data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "A fair die is rolled twice in succession, and the numbers facing up are observed and recorded as $x$ and $y$ respectively. \n$(1)$ If the event \"$x+y=8$\" is denoted as event $A$, find the probability of event $A$ occurring; \n$(2)$ If the event \"$x^{2}+y^{2} \\leqslant 12$... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{6}"
} | {
"index": 1700,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair die, numbered 1, 2, 3, 4, 5, 6, is thrown three times. The numbers obtained are recorded sequentially as $a$, $b$, and $c$. The probability that $a+bi$ (where $i$ is the imaginary unit) is a root of the equation $x^{2}-2x+c=0$ is $\\_\\_\\_\\_\\_\\_$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{108}"
} | {
"index": 1701,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair six-sided die has faces numbered $1, 2, 3, 4, 5, 6$. The die is rolled four times, and the results are $a, b, c, d$. What is the probability that one of the numbers in the set $\\{a, a+b, a+b+c, a+b+c+d\\}$ is equal to 4?"
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{343}{1296}$"
} | {
"index": 1702,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair six-sided die is rolled 3 times. If the sum of the numbers rolled on the first two rolls is equal to the number rolled on the third roll, what is the probability that at least one of the numbers rolled is 2?"
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{8}{15}$"
} | {
"index": 1703,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair six-sided die is rolled twice, and the resulting numbers are denoted as $a$ and $b$.\n\n(1) Find the probability that $a^2 + b^2 = 25$.\n\n(2) Given three line segments with lengths $a$, $b$, and $5$, find the probability that they can form an isosceles triangle (includi... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{18}"
} | {
"index": 1704,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair six-sided die is rolled twice. Let $a$ and $b$ be the numbers obtained from the first and second roll respectively. Determine the probability that three line segments of lengths $a$, $b$, and $5$ can form an isosceles triangle."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{18}"
} | {
"index": 1705,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair six-sided die with faces numbered 1 to 6 is rolled twice. Let $a$ and $b$ denote the numbers obtained in the two rolls.\n\n1. Find the probability that $a + b \\geq 9$.\n2. Find the probability that the line $ax + by + 5 = 0$ is tangent to the circle $x^2 + y^2 = 1$.\n3.... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{18}"
} | {
"index": 1706,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair six-sided die with faces numbered 1, 2, 3, 4, 5, and 6 is rolled twice. Let $a$ and $b$ denote the outcomes of the first and second rolls, respectively.\n\n(1) Find the probability that the line $ax + by + 5 = 0$ is tangent to the circle $x^2 + y^2 = 1$.\n\n(2) Find the ... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{18}"
} | {
"index": 1707,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair six-sided die with uniform quality is rolled twice in succession. Let $a$ and $b$ denote the respective outcomes. Find the probability that the function $f(x) = \\frac{1}{3}x^3 + \\frac{1}{2}ax^2 + bx$ has an extreme value."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{36}"
} | {
"index": 1708,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair standard four-sided die is tossed four times. Given that the sum of the first three tosses equals the fourth toss, what is the probability that at least one \"3\" is rolled?\nA) $\\frac{1}{12}$\nB) $\\frac{1}{6}$\nC) $\\frac{3}{10}$\nD) $\\frac{1}{4}$\nE) $\\frac{1}{2}$"... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{10}"
} | {
"index": 1709,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair standard six-sided dice is tossed four times. Given that the sum of the first three tosses equals the fourth toss, what is the probability that at least one \"3\" is tossed?\nA) $\\frac{1}{6}$\nB) $\\frac{6}{17}$\nC) $\\frac{9}{17}$\nD) $\\frac{1}{2}$\nE) $\\frac{1}{3}$"... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{17}"
} | {
"index": 1710,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fair standard six-sided dice is tossed three times. Given that the sum of the first two tosses equal the third, what is the probability that at least one \"2\" is tossed?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{15}"
} | {
"index": 1711,
"solution": "1. **Identify possible outcomes for the third toss**: The third toss, denoted as $X_3$, must be at least 2 because the minimum sum of two dice ($X_1$ and $X_2$) is 2. Thus, $X_3$ can be 2, 3, 4, 5, or 6.\n\n2. **List possible combinations for each value of $X_3$**:\n - If $X_3 = 2$, ... |
DeepScaleR | [
{
"role": "user",
"content": "A fair, twenty-faced die has $19$ of its faces numbered from $1$ through $19$ and has one blank face. Another fair, twenty-faced die has $19$ of its faces numbered from $1$ through $8$ and $10$ through $20$ and has one blank face. When the two dice are rolled, what is the proba... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{80}"
} | {
"index": 1712,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A famous theorem states that given any five points in the plane, with no three on the same line, there is a unique conic section (ellipse, hyperbola, or parabola) which passes through all five points. The conic section passing through the five points \\[(-\\tfrac32, 1), \\; (0,... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt3}{3}"
} | {
"index": 1713,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A farmer bought $749$ sheep. He sold $700$ of them for the price paid for the $749$ sheep. The remaining $49$ sheep were sold at the same price per head as the other $700$. Based on the cost, the percent gain on the entire transaction is:"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 1714,
