data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Call a positive integer $n$ quixotic if the value of $\\operatorname{lcm}(1,2,3, \\ldots, n) \\cdot\\left(\\frac{1}{1}+\\frac{1}{2}+\\frac{1}{3}+\\ldots+\\frac{1}{n}\\right)$ is divisible by 45 . Compute the tenth smallest quixotic integer."
}
] | math | {
"style": "rule",
"ground_truth": "573"
} | {
"index": 6000,
"solution": "Let $L=\\operatorname{lcm}(1,2,3, \\ldots, n)$, and let $E=L\\left(1+\\frac{1}{2}+\\frac{1}{3}+\\cdots+\\frac{1}{n}\\right)$ denote the expression. In order for $n$ to be quixotic, we need $E \\equiv 0(\\bmod 5)$ and $E \\equiv 0(\\bmod 9)$. We consider these two conditions separately.... |
DeepScaleR | [
{
"role": "user",
"content": "Call a positive integer $n$ weird if $n$ does not divide $(n-2)$!. Determine the number of weird numbers between 2 and 100 inclusive."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 6001,
"solution": "We claim that all the weird numbers are all the prime numbers and 4. Since no numbers between 1 and $p-2$ divide prime $p,(p-2)$! will not be divisible by $p$. We also have $2!=2$ not being a multiple of 4. Now we show that all other numbers are not weird. If $n=p q$ where $p \\neq q$ ... |
DeepScaleR | [
{
"role": "user",
"content": "Call a positive integer 'mild' if its base-3 representation never contains the digit 2. How many values of $n(1 \\leq n \\leq 1000)$ have the property that $n$ and $n^{2}$ are both mild?"
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 6002,
"solution": "7 Such a number, which must consist entirely of 0's and 1's in base 3, can never have more than one 1. Indeed, if $n=3^{a}+3^{b}+$ higher powers where $b>a$, then $n^{2}=3^{2 a}+2 \\cdot 3^{a+b}+$ higher powers which will not be mild. On the other hand, if $n$ does just have one 1 in b... |
DeepScaleR | [
{
"role": "user",
"content": "Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. For example, $1357$, $89$, and $5$ are all uphill integers, but $32$, $1240$, and $466$ are not. How many uphill integers are divisible by $15$?"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 6003,
"solution": "1. **Divisibility by 15**: A number is divisible by 15 if and only if it is divisible by both 3 and 5. \n\n2. **Divisibility by 5**: A number is divisible by 5 if it ends in 0 or 5. Since we are considering positive integers and uphill integers cannot end in 0 (as 0 cannot be preceded ... |
DeepScaleR | [
{
"role": "user",
"content": "Call a positive integer monotonous if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, $3$, $23578$, and $987620$ are monotonous, but $88$, $7434$, and $23557$ are not. How ma... | math | {
"style": "rule",
"ground_truth": "1524"
} | {
"index": 6004,
"solution": "1. **Understanding Monotonous Numbers**: A monotonous number is defined as a number whose digits are either strictly increasing or strictly decreasing when read from left to right. This includes all one-digit numbers.\n\n2. **Counting One-Digit Monotonous Numbers**: There are 9 one-dig... |
DeepScaleR | [
{
"role": "user",
"content": "Call a positive integer strictly monotonous if it is a one-digit number or its digits, read from left to right, form a strictly increasing or a strictly decreasing sequence, and no digits are repeated. Determine the total number of strictly monotonous positive integers."
}
] | math | {
"style": "rule",
"ground_truth": "1013"
} | {
"index": 6005,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Call a positive real number special if it has a decimal representation that consists entirely of digits $0$ and $7$. For example, $\\frac{700}{99}= 7.\\overline{07}= 7.070707\\cdots$ and $77.007$ are special numbers. What is the smallest $n$ such that $1$ can be written as a su... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 6006,
"solution": "1. **Understanding the Problem**: We need to find the smallest number of special numbers (numbers with only the digits $0$ and $7$ in their decimal representation) that can sum up to $1$.\n\n2. **Relating to Super-Special Numbers**: We introduce the concept of super-special numbers, wh... |
DeepScaleR | [
{
"role": "user",
"content": "Call a set $S$ product-free if there do not exist $a, b, c \\in S$ (not necessarily distinct) such that $a b = c$. For example, the empty set and the set $\\{16, 20\\}$ are product-free, whereas the sets $\\{4, 16\\}$ and $\\{2, 8, 16\\}$ are not product-free. Find the number o... | math | {
"style": "rule",
"ground_truth": "252"
} | {
"index": 6007,
"solution": "Let $X$ be a product-free subset, and note that 1 is not in $x$. We consider four cases:\n 1.) both 2 and 3 are not in $X$. Then there are $2^7=128$ possible subsets for this case.\n 2.) 2 is in $X$, but 3 is not. Then 4 in not in $X$, so there are $2^6=64$ subsets; however, there is a... |
DeepScaleR | [
{
"role": "user",
"content": "Call a set of integers \"spacy\" if it contains no more than one out of any three consecutive integers. How many subsets of $\\{1, 2,\n3, \\dots, 12\\}$, including the empty set, are spacy?"
}
] | math | {
"style": "rule",
"ground_truth": "129"
} | {
"index": 6008,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Call a set of integers \"spacy\" if it contains no more than one out of any three consecutive integers. How many subsets of $\\{1, 2, 3, \\dots, 15\\}$, including the empty set, are spacy?"
