ combinations of 3 dots. Of these combinations, $\\binom{2+2}{3}+\\binom{2+3}{3}+\\binom{2+7}{3}=4+10+84=98$ do not give triangles because they are collinear (the rest do give triangles). Thus $455-98=357$ different t... |
DeepScaleR | [
{
"role": "user",
"content": "A dragon has 40 piles of gold coins, with the number of coins in any two piles differing. After the dragon plundered a neighboring city and brought back more gold, the number of coins in each pile increased by either 2, 3, or 4 times. What is the minimum number of different pil... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 1642,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A dragon is tethered by a 25-foot golden rope to the base of a sorcerer's cylindrical tower whose radius is 10 feet. The rope is attached to the tower at ground level and to the dragon at a height of 7 feet. The dragon has pulled the rope taut, the end of the rope is 5 feet fro... | math | {
"style": "rule",
"ground_truth": "862"
} | {
"index": 1643,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A drawer contains a mixture of red socks and blue socks, at most $1991$ in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly $\\frac{1}{2}$ that both are red or both are blue. What is the largest possible number... | math | {
"style": "rule",
"ground_truth": "990"
} | {
"index": 1644,
"solution": "Let $r$ and $b$ denote the number of red and blue socks such that $r+b\\le1991$. Then by complementary counting, the number of ways to get a red and a blue sock must be equal to $1-\\frac12=\\frac12=\\frac{2rb}{(r+b)(r+b-1)}\\implies4rb=(r+b)(r+b-1)$ $=(r+b)^2-(r+b)\\implies r^2+2rb+b^... |
DeepScaleR | [
{
"role": "user",
"content": "A drawer in a darkened room contains $100$ red socks, $80$ green socks, $60$ blue socks and $40$ black socks. A youngster selects socks one at a time from the drawer but is unable to see the color of the socks drawn. What is the smallest number of socks that must be selected to... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 1645,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A dress originally priced at $80$ dollars was put on sale for $25\\%$ off. If $10\\%$ tax was added to the sale price, then the total selling price (in dollars) of the dress was"
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"index": 1646,
"solution": "1. **Calculate the Sale Price After Discount:**\n The original price of the dress is $80$ dollars. The discount offered is $25\\%$. The amount of discount in dollars can be calculated as:\n \\[\n \\text{Discount} = 25\\% \\times 80 = 0.25 \\times 80 = 20 \\text{ dollars}\n \\]\... |
DeepScaleR | [
{
"role": "user",
"content": "A driver left point A and headed towards point D, which are 100 km apart. The road from A to D passes through points B and C. At point B, the navigator showed that there were 30 minutes left to drive, and the driver immediately reduced their speed by 10 km/h. At point C, the na... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 1647,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $\\$0.50$ per mile, and her only expense is gasoline at $\\$2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense?"
... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 1648,
"solution": "1. **Calculate the total distance driven**: The driver travels at a speed of $60$ miles per hour for $2$ hours. Therefore, the total distance driven is:\n \\[\n 60 \\text{ miles/hour} \\times 2 \\text{ hours} = 120 \\text{ miles}\n \\]\n\n2. **Calculate the gasoline used**: The c... |
DeepScaleR | [
{
"role": "user",
"content": "A dump truck delivered sand to a construction site. The sand formed a conical pile with a diameter of $8$ feet and a height that was $75\\%$ of the diameter. How many cubic feet of sand were in the pile? Express your answer in terms of $\\pi$."