"solution": "1. **Define the cost and revenue variables:**\n Let the cost of each sheep be $x$ dollars. Therefore, the total cost for $749$ sheep is $749x$ dollars.\n\n2. **Calculate the revenue from selling 700 sheep:**\n The farmer sold $700$ sheep for the same amount he paid for all $749$ sh... |
DeepScaleR | [
{
"role": "user",
"content": "A farmer builds a rectangular chicken coop that leans against a wall, covering an area of 36m<sup>2</sup>. Due to geographical constraints, the length of the side of the chicken coop, $x$, cannot exceed 7m. The wall height is 2m. The cost of constructing the front of the chicke... | math | {
"style": "rule",
"ground_truth": "2760"
} | {
"index": 1715,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A farmer buys 600 cows. He sells 500 of them for the price he paid for all 600 cows. The remaining 100 cows are sold for 10% more per cow than the price of the 500 cows. Calculate the percentage gain on the entire transaction."
}
] | math | {
"style": "rule",
"ground_truth": "22\\%"
} | {
"index": 1716,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A farmer claims that with four rods, he can fence a square plot of land sufficient for one sheep. If this is true, what is the minimum number of rods needed to fence an area for ten sheep? The answer depends on the shape of your fence. \n\nHow many rods are required for 10 shee... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 1717,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A farmer contracted several acres of fruit trees. This year, he invested 13,800 yuan, and the total fruit yield was 18,000 kilograms. The fruit sells for a yuan per kilogram in the market and b yuan per kilogram when sold directly from the orchard (b < a). The farmer transports... | math | {
"style": "rule",
"ground_truth": "20\\%"
} | {
"index": 1718,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A farmer had an enclosure with a fence 50 rods long, which could only hold 100 sheep. Suppose the farmer wanted to expand the enclosure so that it could hold twice as many sheep.\n\nHow many additional rods will the farmer need?"
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 1719,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A farmer has a rectangular field with dimensions $2m+7$ and $m-2$. If the field has an area of 51 square units, what is the value of $m$?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 1720,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A farmer has a right-angled triangular farm with legs of lengths 3 and 4. At the right-angle corner, the farmer leaves an unplanted square area $S$. The shortest distance from area $S$ to the hypotenuse of the triangle is 2. What is the ratio of the area planted with crops to t... | math | {
"style": "rule",
"ground_truth": "$\\frac{145}{147}$"
} | {
"index": 1721,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A farmer sold domestic rabbits. By the end of the market, he sold exactly one-tenth as many rabbits as the price per rabbit in forints. He then distributed the revenue between his two sons. Starting with the older son, the boys alternately received one-hundred forint bills, but... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 1722,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A farmer's rectangular field is partitioned into a $2$ by $2$ grid of $4$ rectangular sections. In each section the farmer will plant one crop: corn, wheat, soybeans, or potatoes. The farmer does not want to grow corn and wheat in any two sections that share a border, and the f... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"index": 1723,
"solution": "We will analyze the problem by considering different cases based on the distribution of crops in the sections. The key restrictions are that corn cannot be adjacent to wheat, and soybeans cannot be adjacent to potatoes.\n\n#### Case 1: All sections have the same crop.\n- There are 4 ty... |
DeepScaleR | [
{
"role": "user",
"content": "A fast train with a weight of $P=150$ tons travels at a maximum speed of $v=72 \\frac{\\text{km}}{\\text{hour}}$ on a horizontal track with a friction coefficient of $\\rho=0,005$. What speed can it reach on a track with the same friction conditions but with an incline having $... | math | {
"style": "rule",
"ground_truth": "10.3"
} | {
"index": 1724,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fat coin is one which, when tossed, has a $2 / 5$ probability of being heads, $2 / 5$ of being tails, and $1 / 5$ of landing on its edge. Mr. Fat starts at 0 on the real line. Every minute, he tosses a fat coin. If it's heads, he moves left, decreasing his coordinate by 1; if... | math | {
"style": "rule",
"ground_truth": "\\[\n\\frac{1}{3}\n\\]"
} | {
"index": 1725,
"solution": "For $n \\in \\mathbb{Z}$, let $a_{n}$ be the fraction of the time Mr. Fat spends at $n$. By symmetry, $a_{n}=a_{-n}$ for all $n$. For $n>0$, we have $a_{n}=\\frac{2}{5} a_{n-1}+\\frac{2}{5} a_{n+1}$, or $a_{n+1}=\\frac{5}{2} a_{n}-a_{n-1}$. This Fibonacci-like recurrence can be solved ... |
DeepScaleR | [
{
"role": "user",
"content": "A father and son were walking one after the other along a snow-covered road. The father's step length is $80 \\mathrm{~cm}$, and the son's step length is $60 \\mathrm{~cm}$. Their steps coincided 601 times, including at the very beginning and at the end of the journey. What dis... | math | {
"style": "rule",
"ground_truth": "1440"
} | {
"index": 1726,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A faulty car odometer proceeds from digit 3 to digit 5, always skipping the digit 4, regardless of position. If the odometer now reads 002005, how many miles has the car actually traveled?"