}
] | math | {
"style": "rule",
"ground_truth": "406"
} | {
"index": 6009,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Call a set of positive integers \"conspiratorial\" if no three of them are pairwise relatively prime. What is the largest number of elements in any \"conspiratorial\" subset of the integers $1$ to $16$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 6010,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Call a set of positive integers good if there is a partition of it into two sets $S$ and $T$, such that there do not exist three elements $a, b, c \\in S$ such that $a^{b}=c$ and such that there do not exist three elements $a, b, c \\in T$ such that $a^{b}=c$ ( $a$ and $b$ need... | math | {
"style": "rule",
"ground_truth": "65536"
} | {
"index": 6011,
"solution": "First, we claim that the set $\\{2,4,8,256,65536\\}$ is not good. Assume the contrary and say $2 \\in S$. Then since $2^{2}=4$, we have $4 \\in T$. And since $4^{4}=256$, we have $256 \\in S$. Then since $256^{2}=65536$, we have $65536 \\in T$. Now, note that we cannot place 8 in eithe... |
DeepScaleR | [
{
"role": "user",
"content": "Call a string of letters $S$ an almost palindrome if $S$ and the reverse of $S$ differ in exactly two places. Find the number of ways to order the letters in $H M M T T H E M E T E A M$ to get an almost palindrome."
}
] | math | {
"style": "rule",
"ground_truth": "2160"
} | {
"index": 6012,
"solution": "Note that $T, E, A$ are used an odd number of times. Therefore, one must go in the middle spot and the other pair must match up. There are are $3 \\cdot 2\\left(\\frac{6!}{2!}\\right)=2160$ ways to fill in the first six spots with the letters $T, H, E, M, M$ and a pair of different let... |
DeepScaleR | [
{
"role": "user",
"content": "Call a three-digit number $\\overline{ABC}$ $\\textit{spicy}$ if it satisfies $\\overline{ABC}=A^3+B^3+C^3$ . Compute the unique $n$ for which both $n$ and $n+1$ are $\\textit{spicy}$ ."
}
] | math | {
"style": "rule",
"ground_truth": "370"
} | {
"index": 6013,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Call a three-term strictly increasing arithmetic sequence of integers special if the sum of the squares of the three terms equals the product of the middle term and the square of the common difference. Find the sum of the third terms of all special sequences."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 6014,
"solution": "Proceed as in solution 2, until we reach \\[3x^2+2d^2=xd^2,\\] Write\n$d^2=\\frac{3x^2}{x-2}$, it follows that $x-2=3k^2$ for some (positive) integer k and $k \\mid x$.\n Taking both sides modulo $k$, $-2 \\equiv 0 \\pmod{k}$, so $k \\mid 2 \\rightarrow k=1,2$.\n When $k=1$, we have $x... |
DeepScaleR | [
{
"role": "user",
"content": "Call a triangle nice if the plane can be tiled using congruent copies of this triangle so that any two triangles that share an edge (or part of an edge) are reflections of each other via the shared edge. How many dissimilar nice triangles are there?"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 6015,
"solution": "The triangles are 60-60-60, 45-45-90, 30-60-90, and 30-30-120. We make two observations. - By reflecting \"around\" the same point, any angle of the triangle must be an integer divisor of $360^{\\circ}$. - if any angle is an odd divisor of $360^{\\circ}$, i.e equals \\frac{360}{k}$ for... |
DeepScaleR | [
{
"role": "user",
"content": "Call an integer $n$ oddly powerful if there exist positive integers $a$ and $b$, where $b>1$, $b$ is odd, and $a^b = n$. How many oddly powerful integers are less than $2010$?"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 6016,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Call an integer $n>1$ radical if $2^{n}-1$ is prime. What is the 20th smallest radical number? If $A$ is your answer, and $S$ is the correct answer, you will get $\\max \\left(25\\left(1-\\frac{|A-S|}{S}\\right), 0\\right)$ points, rounded to the nearest integer."
}
] | math | {
"style": "rule",
"ground_truth": "4423"
} | {
"index": 6017,
"solution": "The answer is 4423."
} |
DeepScaleR | [
{
"role": "user",
"content": "Call an integer \\( n > 1 \\) radical if \\( 2^n - 1 \\) is prime. What is the 20th smallest radical number?"
}
] | math | {
"style": "rule",
"ground_truth": "4423"
} | {
"index": 6018,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Call an ordered pair $(a, b)$ of positive integers fantastic if and only if $a, b \\leq 10^{4}$ and $\\operatorname{gcd}(a \\cdot n!-1, a \\cdot(n+1)!+b)>1$ for infinitely many positive integers $n$. Find the sum of $a+b$ across all fantastic pairs $(a, b)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\[ 5183 \\]"
} | {
"index": 6019,
"solution": "We first prove the following lemma, which will be useful later. Lemma: Let $p$ be a prime and $1 \\leq n \\leq p-1$ be an integer. Then, $n!(p-1-n)!\\equiv(-1)^{n-1}(\\bmod p)$. Proof. Write $$\\begin{aligned} n!(p-n-1)! & =(1 \\cdot 2 \\cdots n)((p-n-1) \\cdots 2 \\cdot 1) \\\\ & \\eq... |
DeepScaleR | [
{
"role": "user",
"content": "Call an ordered triple $(a, b, c)$ of integers feral if $b -a, c - a$ and $c - b$ are all prime. \nFind the number of feral triples where $1 \\le a < b < c \\le 20$ ."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"index": 6020,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Call an positive integer almost-square if it can be written as $a \\cdot b$, where $a$ and $b$ are integers and $a \\leq b \\leq \\frac{4}{3} a$. How many almost-square positive integers are less than or equal to 1000000 ? Your score will be equal to $25-65 \\frac{|A-C|}{\\min ... | math | {
"style": "rule",
"ground_truth": "130348"
} | {
"index": 6021,
"solution": "To get a good estimate for the number of almost-square integers, note that any number of the form $a \\cdot b$, with $b \\leq \\frac{4}{3} a$, will be by definition almost-square. Let's assume that it's relatively unlikely that a number is almost-square in more than one way. Then the n... |
DeepScaleR | [
{
"role": "user",
"content": "Call the pentominoes found in the last problem square pentominoes. Just like dominos and ominos can be used to tile regions of the plane, so can square pentominoes. In particular, a square pentomino tiling of a region of the plane is a way of covering it (and only it) completel... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 6022,
"solution": "Since 5 does not divide 144, there are 0."