}
] | math | {
"style": "rule",
"ground_truth": "32 \\pi"
} | {
"index": 1649,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A facility has 7 consecutive parking spaces, and there are 3 different models of cars to be parked. If it is required that among the remaining 4 parking spaces, exactly 3 are consecutive, then the number of different parking methods is \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 1650,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A factory arranges $5$ workers to three duty positions, with each worker assigned to only one position. Each position must have at least $1$ worker. Calculate the number of ways to assign workers A and B to the same position."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 1651,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A factory assigns five newly recruited employees, including A and B, to three different workshops. Each workshop must be assigned at least one employee, and A and B must be assigned to the same workshop. The number of different ways to assign the employees is \\_\\_\\_\\_\\_\\_... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 1652,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A factory conducted a survey on the defective parts produced by a team. The number of defective parts produced by the team each day in 7 days is as follows (unit: pieces): 3, 3, 0, 2, 3, 0, 3. What is the value of the variance of the number of defective parts produced by the te... | math | {
"style": "rule",
"ground_truth": "\\frac{12}{7}"
} | {
"index": 1653,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A factory estimates that the total demand for a particular product in the first $x$ months starting from the beginning of 2016, denoted as $f(x)$ (in units of 'tai'), is approximately related to the month $x$ as follows: $f(x)=x(x+1)(35-2x)$, where $x \\in \\mathbb{N}^*$ and $x... | math | {
"style": "rule",
"ground_truth": "171"
} | {
"index": 1654,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A factory has a fixed daily cost of 20,000 yuan, and the maximum daily production capacity is 360 units. The cost increases by 100 yuan for each unit produced. The revenue function for producing $x$ units of product per day is $R(x) = -\\frac{1}{2}x^2 + 400x$. Let $L(x)$ and $P... | math | {
"style": "rule",
"ground_truth": "95"
} | {
"index": 1655,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A factory has two branches, one in location A and the other in location B, producing 12 and 6 machines respectively. Now, they need to distribute 10 machines to area A and 8 machines to area B. It is known that the transportation cost for moving one machine from location A to a... | math | {
"style": "rule",
"ground_truth": "8600"
} | {
"index": 1656,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A factory implements a time-based wage system, where each worker is paid 6 yuan for every hour worked, for a total of 8 hours per day. However, the clock used for timing is inaccurate: it takes 69 minutes for the minute hand to coincide with the hour hand once. Calculate the am... | math | {
"style": "rule",
"ground_truth": "2.60"
} | {
"index": 1657,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A factory makes chocolate bars. Five boxes, labelled $V, W, X, Y, Z$, are each packed with 20 bars. Each of the bars in three of the boxes has a mass of 100 g. Each of the bars in the other two boxes has a mass of 90 g. One bar is taken from box $V$, two bars are taken from box... | math | {
"style": "rule",
"ground_truth": "W \\text{ and } Z"
} | {
"index": 1658,
"solution": "The number of bars taken from the boxes is $1+2+4+8+16=31$. If these bars all had mass 100 g, their total mass would be 3100 g. Since their total mass is 2920 g, they are $3100 \\mathrm{~g}-2920 \\mathrm{~g}=180 \\mathrm{~g}$ lighter. Since all of the bars have a mass of 100 g or of 90... |
DeepScaleR | [
{
"role": "user",
"content": "A factory must filter its emissions before discharging them. The relationship between the concentration of pollutants $p$ (in milligrams per liter) and the filtration time $t$ (in hours) during the filtration process is given by the equation $p(t) = p_0e^{-kt}$. Here, $e$ is th... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 1659,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A factory packs its products in cubic boxes. In one store, they put $512$ of these cubic boxes together to make a large $8\\times 8 \\times 8$ cube. When the temperature goes higher than a limit in the store, it is necessary to separate the $512$ set of boxes using horizo... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 1660,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A factory produced an original calculator that performs two operations:\n\n(a) the usual addition, denoted by \\( + \\) \n\n(b) an operation denoted by \\( \\circledast \\).\n\nWe know that, for any natural number \\( a \\), the following hold:\n\n\\[ \n(i) \\quad a \\circledas... | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"index": 1661,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A factory produces a certain product for the Brazilian Olympic Games with an annual fixed cost of 2.5 million yuan. For every $x$ thousand units produced, an additional cost of $C(x)$ (in ten thousand yuan) is incurred. When the annual production is less than 80 thousand units,... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 1662,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A factory produces a certain product with an annual fixed cost of $250 million. When producing $x$ million units, an additional cost of $C(x)$ million is required. When the annual production volume is less than 80 million units, $C(x)= \\frac {1}{3}x^{2}+10x$; when the annual p... | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"index": 1663,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A factory produces a certain type of component that undergoes two processes. The probability of producing a defective component in the first and second processes is 0.01 and 0.03, respectively. The production of defective components in each process is independent. What is the p... | math | {