}
] | math | {
"style": "rule",
"ground_truth": "1462"
} | {
"index": 1727,
"solution": "\nTo solve this problem, we need to determine how many miles the car has actually traveled when the odometer reads 002005, given that it skips every occurrence of the digit 4.\n\n#### Method 1: Counting Skipped Numbers\n\n1. **Counting Numbers with Digit 4**:\n - **Hundreds Place**: ... |
DeepScaleR | [
{
"role": "user",
"content": "A fenced, rectangular field measures $24$ meters by $52$ meters. An agricultural researcher has 1994 meters of fence that can be used for internal fencing to partition the field into congruent, square test plots. The entire field must be partitioned, and the sides of the square... | math | {
"style": "rule",
"ground_truth": "702"
} | {
"index": 1728,
"solution": "Suppose there are $n$ squares in every column of the grid, so there are $\\frac{52}{24}n = \\frac {13}6n$ squares in every row. Then $6|n$, and our goal is to maximize the value of $n$.\n Each vertical fence has length $24$, and there are $\\frac{13}{6}n - 1$ vertical fences; each hori... |
DeepScaleR | [
{
"role": "user",
"content": "A ferry boat begins shuttling tourists to an island every hour starting at 10 AM, with its last trip starting at 4 PM. On the 10 AM trip, there were 100 tourists on the ferry, and on each successive trip, the number of tourists decreased by 2 from the previous trip. Calculate t... | math | {
"style": "rule",
"ground_truth": "658"
} | {
"index": 1729,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A ferry boat shuttles tourists to an island every half-hour from 10 AM to 3 PM, with 100 tourists on the first trip and 2 fewer tourists on each successive trip. Calculate the total number of tourists taken to the island that day."
}
] | math | {
"style": "rule",
"ground_truth": "990"
} | {
"index": 1730,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 3 PM. One day the boat captain notes that on the 10 AM trip there were 100 tourists on the ferry boat, and that on each successive trip, the number of tourists was 1 fe... | math | {
"style": "rule",
"ground_truth": "585"
} | {
"index": 1731,
"solution": "1. **Identify the number of trips**: The ferry starts at 10 AM and makes its last trip at 3 PM. Since the trips are hourly, we can calculate the number of trips as follows:\n - 10 AM, 11 AM, 12 PM, 1 PM, 2 PM, 3 PM\n - This gives us a total of 6 trips.\n\n2. **Calculate the number ... |
DeepScaleR | [
{
"role": "user",
"content": "A fifteen-digit integer is formed by repeating a positive five-digit integer three times. For example, 42563,42563,42563 or 60786,60786,60786 are integers of this form. What is the greatest common divisor of all fifteen-digit integers in this form?"
}
] | math | {
"style": "rule",
"ground_truth": "10000100001"
} | {
"index": 1732,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fifth number, $n$, is added to the set $\\{ 3,6,9,10 \\}$ to make the mean of the set of five numbers equal to its median. The number of possible values of $n$ is"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 1733,
"solution": "We are given the set $\\{3, 6, 9, 10\\}$ and we need to add a number $n$ such that the mean of the new set equals its median. Let's analyze the problem step by step.\n\n#### Step 1: Understand the properties of mean and median\n- **Mean** of a set is the average of all the numbers in t... |
DeepScaleR | [
{
"role": "user",
"content": "A figure is constructed from unit cubes. Each cube shares at least one face with another cube. What is the minimum number of cubes needed to build a figure with the front and side views shown? [asy]\n/* AMC8 2003 #15 Problem */\ndraw((0,0)--(2,0)--(2,1)--(1,1)--(1,2)--(0,2)-... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 1734,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A finite arithmetic progression \\( a_1, a_2, \\ldots, a_n \\) with a positive common difference has a sum of \\( S \\), and \\( a_1 > 0 \\). It is known that if the common difference of the progression is increased by 3 times while keeping the first term unchanged, the sum \\(... | math | {
"style": "rule",
"ground_truth": "5/2"
} | {
"index": 1735,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A finite increasing sequence \\(a_{1}, a_{2}, \\ldots, a_{n}\\) of natural numbers is given, where \\(n \\geq 3\\), and for all \\(k \\leq n-2\\) the following equality holds: \\(a_{k+2} = 3a_{k+1} - 2a_{k} - 2\\). The sequence must include \\(a_{k} = 2022\\). Determine the max... | math | {
"style": "rule",
"ground_truth": "225"
} | {
"index": 1736,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A finite non-empty set of integers is called $3$ -*good* if the sum of its elements is divisible by $3$ . Find the number of $3$ -good subsets of $\\{0,1,2,\\ldots,9\\}$ ."