} |
DeepScaleR | [
{
"role": "user",
"content": "Calvin has a bag containing 50 red balls, 50 blue balls, and 30 yellow balls. Given that after pulling out 65 balls at random (without replacement), he has pulled out 5 more red balls than blue balls, what is the probability that the next ball he pulls out is red?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9}{26}"
} | {
"index": 6023,
"solution": "Solution 1. The only information this gives us about the number of yellow balls left is that it is even. A bijection shows that the probability that there are $k$ yellow balls left is equal to the probability that there are $30-k$ yellow balls left (flip the colors of the red and blue ... |
DeepScaleR | [
{
"role": "user",
"content": "Cameron writes down the smallest positive multiple of 20 that is a perfect square, the smallest positive multiple of 20 that is a perfect cube, and all the multiples of 20 between them. How many integers are in Cameron's list?"
}
] | math | {
"style": "rule",
"ground_truth": "46"
} | {
"index": 6024,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cameron writes down the smallest positive multiple of 30 that is a perfect square, the smallest positive multiple of 30 that is a perfect cube, and all the multiples of 30 between them. How many integers are in Cameron's list?"
}
] | math | {
"style": "rule",
"ground_truth": "871"
} | {
"index": 6025,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Camilla had three times as many blueberry jelly beans as cherry jelly beans. She also had some raspberry jelly beans, the number of which is not initially given. After eating 15 blueberry and 5 cherry jelly beans, she now has five times as many blueberry jelly beans as cherry j... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 6026,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Camilla had twice as many blueberry jelly beans as cherry jelly beans. After eating 10 pieces of each kind, she now has three times as many blueberry jelly beans as cherry jelly beans. How many blueberry jelly beans did she originally have?"
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"index": 6027,
"solution": "1. **Define Variables:**\n Let $b$ represent the number of blueberry jelly beans Camilla originally had, and $c$ represent the number of cherry jelly beans she originally had.\n\n2. **Set Up Initial Equations:**\n According to the problem, Camilla had twice as many blueberry jelly ... |
DeepScaleR | [
{
"role": "user",
"content": "Camille the snail lives on the surface of a regular dodecahedron. Right now he is on vertex $P_{1}$ of the face with vertices $P_{1}, P_{2}, P_{3}, P_{4}, P_{5}$. This face has a perimeter of 5. Camille wants to get to the point on the dodecahedron farthest away from $P_{1}$. T... | math | {
"style": "rule",
"ground_truth": "\\frac{17+7 \\sqrt{5}}{2}"
} | {
"index": 6028,
"solution": "Consider the net of the dodecahedron. It suffices to look at three pentagons $A B C D E, E D F G H$, and $G F I J K$, where $A J=L$. This can be found by the law of cosines on triangle $A E J$. We have $A E=1$, $E J=\\tan 72^{\\circ}$, and $\\angle A E J=162^{\\circ}$. Thus $L^{2}=1+\\... |
DeepScaleR | [
{
"role": "user",
"content": "Camp Koeller offers exactly three water activities: canoeing, swimming, and fishing. None of the campers is able to do all three of the activities. In total, 15 of the campers go canoeing, 22 go swimming, 12 go fishing, and 9 do not take part in any of these activities. Determi... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"index": 6029,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Can three persons, having one double motorcycle, overcome the distance of $70$ km in $3$ hours? Pedestrian speed is $5$ km / h and motorcycle speed is $50$ km / h."
}
] | math | {
"style": "rule",
"ground_truth": "\\text{No}"
} | {
"index": 6030,
"solution": "\nConsider the scenario where three persons need to cover a distance of 70 km within 3 hours using one double-seater motorcycle. The speeds involved are pedestrian speed of 5 km/h and motorcycle speed of 50 km/h.\n\nTo determine whether this is possible, let us calculate the maximum po... |
DeepScaleR | [
{
"role": "user",
"content": "Can we find \\( N \\) such that all \\( m \\times n \\) rectangles with \\( m, n > N \\) can be tiled with \\( 4 \\times 6 \\) and \\( 5 \\times 7 \\) rectangles?"
}
] | math | {
"style": "rule",
"ground_truth": "840"
} | {
"index": 6031,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Can you multiply 993 and 879 in your head? Interestingly, if we have two two-digit numbers containing the same number of tens, and the sum of the digits of their units place equals 10, then such numbers can always be multiplied mentally as follows:\nSuppose we need to multiply ... | math | {
"style": "rule",
"ground_truth": "872847"
} | {
"index": 6032,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Can you use the four basic arithmetic operations (addition, subtraction, multiplication, division) and parentheses to write the number 2016 using the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 in sequence?"