"style": "rule",
"ground_truth": "0.960"
} | {
"index": 1664,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A factory produces a certain type of component, and the inspector randomly selects 16 of these components from the production line each day to measure their dimensions (in cm). The dimensions of the 16 components selected in one day are as follows:\n\n10.12, 9.97, 10.01, 9.95, ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"index": 1665,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A factory produces a type of instrument. Due to limitations in production capacity and technical level, some defective products are produced. According to experience, the defect rate $p$ of the factory producing this instrument is generally related to the daily output $x$ (piec... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"index": 1666,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A factory produces two types of products, A and B, with profits P and Q (in ten thousand yuan), respectively. The relationship between the profits and the invested capital m (in ten thousand yuan) follows the empirical formulas P = (1/3)m + 65 and Q = 76 + 4√m. Now, 150 ten tho... | math | {
"style": "rule",
"ground_truth": "203"
} | {
"index": 1667,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A factory received a task to process 6000 pieces of part P and 2000 pieces of part Q. The factory has 214 workers. Each worker spends the same amount of time processing 5 pieces of part P as they do processing 3 pieces of part Q. The workers are divided into two groups to work ... | math | {
"style": "rule",
"ground_truth": "137"
} | {
"index": 1668,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A factory uses radiation to sterilize food and is now planning to build a dormitory for its workers near the factory, with radiation protection measures for the dormitory. The choice of radiation protection materials for the building and the distance of the dormitory from the f... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 1669,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A factory's annual fixed cost for producing a certain product is 2.5 million yuan. For every $x$ thousand units produced, an additional cost $C(x)$ is incurred. When the annual output is less than 80 thousand units, $C(x)=\\frac{1}{3}x^2+10x$ (in million yuan). When the annual ... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 1670,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A factory's total cost of producing $x$ units of a product is $c(x) = 1200 + \\frac{2}{75}x^{3}$ (in ten thousand yuan). It is known that the unit price $P$ (in ten thousand yuan) of the product and the number of units $x$ satisfy: $P^{2} = \\frac{k}{x}$. The unit price for pro... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 1671,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair $6$ sided die is rolled twice. What is the probability that the first number that comes up is greater than or equal to the second number?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{12}"
} | {
"index": 1672,
"solution": "1. **Define the Event**: Let's define the event $A$ as the event where the number on the first die roll is greater than or equal to the number on the second die roll.\n\n2. **Total Possible Outcomes**: Since each die has 6 faces, and the die is rolled twice, the total number of outcome... |
DeepScaleR | [
{
"role": "user",
"content": "A fair $6$-sided die is repeatedly rolled until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{20}"
} | {
"index": 1673,
"solution": "1. **Understanding the Problem**: We need to find the probability that all even numbers (2, 4, 6) appear at least once before the first odd number (1, 3, 5) appears when rolling a fair 6-sided die.\n\n2. **Setting Up the Probability for Each Roll**:\n - The probability of rolling an ... |
DeepScaleR | [
{
"role": "user",
"content": "A fair 10-sided die is rolled repeatedly until an odd number appears. What is the probability that every even number appears at least once before the first occurrence of an odd number?\n- **A** $\\frac{1}{300}$\n- **B** $\\frac{1}{252}$\n- **C** $\\frac{1}{500}$\n- **D** $\\fra... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{252}"
} | {
"index": 1674,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair 6-sided die and a fair 8-sided die are rolled once each. Calculate the probability that the sum of the numbers rolled is greater than 10."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{16}"
} | {
"index": 1675,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair 6-sided die is rolled once. If I roll $n$, then I win $6-n$ dollars. What is the expected value of my win, in dollars?"
}
] | math | {
"style": "rule",
"ground_truth": "2.50"
} | {
"index": 1676,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair 6-sided die is rolled. If I roll $n$, then I win $n^2$ dollars. What is the expected value of my win? Express your answer as a dollar value rounded to the nearest cent."
}
] | math | {
"style": "rule",
"ground_truth": "\\$15.17"
} | {
"index": 1677,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair 6-sided die is rolled. If the roll is even, then you win that amount of dollars (so that, for example, if you roll 4, then you win $\\$4$). If the roll is odd, you win nothing. What is the expected value of your winnings? Express your answer as a dollar value."
}
] | math | {
"style": "rule",
"ground_truth": "\\$2"
} | {
"index": 1678,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair 6-sided die is rolled. What is the probability that the number rolled is a divisor of 6?"