}
] | math | {
"style": "rule",
"ground_truth": "351"
} | {
"index": 1737,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first t... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"index": 1738,
"solution": "1. **Understanding the Sequence Property**: The sequence is such that the tens and units digits of each term become the hundreds and tens digits of the next term. This implies that each digit in a position (hundreds, tens, units) of one term will appear in the next position in the next... |
DeepScaleR | [
{
"role": "user",
"content": "A finite set $\\{a_1, a_2, ... a_k\\}$ of positive integers with $a_1 < a_2 < a_3 < ... < a_k$ is named *alternating* if $i+a$ for $i = 1, 2, 3, ..., k$ is even. The empty set is also considered to be alternating. The number of alternating subsets of $\\{1, 2, 3,..., ... | math | {
"style": "rule",
"ground_truth": "5702887"
} | {
"index": 1739,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A finite set $S$ of points in the coordinate plane is called [i]overdetermined[/i] if $|S|\\ge 2$ and there exists a nonzero polynomial $P(t)$, with real coefficients and of degree at most $|S|-2$, satisfying $P(x)=y$ for every point $(x,y)\\in S$. \n \nFor each integer $n\\ge ... | math | {
"style": "rule",
"ground_truth": "2^{n-1} - n"
} | {
"index": 1740,
"solution": "\nGiven a finite set \\( S \\) of points in the coordinate plane, a set \\( S \\) is called \\textit{overdetermined} if \\( |S| \\ge 2 \\) and there exists a nonzero polynomial \\( P(t) \\) with real coefficients of degree at most \\( |S| - 2 \\), such that \\( P(x) = y \\) for every p... |
DeepScaleR | [
{
"role": "user",
"content": "A finite set $\\mathcal{S}$ of distinct real numbers has the following properties: the mean of $\\mathcal{S}\\cup\\{1\\}$ is $13$ less than the mean of $\\mathcal{S}$, and the mean of $\\mathcal{S}\\cup\\{2001\\}$ is $27$ more than the mean of $\\mathcal{S}$. Find the mean of $... | math | {
"style": "rule",
"ground_truth": "651"
} | {
"index": 1741,
"solution": "Since this is a weighted average problem, the mean of $S$ is $\\frac{13}{27}$ as far from $1$ as it is from $2001$. Thus, the mean of $S$ is $1 + \\frac{13}{13 + 27}(2001 - 1) = \\boxed{651}$.\n ~ pi_is_3.14"
} |
DeepScaleR | [
{
"role": "user",
"content": "A firecracker was thrown vertically upwards with a speed of $20 \\mathrm{~m/s}$. Three seconds after the start of the flight, it exploded into two fragments of equal mass. The first fragment flew horizontally immediately after the explosion with a speed of $48 \\mathrm{~m/s}$. ... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"index": 1742,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A firecracker was thrown vertically upwards with a speed of 20 m/s. Three seconds after the start of its flight, it exploded into two unequal parts, the mass ratio of which is $1: 2$. The smaller fragment immediately after the explosion flew horizontally at a speed of $16 \\mat... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 1743,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A five-character license plate is composed of English letters and digits. The first four positions must contain exactly two English letters (letters $I$ and $O$ cannot be used). The last position must be a digit. Xiao Li likes the number 18, so he hopes that his license plate c... | math | {
"style": "rule",
"ground_truth": "23040"
} | {
"index": 1744,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A five-digit integer is chosen at random from all possible positive five-digit integers. What is the probability that the number's units digit is an even number and less than 6? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{10}"
} | {
"index": 1745,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A five-digit integer will be chosen at random from all possible positive five-digit integers. What is the probability that the number's units digit will be less than 5? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 1746,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A five-digit natural number $\\overline{a_1a_2a_3a_4a_5}$ is considered a \"concave number\" if and only if $a_1 > a_2 > a_3$ and $a_3 < a_4 < a_5$, with each $a_i \\in \\{0,1,2,3,4,5\\}$ for $i=1,2,3,4,5$. Calculate the number of possible \"concave numbers\"."