}
] | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"index": 6033,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Canadian currency has coins with values $2.00, 1.00, 0.25, 0.10,$ and $0.05. Barry has 12 coins including at least one of each of these coins. Find the smallest total amount of money that Barry could have."
}
] | math | {
"style": "rule",
"ground_truth": "3.75"
} | {
"index": 6034,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Candice starts driving home from work at 5:00 PM. Starting at exactly 5:01 PM, and every minute after that, Candice encounters a new speed limit sign and slows down by 1 mph. Candice's speed, in miles per hour, is always a positive integer. Candice drives for \\(2/3\\) of a mil... | math | {
"style": "rule",
"ground_truth": "5:05(PM)"
} | {
"index": 6035,
"solution": "Suppose that Candice starts driving at \\(n\\) miles per hour. Then she slows down and drives \\((n-1)\\) mph, \\((n-2)\\) mph, and so on, with her last speed being \\((m+1)\\) mph. Then the total distance traveled is \\(\\frac{1}{60}\\left(\\frac{n(n+1)}{2}-\\frac{m(m+1)}{2}\\right) =... |
DeepScaleR | [
{
"role": "user",
"content": "Candy sales from the Boosters Club from January through April are shown. What were the average sales per month in dollars?"
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 6036,
"solution": "1. **Identify the total sales**: The problem provides the sales for each month from January through April as $100$, $60$, $40$, and $120$ dollars respectively.\n\n2. **Calculate the total sales**: Add the sales for each month to find the total sales over the four months.\n \\[\n 10... |
DeepScaleR | [
{
"role": "user",
"content": "Captain Billy the Pirate looted 1010 gold doubloons and set sail on his ship to a deserted island to bury his treasure. Each evening of their voyage, he paid each of his pirates one doubloon. On the eighth day of sailing, the pirates plundered a Spanish caravel, doubling Billy'... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 6037,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Captain Zarnin of Planet Hvan has four job openings for his battle station: Assistant Engineer, Weapons Maintenance, Field Technician, and Radio Specialist. Zarnin has received 24 resumes and found himself less than thrilled about half of them and does not hire them. The rest, ... | math | {
"style": "rule",
"ground_truth": "11,\\!880"
} | {
"index": 6038,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Car $A$ departs from Station $J$ towards Station $Y$, while cars $B$ and $C$ depart from Station $Y$ towards Station $J$ simultaneously, and move in opposite directions towards car $A$. Car $A$ meets car $B$ first, then 20 minutes later it meets car $C$. Given the speeds of car... | math | {
"style": "rule",
"ground_truth": "425"
} | {
"index": 6039,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Car A and Car B are traveling from point A to point B. Car A departs 6 hours later than Car B. The speed ratio of Car A to Car B is 4:3. 6 hours after Car A departs, its speed doubles, and both cars arrive at point B simultaneously. How many hours in total did Car A take to tra... | math | {
"style": "rule",
"ground_truth": "8.4"
} | {
"index": 6040,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Car A and Car B are traveling in opposite directions on a road parallel to a railway. A 180-meter-long train is moving in the same direction as Car A at a speed of 60 km/h. The time from when the train catches up with Car A until it meets Car B is 5 minutes. If it takes the tra... | math | {
"style": "rule",
"ground_truth": "1.25"
} | {
"index": 6041,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Car A and Car B start simultaneously from points $A$ and $B$ respectively, traveling towards each other. The initial speed ratio of car A to car B is 5:4. Shortly after departure, car A has a tire blowout, stops to replace the tire, and then resumes the journey, increasing its ... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"index": 6042,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Car A departs from point $A$ heading towards point $B$ and returns; Car B departs from point $B$ at the same time heading towards point $A$ and returns. After the first meeting, Car A continues for 4 hours to reach $B$, and Car B continues for 1 hour to reach $A$. If the distan... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 6043,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Car X is traveling at a constant speed of 90 km/h and has a length of 5 meters, while Car Y is traveling at a constant speed of 91 km/h and has a length of 6 meters. Given that Car Y starts behind Car X and eventually passes Car X, calculate the length of time between the insta... | math | {
"style": "rule",
"ground_truth": "39.6"
} | {
"index": 6044,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cara is in a group photo with her seven friends. If Cara must stand between two of her friends, how many different possible pairs of friends could she be standing between?"
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 6045,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cara is sitting at a circular table with her five friends as shown below. How many different possible pairs of people could Cara be sitting between?\n\n[asy]\ndraw(circle((0,0),1));\nlabel(\"$\\_$\",1.5dir(0));\nlabel(\"$\\_$\",1.5dir(60));\nlabel(\"Cara\",1.5dir(120));\nlabel(... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 6046,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cara is sitting at a circular table with her seven friends. Determine the different possible pairs of people Cara could be sitting between on this table."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 6047,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cara is sitting at a circular table with her seven friends. How many different possible pairs of people could Cara be sitting between?"