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac23"
} | {
"index": 1679,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair coin is flipped $7$ times. What is the probability that at least $5$ consecutive flips come up heads?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{16}"
} | {
"index": 1680,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair coin is flipped $8$ times. What is the probability that at least $6$ consecutive flips come up heads?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{128}"
} | {
"index": 1681,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair coin is flipped 7 times. What is the probability that at least 5 of the flips come up heads?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{29}{128}"
} | {
"index": 1682,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair coin is flipped 8 times. What is the probability that at least 6 consecutive flips come up heads?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{128}"
} | {
"index": 1683,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair coin is flipped 8 times. What is the probability that exactly 6 of the flips come up heads?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{64}"
} | {
"index": 1684,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair coin is flipped 8 times. What is the probability that fewer than 3 of the flips come up heads?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{37}{256}"
} | {
"index": 1685,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair coin is flipped 9 times. What is the probability that at least 6 of the flips result in heads?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{65}{256}"
} | {
"index": 1686,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair coin is flipped eight times in a row. Let $p$ be the probability that there is exactly one pair of consecutive flips that are both heads and exactly one pair of consecutive flips that are both tails. If $p=\\frac{a}{b}$, where $a, b$ are relatively prime positive integer... | math | {
"style": "rule",
"ground_truth": "1028"
} | {
"index": 1687,
"solution": "Separate the sequence of coin flips into alternating blocks of heads and tails. Of the blocks of heads, exactly one block has length 2, and all other blocks have length 1. The same statement applies to blocks of tails. Thus, if there are $k$ blocks in total, there are $k-2$ blocks of l... |
DeepScaleR | [
{
"role": "user",
"content": "A fair coin is flipped every second and the results are recorded with 1 meaning heads and 0 meaning tails. What is the probability that the sequence 10101 occurs before the first occurrence of the sequence 010101?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{21}{32}"
} | {
"index": 1688,
"solution": "Call it a win if we reach 10101, a loss if we reach 010101. Let $x$ be the probability of winning if the first flip is a 1, let $y$ be the probability of winning if the first flip is a 0. Then the probability of winning is $(x+y) / 2$ since the first flip is 1 or 0, each with probabili... |
DeepScaleR | [
{
"role": "user",
"content": "A fair coin is to be tossed $10_{}^{}$ times. Let $\\frac{i}{j}^{}_{}$, in lowest terms, be the probability that heads never occur on consecutive tosses. Find $i+j_{}^{}$."
}
] | math | {
"style": "rule",
"ground_truth": "73"
} | {
"index": 1689,
"solution": "We can also split the problem into casework.\n Case 1: 0 Heads\n There is only one possibility.\n Case 2: 1 Head\n There are 10 possibilities.\n Case 3: 2 Heads\n There are 36 possibilities.\n Case 4: 3 Heads\n There are 56 possibilities.\n Case 5: 4 Heads\n There are 35 possibilities.... |
DeepScaleR | [
{
"role": "user",
"content": "A fair coin is tossed 3 times. What is the probability of at least two consecutive heads?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 1690,
"solution": "1. **Total Outcomes**: A fair coin tossed 3 times can result in $2^3 = 8$ possible outcomes. These outcomes are: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.\n\n2. **Unfavorable Outcomes**: We need to find the outcomes where there are no two consecutive heads. These are:\n - TTT: No heads... |
DeepScaleR | [
{
"role": "user",
"content": "A fair coin is tossed 4 times. Calculate the probability of getting at least two consecutive heads."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 1691,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair coin is tossed 4 times. What is the probability of at least two consecutive heads?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{8}"
} | {
"index": 1692,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair coin is tossed 4 times. What is the probability of getting at least two consecutive heads?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{8}"
} | {
"index": 1693,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair coin is tossed six times and the sequence of heads and tails is recorded. What is the probability that the sequence contains exactly two heads? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{15}{64}"
} | {
"index": 1694,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair coin is tossed. If the first toss shows heads, it is tossed again: if the second toss shows heads, record 0; if the second toss shows tails, record 1. If the first toss shows tails, record any random real number in the closed interval $[0, 1]$. This process is repeated t... | math | {
"style": "rule",
"ground_truth": "$\\frac{7}{16}$"
} | {
"index": 1695,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair dice is rolled twice, and the scores obtained are denoted as $m$ and $n$ respectively. Let the angle between vector $a=(m,n)$ and vector $b=(1,-1)$ be $\\theta$. The probability that $\\theta$ is an acute angle is $\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{12}"
} | {
"index": 1696,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair die (a cube toy with 1, 2, 3, 4, 5, 6 dots on each face) is rolled twice. Let 'm' be the number of dots that appear on the first roll, and 'n' be the number of dots that appear on the second roll. The probability that m = kn (k ∈ N*) is _____."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{18}"
} | {
"index": 1697,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair die is rolled four times. The probability that each of the final three rolls is at least as large as the roll preceding it may be expressed in the form $\\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$."
}
] | math | {
"style": "rule",
"ground_truth": "79"
} | {
"index": 1698,
"solution": "Recast the problem entirely as a block-walking problem. Call the respective dice $a, b, c, d$. In the diagram below, the lowest $y$-coordinate at each of $a$, $b$, $c$, and $d$ corresponds to the value of the roll.\n The red path corresponds to the sequence of rolls $2, 3, 5, 5$. This ... |
DeepScaleR | [
{
"role": "user",
"content": "A fair die is rolled six times. The probability of rolling at least a five at least five times is"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{729}"
} | {
"index": 1699,
"solution": "1. **Determine the probability of rolling at least a five on a single roll**: \n A fair die has six faces, and rolling at least a five corresponds to rolling either a five or a six. There are 2 favorable outcomes (5 and 6) out of 6 possible outcomes. Thus, the probability of rolling ... |
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