}
] | math | {
"style": "rule",
"ground_truth": "146"
} | {
"index": 1747,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A five-digit number \\(abcde\\) satisfies:\n\\[ a < b, \\, b > c > d, \\, d < e, \\, \\text{and} \\, a > d, \\, b > e. \\]\n\nFor example, 34 201, 49 412. If the digit order's pattern follows a variation similar to the monotonicity of a sine function over one period, then the f... | math | {
"style": "rule",
"ground_truth": "2892"
} | {
"index": 1748,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A five-digit number has one of its digits crossed out, and the resulting four-digit number is added to the original number. The sum is 54321. Find the original number."
}
] | math | {
"style": "rule",
"ground_truth": "49383"
} | {
"index": 1749,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A five-digit number is called a \"hill\" if its first three digits are in ascending order and its last three digits are in descending order. For example, 13760 and 28932 are hills, whereas 78821 and 86521 are not hills. How many hills exist that are greater than the number 7777... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 1750,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A five-digit number is called a \"pit\" if its first three digits are in descending order and the last three digits are in ascending order. For example, 73016 and 98346 are pits, while 88012 and 56821 are not pits. How many pits are there that are less than the number 22222?"
... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 1751,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A five-digit number is called irreducible if it cannot be expressed as a product of two three-digit numbers.\n\nWhat is the greatest number of consecutive irreducible five-digit numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "99"
} | {
"index": 1752,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A five-digit palindrome is a positive integer with respective digits $abcba$, where $a$ is non-zero. Let $S$ be the sum of all five-digit palindromes. What is the sum of the digits of $S$?"
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 1753,
"solution": "1. **Define the form of a five-digit palindrome**: A five-digit palindrome can be represented as $\\overline{abcba}$, where $a, b, c$ are digits and $a \\neq 0$ (since it is a five-digit number).\n\n2. **Calculate the total number of five-digit palindromes**: \n - $a$ can be any digi... |
DeepScaleR | [
{
"role": "user",
"content": "A five-digit positive integer is created using each of the odd digits $1, 3, 5, 7, 9$ once so that the thousands digit is larger than the hundreds digit, the thousands digit is larger than the ten thousands digit, the tens digit is larger than the hundreds digit, and the tens d... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 1754,
"solution": "We write such a five-digit positive integer with digits $V W X Y Z$. We want to count the number of ways of assigning $1, 3, 5, 7, 9$ to the digits $V, W, X, Y, Z$ in such a way that the given properties are obeyed. From the given conditions, $W > X, W > V, Y > X$, and $Y > Z$. The dig... |
DeepScaleR | [
{
"role": "user",
"content": "A flag consists of three horizontal strips of fabric, each of a solid color, from the choices of red, white, blue, green, or yellow. If no two adjacent strips can be the same color, and an additional rule that no color can be used more than twice, how many distinct flags are po... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 1755,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A flag in the shape of a square has a symmetrical cross of uniform width, featuring a center decorated with a green square, and set against a yellow background. The entire cross (comprising both the arms and the green center) occupies 49% of the total area of the flag. What per... | math | {
"style": "rule",
"ground_truth": "25\\%"
} | {
"index": 1756,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A flag is made of three horizontal strips of fabric, each of a solid color, either red, white, blue or green. If no two adjacent strips can be the same color, how many distinct flags are possible? These two flags are different.\n\n[asy]draw((0,0)--(0,6),linewidth(1));\nfilldraw... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 1757,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A flagpole is originally $5$ meters tall. A hurricane snaps the flagpole at a point $x$ meters above the ground so that the upper part, still attached to the stump, touches the ground $1$ meter away from the base. What is $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "2.4"
} | {
"index": 1758,
"solution": "1. **Identify the problem setup**: Let $AB$ represent the flagpole, which is $5$ meters tall. The flagpole breaks at point $D$, and the top part touches the ground at point $C$, $1$ meter away from the base $B$. We need to find the height $x = AD$ where the flagpole breaks.\n\n2. **Use... |
DeepScaleR | [
{
"role": "user",
"content": "A flat board has a circular hole with radius $1$ and a circular hole with radius $2$ such that the distance between the centers of the two holes is $7.$ Two spheres with equal radii sit in the two holes such that the spheres are tangent to each other. The square of the radius o... | math | {
"style": "rule",
"ground_truth": "173"
} | {
"index": 1759,
"solution": "[asy] size(10cm); pair A, B, C, D, O, P, H, L, X, Y; A = (-1, 0); B = (1, 0); H = (0, 0); C = (5, 0); D = (9, 0); L = (7, 0); O = (0, sqrt(160/13 - 1)); P = (7, sqrt(160/13 - 4)); X = (0, sqrt(160/13 - 4)); Y = (O + P) / 2; draw(A -- O -- B -- cycle); draw(C -- P -- D -- cycle); draw(B... |
DeepScaleR | [
{
"role": "user",
"content": "A flea is jumping on the vertices of square \\(ABCD\\), starting from vertex \\(A\\). With each jump, it moves to an adjacent vertex with a probability of \\(\\frac{1}{2}\\). The flea stops when it reaches the last vertex it has not yet visited. Determine the probability that e... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 1760,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A flock of geese is flying, and a lone goose flies towards them and says, \"Hello, a hundred geese!\" The leader of the flock responds, \"No, we are not a hundred geese! If there were as many of us as there are now, plus the same amount, plus half of that amount, plus a quarter... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 1761,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A flock of geese was flying over several lakes. On each lake, half of the geese and an additional half goose landed, while the rest continued flying. All the geese landed after seven lakes. How many geese were in the flock?"