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 6048,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cara is sitting at a circular table with her seven friends. Two of her friends, Alice and Bob, insist on sitting together but not next to Cara. How many different possible pairs of people could Cara be sitting between?"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 6049,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cara is sitting at a circular table with her six friends. How many different sets of two friends can Cara be directly sitting between?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 6050,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cara is sitting at a circular table with six friends. Assume there are three males and three females among her friends. How many different possible pairs of people could Cara sit between if each pair must include at least one female friend?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 6051,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cara is sitting at a round table with her eight friends. How many different pairs of friends could Cara be potentially sitting between?"
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 6052,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cards numbered 1 and 2 must be placed into the same envelope, and three cards are left to be placed into two remaining envelopes, find the total number of different methods."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 6053,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Carefully observe the arrangement pattern of the following hollow circles ($○$) and solid circles ($●$): $○●○○●○○○●○○○○●○○○○○●○○○○○○●…$. If this pattern continues, a series of $○$ and $●$ will be obtained. The number of $●$ in the first $100$ of $○$ and $●$ is $\\_\\_\\_\\_\\_\... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 6054,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Carefully observe the following three rows of related numbers:<br/>First row: $-2$, $4$, $-8$, $16$, $-32$, $\\ldots$;<br/>Second row: $0$, $6$, $-6$, $18$, $-30$, $\\ldots$;<br/>Third row: $-1$, $2$, $-4$, $8$, $-16$, $\\ldots$;<br/>Answer the following questions:<br/>$(1)$ Th... | math | {
"style": "rule",
"ground_truth": "256"
} | {
"index": 6055,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Carina has three pins, labeled $A, B$ , and $C$ , respectively, located at the origin of the coordinate plane. In a move, Carina may move a pin to an adjacent lattice point at distance $1$ away. What is the least number of moves that Carina can make in order for triangle $ABC$ ... | math | {
"style": "rule",
"ground_truth": "\\[ 128 \\]"
} | {
"index": 6056,
"solution": "The answer is $128$ , achievable by $A=(10,0), B=(0,-63), C=(-54,1)$ . We now show the bound.\nWe first do the following optimizations:\n-if you have a point goes both left and right, we may obviously delete both of these moves and decrease the number of moves by $2$ .\n-if all of $A,B... |
DeepScaleR | [
{
"role": "user",
"content": "Carina is in a tournament in which no game can end in a tie. She continues to play games until she loses 2 games, at which point she is eliminated and plays no more games. The probability of Carina winning the first game is $\frac{1}{2}$. After she wins a game, the probability ... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 6057,
"solution": "We want to determine the probability that Carina wins 3 games before she loses 2 games. This means that she either wins 3 and loses 0, or wins 3 and loses 1. If Carina wins her first three games, we do not need to consider the case of Carina losing her fourth game, because we can stop ... |
DeepScaleR | [
{
"role": "user",
"content": "Carl and André are running a race. Carl runs at a constant speed of $x \\mathrm{~m} / \\mathrm{s}$. André runs at a constant speed of $y \\mathrm{~m} / \\mathrm{s}$. Carl starts running, and then André starts running 20 s later. After André has been running for 10 s, he catches... | math | {
"style": "rule",
"ground_truth": "3:1"
} | {
"index": 6058,
"solution": "André runs for 10 seconds at a speed of $y \\mathrm{~m} / \\mathrm{s}$. Therefore, André runs $10y \\mathrm{~m}$. Carl runs for 20 seconds before André starts to run and then 10 seconds while André is running. Thus, Carl runs for 30 seconds. Since Carl runs at a speed of $x \\mathrm{~m... |
DeepScaleR | [
{
"role": "user",
"content": "Carl and Bob can demolish a building in 6 days, Anne and Bob can do it in 3, Anne and Carl in 5. How many days does it take all of them working together if Carl gets injured at the end of the first day and can't come back? Express your answer as a fraction in lowest terms."
}... | math | {
"style": "rule",
"ground_truth": "\\frac{59}{20}"
} | {
"index": 6059,
"solution": "Let $a$ be the portion of the work that Anne does in one day, similarly $b$ for Bob and $c$ for Carl. Then what we are given is the system of equations $b+c=1 / 6, a+b=1 / 3$, and $a+c=1 / 5$. Thus in the first day they complete $a+b+c=\\frac{1}{2}(1 / 6+1 / 3+1 / 5)=7 / 20$, leaving $... |
DeepScaleR | [
{
"role": "user",
"content": "Carl and Dina each arrive at a café at a random time between 1:00 PM and 1:45 PM. Each stays for 30 minutes before leaving. What is the probability that Carl and Dina meet each other at the café?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8}{9}"
} | {
"index": 6060,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Carl decided to fence his rectangular flowerbed using 24 fence posts, including one on each corner. He placed the remaining posts spaced exactly 3 yards apart along the perimeter of the bed. The bed’s longer side has three times as many posts compared to the shorter side, inclu... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 6061,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Carl decided to fence in his rectangular garden. He bought $20$ fence posts, placed one on each of the four corners, and spaced out the rest evenly along the edges of the garden, leaving exactly $4$ yards between neighboring posts. The longer side of his garden, including the c... | math | {
"style": "rule",
"ground_truth": "336"
} | {
"index": 6062,
"solution": "1. **Determine the number of posts on each side**: \n Let $x$ be the number of posts on each of the shorter sides, including the corners. Since the longer side has twice as many posts as the shorter side, it has $2x$ posts, including the corners.\n\n2. **Account for the total number ... |
DeepScaleR | [
{
"role": "user",
"content": "Carl drove continuously from 7:30 a.m. until 2:15 p.m. of the same day and covered a distance of 234 miles. What was his average speed in miles per hour?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{936}{27}"
} | {
"index": 6063,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Carl has 5 cubes each having side length 1, and Kate has 5 cubes each having side length 2. What is the total volume of these 10 cubes?"