}
] | math | {
"style": "rule",
"ground_truth": "127"
} | {
"index": 1762,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A floor 9 feet by 12 feet, is to be tiled with 4-inch-by-6-inch tiles. How many tiles are needed to cover the floor?"
}
] | math | {
"style": "rule",
"ground_truth": "648"
} | {
"index": 1763,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A floor decoration is a circle with eight rays pointing from the center. The rays form eight congruent central angles. One of the rays points due north. What is the measure in degrees of the smaller angle formed between the ray pointing East and the ray pointing Southwest?\n... | math | {
"style": "rule",
"ground_truth": "135"
} | {
"index": 1764,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?"
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 1765,
"solution": "Let's denote the total number of flowers as $F$. We are given several pieces of information about the distribution of these flowers:\n\n1. Six tenths of the flowers are pink, so the number of pink flowers is $\\frac{6}{10}F$.\n2. One third of the pink flowers are roses, so the number o... |
DeepScaleR | [
{
"role": "user",
"content": "A flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What percent of the flowers are carnations?"
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 1766,
"solution": "1. **Identify the fraction of pink and red flowers**: \n Given that six tenths (or $\\frac{6}{10}$) of the flowers are pink, we can simplify this fraction to $\\frac{3}{5}$. Consequently, the remaining flowers must be red, which is $\\frac{2}{5}$ of the total flowers (since the total... |
DeepScaleR | [
{
"role": "user",
"content": "A flower shop buys a number of roses from a farm at a price of 5 yuan per rose each day and sells them at a price of 10 yuan per rose. If the roses are not sold by the end of the day, they are discarded.\n\n(1) If the shop buys 16 roses in one day, find the profit function \\( ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 1767,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fly is being chased by three spiders on the edges of a regular octahedron. The fly has a speed of $50$ meters per second, while each of the spiders has a speed of $r$ meters per second. The spiders choose their starting positions, and choose the fly's starting position, w... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 1768,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fly is on the edge of a ceiling of a circular room with a radius of 58 feet. The fly walks straight across the ceiling to the opposite edge, passing through the center of the circle. It then walks straight to another point on the edge of the circle but not back through the ce... | math | {
"style": "rule",
"ground_truth": "280"
} | {
"index": 1769,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A fly trapped inside a cubical box with side length $1$ meter decides to relieve its boredom by visiting each corner of the box. It will begin and end in the same corner and visit each of the other corners exactly once. To get from a corner to any other corner, it will either f... | math | {
"style": "rule",
"ground_truth": "$4\\sqrt{2}+4\\sqrt{3}$"
} | {
"index": 1770,
"solution": "1. **Understanding the Problem**: A fly is inside a cubical box with side length $1$ meter. It starts at one corner, visits each of the other corners exactly once, and returns to the starting corner. The fly moves in straight lines between corners. We need to find the maximum possible ... |
DeepScaleR | [
{
"role": "user",
"content": "A fly trapped inside a rectangular prism with dimensions $1$ meter, $2$ meters, and $3$ meters decides to tour the corners of the box. It starts from the corner $(0,0,0)$ and ends at the corner $(0,0,3)$, visiting each of the other corners exactly once. Determine the maximum po... | math | {
"style": "rule",
"ground_truth": "\\sqrt{14} + 6 + \\sqrt{13} + \\sqrt{5}"
} | {
"index": 1771,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A food factory has made 4 different exquisite cards. Each bag of food produced by the factory randomly contains one card. If all 4 different cards are collected, a prize can be won. Xiaoming buys 6 bags of this food at once. What is the probability that Xiaoming will win the pr... | math | {
"style": "rule",
"ground_truth": "195/512"
} | {
"index": 1772,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A food factory processes mushrooms, with a cost of 20 yuan per kilogram of mushrooms, and a processing fee of t yuan per kilogram (t is a constant, and $2 \\leq t \\leq 5$). Assume the factory price per kilogram of mushrooms is x yuan ($25 \\leq x \\leq 40$). According to marke... | math | {
"style": "rule",
"ground_truth": "100e^4"
} | {
"index": 1773,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A food factory regularly purchases flour. It is known that the factory needs 6 tons of flour per day, the price of each ton of flour is 1800 yuan, and the storage and other costs for flour are an average of 3 yuan per ton per day. Each time flour is purchased, a shipping fee of... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 1774,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A football association stipulates that in the league, a team earns $a$ points for a win, $b$ points for a draw, and 0 points for a loss, where $a$ and $b$ are real numbers such that $a > b > 0$. If a team has 2015 possible total scores after $n$ games, find the minimum value of... | math | {
"style": "rule",
"ground_truth": "62"
} | {
"index": 1775,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 points. How many points did the Panthers score?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 1776,
"solution": "\nWe are given that the total points scored by both teams is 34 and the Cougars won by a margin of 14 points. We need to find the number of points scored by the Panthers.\n\n#### Solution 1: Using the margin and total points\n1. Let $C$ be the points scored by the Cougars and $P$ be th... |
DeepScaleR | [
{
"role": "user",
"content": "A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of $34$ points, and the Cougars won by a margin of $14$ points. How many points did the Panthers score?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 1777,
"solution": "1. **Define Variables:**\n Let $x$ be the number of points scored by the Cougars, and $y$ be the number of points scored by the Panthers.\n\n2. **Set Up Equations:**\n From the problem, we have two equations based on the total points scored and the margin by which the Cougars won:\... |
DeepScaleR | [
{
"role": "user",
"content": "A force of $60 \\mathrm{H}$ stretches a spring by 2 cm. The initial length of the spring is $14 \\mathrm{~cm}$. How much work is required to stretch it to 20 cm?"
}
] | math | {
"style": "rule",
"ground_truth": "5.4"
} | {
"index": 1778,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A foreign investor plans to invest in 3 different projects among 4 candidate cities, with no more than 2 projects in the same city. How many different investment plans are there?"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 1779,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A four digit number is called *stutterer* if its first two digits are the same and its last two digits are also the same, e.g. $3311$ and $2222$ are stutterer numbers. Find all stutterer numbers that are square numbers."
}
] | math | {
"style": "rule",
"ground_truth": "7744"
} | {
"index": 1780,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A four-digit integer $m$ and the four-digit integer obtained by reversing the order of the digits of $m$ are both divisible by 45. If $m$ is divisible by 7, what is the greatest possible value of $m$?"
}
] | math | {
"style": "rule",
"ground_truth": "5985"
} | {
"index": 1781,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A four-digit integer $m$ and the four-digit integer obtained by reversing the order of the digits of $m$ are both divisible by 63. If $m$ is also divisible by 11, what is the greatest possible value of $m$?"
}
] | math | {
"style": "rule",
"ground_truth": "9702"
} | {
"index": 1782,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A four-digit integer \\( p \\) and the four-digit integer obtained by reversing the order of the digits of \\( p \\) are both divisible by \\( 63 \\) (product of primes \\( 7 \\) and \\( 9 \\)). Additionally, \\( p \\) must be divisible by \\( 19 \\). Determine the greatest pos... | math | {
"style": "rule",
"ground_truth": "5985"
} | {
"index": 1783,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A four-digit natural number $M$, where the digits in each place are not $0$, we take its hundreds digit as the tens digit and the tens digit as the units digit to form a new two-digit number. If this two-digit number is greater than the sum of the thousands digit and units digi... | math | {
"style": "rule",
"ground_truth": "5883"
} | {
"index": 1784,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A four-digit number $2\\Box\\Box5$ is divisible by $45$. How many such four-digit numbers are there?"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 1785,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A four-digit number $\\overline{a b c d}$ (where digits can repeat and are non-zero) is called a \"good number\" if it satisfies the conditions $\\overline{a b}<20$, $\\overline{b c}<21$, and $\\overline{c d}<22$. How many such \"good numbers\" are there?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 1786,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A four-digit number $\\overline{abcd}$ has the properties that $a + b + c + d = 26$, the tens digit of $b \\cdot d$ equals $a + c$, and $bd - c^2$ is a multiple of 2. Find this four-digit number (provide justification)."