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 6064,
"solution": "1. **Calculate the volume of one of Carl's cubes**: \n The volume $V$ of a cube with side length $s$ is given by the formula:\n \\[\n V = s^3\n \\]\n For Carl's cubes, each has a side length of $1$. Therefore, the volume of one cube is:\n \\[\n V = 1^3 = 1\n \\]\n\n2. *... |
DeepScaleR | [
{
"role": "user",
"content": "Carl is on a vertex of a regular pentagon. Every minute, he randomly selects an adjacent vertex (each with probability $\\frac{1}{2}$ ) and walks along the edge to it. What is the probability that after 10 minutes, he ends up where he had started?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{127}{512}"
} | {
"index": 6065,
"solution": "Let A denote a clockwise move and B denote a counterclockwise move. We want to have some combination of 10 A's and B's, with the number of A's and the number of B's differing by a multiple of 5. We have $\\binom{10}{0}+\\binom{10}{5}+\\binom{10}{10}=254$. Hence the answer is $\\frac{25... |
DeepScaleR | [
{
"role": "user",
"content": "Carl only eats food in the shape of equilateral pentagons. Unfortunately, for dinner he receives a piece of steak in the shape of an equilateral triangle. So that he can eat it, he cuts off two corners with straight cuts to form an equilateral pentagon. The set of possible peri... | math | {
"style": "rule",
"ground_truth": "4 \\sqrt{3}-6"
} | {
"index": 6066,
"solution": "Assume that the triangle has side length 1. We will show the pentagon side length $x$ is in $\\left[2 \\sqrt{3}-3, \\frac{1}{2}\\right)$. Call the triangle $A B C$ and let corners $B, C$ be cut. Choose $P$ on $A B, Q, R$ on $B C$, and $S$ on $A C$ such that $A P Q R S$ is equilateral. ... |
DeepScaleR | [
{
"role": "user",
"content": "Carl wrote a list of 10 distinct positive integers on a board. Each integer in the list, apart from the first, is a multiple of the previous integer. The last of the 10 integers is between 600 and 1000. What is this last integer?"
}
] | math | {
"style": "rule",
"ground_truth": "768"
} | {
"index": 6067,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Carla rotated point $A$ 420 degrees clockwise about point $B$ and it landed at point $C$. Devon rotated the original point $A$ $x$ degrees counterclockwise about point $B$ and it also landed at point $C$. If $x<360$, what is the value of $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "300^{\\circ}"
} | {
"index": 6068,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Carlos took $70\\%$ of a whole pie. Maria took one third of the remainder. What portion of the whole pie was left?"
}
] | math | {
"style": "rule",
"ground_truth": "20\\%"
} | {
"index": 6069,
"solution": "1. **Calculate the remaining pie after Carlos's share:**\n Carlos took $70\\%$ of the pie. Therefore, the remaining part of the pie is:\n \\[\n 100\\% - 70\\% = 30\\%\n \\]\n\n2. **Determine Maria's share of the remaining pie:**\n Maria took one third of the remaining $30\\%$... |
DeepScaleR | [
{
"role": "user",
"content": "Carmen takes a long bike ride on a hilly highway. The graph indicates the miles traveled during the time of her ride. What is Carmen's average speed for her entire ride in miles per hour?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 6070,
"solution": "To find Carmen's average speed for her entire ride, we need to use the formula for average speed, which is given by:\n\\[\n\\text{Average Speed} = \\frac{\\text{Total Distance}}{\\text{Total Time}}\n\\]\n\nFrom the problem statement, we are given:\n- Total Distance = 35 miles\n- Total ... |
DeepScaleR | [
{
"role": "user",
"content": "Caroline can make eleven lassis out of two mangoes. How many lassis can she make out of twelve mangoes?"
}
] | math | {
"style": "rule",
"ground_truth": "66"
} | {
"index": 6071,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Caroline starts with the number 1, and every second she flips a fair coin; if it lands heads, she adds 1 to her number, and if it lands tails she multiplies her number by 2. Compute the expected number of seconds it takes for her number to become a multiple of 2021."
}
] | math | {
"style": "rule",
"ground_truth": "4040"
} | {
"index": 6072,
"solution": "Consider this as a Markov chain on $\\mathbb{Z} / 2021 \\mathbb{Z}$. This Markov chain is aperiodic (since 0 can go to 0) and any number can be reached from any other number (by adding 1), so it has a unique stationary distribution $\\pi$, which is uniform (since the uniform distributi... |
DeepScaleR | [
{
"role": "user",
"content": "Carolyn and Paul are playing a game starting with a list of the integers $1$ to $n.$ The rules of the game are:\n\n$\\bullet$ Carolyn always has the first turn.\n\n$\\bullet$ Carolyn and Paul alternate turns.\n\n$\\bullet$ On each of her turns, Carolyn must remove one number... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 6073,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Carolyn and Paul are playing a game starting with a list of the integers $1$ to $n.$ The rules of the game are:\n\n$\\bullet$ Carolyn always has the first turn.\n\n$\\bullet$ Carolyn and Paul alternate turns.\n\n$\\bullet$ On each of her turns, Carolyn must remove one number... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 6074,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Carrie has a rectangular garden that measures $6$ feet by $8$ feet. She plants the entire garden with strawberry plants. Carrie is able to plant $4$ strawberry plants per square foot, and she harvests an average of $10$ strawberries per plant. How many strawberries can she expe... | math | {
"style": "rule",
"ground_truth": "1920"
} | {
"index": 6075,
"solution": "1. **Calculate the area of the garden**: \n The garden is a rectangle with dimensions $6$ feet by $8$ feet. The area \\( A \\) of a rectangle is given by the formula:\n \\[\n A = \\text{length} \\times \\text{width}\n \\]\n Substituting the given dimensions:\n \\[\n A = ... |
DeepScaleR | [
{
"role": "user",
"content": "Carrie sends five text messages to her brother each Saturday and Sunday, and two messages on other days. Over four weeks, how many text messages does Carrie send?"