}
] | math | {
"style": "rule",
"ground_truth": "1979"
} | {
"index": 1787,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A four-digit number \\((xyzt)_B\\) is called a stable number in base \\(B\\) if \\((xyzt)_B = (dcba)_B - (abcd)_B\\), where \\(a \\leq b \\leq c \\leq d\\) are the digits \\(x, y, z, t\\) arranged in ascending order. Determine all the stable numbers in base \\(B\\).\n\n(Problem... | math | {
"style": "rule",
"ground_truth": "(1001)_2, (3021)_4, (3032)_5, (3B/5, B/5-1, 4B/5-1, 2B/5)_B, 5 | B"
} | {
"index": 1788,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A four-digit number \\(\\overline{abcd} (1 \\leqslant a \\leqslant 9, 0 \\leqslant b, c, d \\leqslant 9)\\) is called a \\(P\\) type number if \\(a > b, b < c, c > d\\). It is called a \\(Q\\) type number if \\(a < b, b > c, c < d\\). Let \\(N(P)\\) and \\(N(Q)\\) be the number... | math | {
"style": "rule",
"ground_truth": "285"
} | {
"index": 1789,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A four-digit number has the following properties:\r\n\r\n(a) It is a perfect square;\r\n\r\n(b) Its first two digits are equal\r\n\r\n(c) Its last two digits are equal.\r\n\r\nFind all such four-digit numbers."
}
] | math | {
"style": "rule",
"ground_truth": "7744"
} | {
"index": 1790,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A four-digit number satisfies the following conditions:\n(1) If you simultaneously swap its unit digit with the hundred digit and the ten digit with the thousand digit, the value increases by 5940;\n(2) When divided by 9, the remainder is 8.\nFind the smallest odd four-digit nu... | math | {
"style": "rule",
"ground_truth": "1979"
} | {
"index": 1791,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A four-digit number with digits in the thousands, hundreds, tens, and units places respectively denoted as \\(a, b, c, d\\) is formed by \\(10 \\cdot 23\\). The sum of these digits is 26. The tens digit of the product of \\(b\\) and \\(d\\) equals \\((a+c)\\). Additionally, \\(... | math | {
"style": "rule",
"ground_truth": "1979"
} | {
"index": 1792,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A four-digit palindrome between $1000$ and $10,000$ is chosen at random. What is the probability that it is divisible by $9$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{10}"
} | {
"index": 1793,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A four-digit palindrome is defined as any four-digit natural number that has the same digit in the units place as in the thousands place, and the same digit in the tens place as in the hundreds place. How many pairs of four-digit palindromes exist whose difference is 3674?"
}... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 1794,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A four-digit positive integer is called [i]virtual[/i] if it has the form $\\overline{abab}$, where $a$ and $b$ are digits and $a \\neq 0$. For example 2020, 2121 and 2222 are virtual numbers, while 2002 and 0202 are not. Find all virtual numbers of the form $n^2+1$, for some p... | math | {
"style": "rule",
"ground_truth": "8282"
} | {
"index": 1795,
"solution": "\nTo solve the problem of finding all virtual numbers of the form \\( n^2 + 1 \\), we need to express a virtual number in the required form and establish conditions for \\( n \\).\n\nA virtual number \\(\\overline{abab}\\) can be expressed mathematically as:\n\\[\n101a + 10b + 10a + b ... |
DeepScaleR | [
{
"role": "user",
"content": "A four-dimensional rectangular hyper-box has side lengths $W$, $X$, $Y$, and $Z$. It has \"faces\" (three-dimensional volumes) whose measures are $60$, $80$, $120$, $60$, $80$, $120$ cubic units. What is $W$ + $X$ + $Y$ + $Z$?\n**A)** 200\n**B)** 250\n**C)** 300\n**D)** 318.5\n... | math | {
"style": "rule",
"ground_truth": "318.5"
} | {
"index": 1796,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A four-meter gas pipe has rusted in two places. Determine the probability that all three resulting parts can be used as connections to gas stoves, if according to standards, a stove should not be located closer than 1 meter from the main gas pipe."
}
] | math | {
"style": "rule",
"ground_truth": "1/4"
} | {
"index": 1797,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A four-meter gas pipe has rusted in two places. Determine the probability that all three resulting pieces can be used as connections to gas stoves, given that regulations require a stove to be at least 1 meter away from the main gas pipeline."
}
] | math | {
"style": "rule",
"ground_truth": "1/8"
} | {
"index": 1798,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A freight train travels 1 mile in 1 minute 30 seconds. At this rate, how many miles will the train travel in 1 hour?"
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 1799,
"solution": ""
} |
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