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 6076,
"solution": "Each week, Carrie sends 5 messages to her brother on each of 2 days, for a total of 10 messages. Each week, Carrie sends 2 messages to her brother on each of the remaining 5 days, for a total of 10 messages. Therefore, Carrie sends $10+10=20$ messages per week. In four weeks, Carrie se... |
DeepScaleR | [
{
"role": "user",
"content": "Cars A and B simultaneously depart from locations $A$ and $B$, and travel uniformly towards each other. When car A has traveled 12 km past the midpoint of $A$ and $B$, the two cars meet. If car A departs 10 minutes later than car B, they meet exactly at the midpoint of $A$ and ... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 6077,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cars A and B travel the same distance. Car A travels half that distance at $u$ miles per hour and half at $v$ miles per hour. Car B travels half the time at $u$ miles per hour and half at $v$ miles per hour. The average speed of Car A is $x$ miles per hour and that of Car B is ... | math | {
"style": "rule",
"ground_truth": "$x \\leq y$"
} | {
"index": 6078,
"solution": "1. **Define the average speed for Car A**:\n Car A travels half the distance at speed $u$ mph and the other half at $v$ mph. Let the total distance be $D$. Then each half is $\\frac{D}{2}$. The time taken to travel the first half at speed $u$ is $\\frac{\\frac{D}{2}}{u} = \\frac{D}{2... |
DeepScaleR | [
{
"role": "user",
"content": "Cassandra sets her watch to the correct time at noon. At the actual time of 1:00 PM, she notices that her watch reads 12:57 and 36 seconds. Assuming that her watch loses time at a constant rate, what will be the actual time when her watch first reads 10:00 PM?"
}
] | math | {
"style": "rule",
"ground_truth": "10:25 PM"
} | {
"index": 6079,
"solution": "1. **Identify the rate of time loss:** Cassandra's watch loses time such that in 1 hour of actual time, her watch shows only 57 minutes and 36 seconds. We convert 36 seconds to minutes: \n \\[\n 36 \\text{ seconds} = \\frac{36}{60} \\text{ minutes} = 0.6 \\text{ minutes}\n \\]\n ... |
DeepScaleR | [
{
"role": "user",
"content": "Cassie leaves Escanaba at 8:30 AM heading for Marquette on her bike. She bikes at a uniform rate of 12 miles per hour. Brian leaves Marquette at 9:00 AM heading for Escanaba on his bike. He bikes at a uniform rate of 16 miles per hour. They both bike on the same 62-mile route b... | math | {
"style": "rule",
"ground_truth": "11:00"
} | {
"index": 6080,
"solution": "1. **Define Variables:**\n Let $x$ be the number of hours after Cassie leaves Escanaba when both of them meet. Cassie leaves at 8:30 AM, so if they meet $x$ hours later, the time will be $8:30$ AM + $x$ hours.\n\n2. **Calculate Distance Traveled by Cassie:**\n Cassie travels at a r... |
DeepScaleR | [
{
"role": "user",
"content": "Cat and Claire are having a conversation about Cat's favorite number.\n \nCat says, \"My favorite number is a two-digit positive integer with distinct nonzero digits, $\\overline{AB}$ , such that $A$ and $B$ are both factors of $\\overline{AB}$ .\"\n \nClaire says, ... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 6081,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cat and Claire are having a conversation about Cat's favorite number. Cat says, \"My favorite number is a two-digit positive prime integer whose first digit is less than its second, and when you reverse its digits, it's still a prime number!\" \n\nClaire asks, \"If you picked a... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 6082,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cat and Claire are having a conversation about Cat’s favorite number. Cat says, “My favorite number is a two-digit perfect square!”\nClaire asks, “If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there an... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 6083,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Cedric has deposited $\\$12,\\!000$ into an account that pays $5\\%$ interest compounded annually.\n\nDaniel has deposited $\\$12,\\!000$ into an account that pays $7\\%$ simple annual interest.\n\nIn $15$ years Cedric and Daniel compare their respective balances. To the neares... | math | {
"style": "rule",
"ground_truth": "\\$347"
} | {
"index": 6084,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Celine has 240 feet of fencing. She needs to enclose a rectangular area such that the area is eight times the perimeter of the rectangle. If she uses up all her fencing material, how many feet is the largest side of the enclosure?"
}
] | math | {
"style": "rule",
"ground_truth": "101"
} | {
"index": 6085,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Centered at each lattice point in the coordinate plane are a circle radius $\\frac{1}{10}$ and a square with sides of length $\\frac{1}{5}$ whose sides are parallel to the coordinate axes. The line segment from $(0,0)$ to $(1001, 429)$ intersects $m$ of the squares and $n$ of t... | math | {
"style": "rule",
"ground_truth": "574"
} | {
"index": 6086,
"solution": "This is mostly a clarification to Solution 1, but let's take the diagram for the origin to $(7,3)$. We have the origin circle and square intersected, then two squares, then the circle and square at $(7,3)$. If we take the circle and square at the origin out of the diagram, we will be a... |
DeepScaleR | [
{
"role": "user",
"content": "Central High School is competing against Northern High School in a backgammon match. Each school has three players, and the contest rules require that each player play two games against each of the other school's players. The match takes place in six rounds, with three games pl... | math | {
"style": "rule",
"ground_truth": "900"
} | {
"index": 6087,
"solution": "Let us label the players of the first team $A$, $B$, and $C$, and those of the second team, $X$, $Y$, and $Z$. \n\n#### Step 1: Understand the game requirements\nEach player from one team must play two games against each player from the other team. This results in $3 \\times 3 \\times ... |
DeepScaleR | [
{
"role": "user",
"content": "Certain integers, when divided by $\\frac{3}{5}, \\frac{5}{7}, \\frac{7}{9}, \\frac{9}{11}$, result in a mixed number where the fractional part is $\\frac{2}{3}, \\frac{2}{5}, \\frac{2}{7}, \\frac{2}{9}$, respectively. Find the smallest integer greater than 1 that satisfies the... | math | {
"style": "rule",
"ground_truth": "316"
} | {
"index": 6088,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Chandler wants to buy a $500$ dollar mountain bike. For his birthday, his grandparents send him $50$ dollars, his aunt sends him $35$ dollars and his cousin gives him $15$ dollars. He earns $16$ dollars per week for his paper route. He will use all of his birthday money and all... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 6089,
"solution": "1. **Identify the total amount of money Chandler needs:** Chandler wants to buy a mountain bike that costs $500.\n\n2. **Calculate the total amount of birthday money Chandler received:**\n - From his grandparents: $50$\n - From his aunt: $35$\n - From his cousin: $15$\n - Total... |
DeepScaleR | [
{
"role": "user",
"content": "Chandra has four bowls. Each one is a different color (red, blue, yellow, green). She also has exactly one glass the same color as each bowl. If she chooses a bowl and a glass from the cupboard, how many pairings are possible? One such pairing is a blue bowl and a yellow gl... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 6090,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Chandra now has five bowls and five glasses, and each expands to a new set of colors: red, blue, yellow, green, and purple. However, she dislikes pairing the same colors; thus, a bowl and glass of the same color cannot be paired together like a red bowl with a red glass. How ma... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"index": 6091,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Chandra pays an on-line service provider a fixed monthly fee plus an hourly charge for connect time. Her December bill was $12.48, but in January her bill was $17.54 because she used twice as much connect time as in December. What is the fixed monthly fee?"
}
] | math | {
"style": "rule",
"ground_truth": "6.24"
} | {
"index": 6092,
"solution": "Let's denote:\n- $x$ as the fixed monthly fee.\n- $y$ as the hourly charge for the connect time used in December.\n\nFrom the problem statement, we have the following equations:\n1. \\( x + y = 12.48 \\) (December bill)\n2. \\( x + 2y = 17.54 \\) (January bill, with twice the connect t... |
DeepScaleR | [
{
"role": "user",
"content": "Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. Chantal starts walking at $4$ miles per hour. Halfway to the tower, the trail becomes really steep, and Chantal slows down to $2$ miles per hour. After reachin... | math | {
"style": "rule",
"ground_truth": "\\frac{12}{13}"
} | {
"index": 6093,
"solution": "Let's analyze the problem step by step:\n\n1. **Define the problem in terms of distance and speed:**\n Let the total distance from the trailhead to the fire tower be $2d$. Chantal walks the first half of the trail, a distance of $d$, at a speed of $4$ miles per hour. The second half ... |
DeepScaleR | [
{
"role": "user",
"content": "Charles has $5q + 1$ quarters and Richard has $q + 5$ quarters. The difference in their money in dimes is:"
}
] | math | {
"style": "rule",
"ground_truth": "10(q - 1)"
} | {
"index": 6094,
"solution": "1. **Identify the number of quarters each person has:**\n - Charles has $5q + 1$ quarters.\n - Richard has $q + 5$ quarters.\n\n2. **Calculate the difference in the number of quarters between Charles and Richard:**\n \\[\n (5q + 1) - (q + 5) = 5q + 1 - q - 5 = 4q - 4\n \\]\n ... |
DeepScaleR | [
{
"role": "user",
"content": "Charles has two six-sided die. One of the die is fair, and the other die is biased so that it comes up six with probability $\\frac{2}{3}$ and each of the other five sides has probability $\\frac{1}{15}$. Charles chooses one of the two dice at random and rolls it three times. G... | math | {
"style": "rule",
"ground_truth": "167"
} | {
"index": 6095,
"solution": "The probability that he rolls a six twice when using the fair die is $\\frac{1}{6}\\times \\frac{1}{6}=\\frac{1}{36}$. The probability that he rolls a six twice using the biased die is $\\frac{2}{3}\\times \\frac{2}{3}=\\frac{4}{9}=\\frac{16}{36}$. Given that Charles rolled two sixes, ... |
DeepScaleR | [
{
"role": "user",
"content": "Charlie and Daisy each arrive at a cafe at a random time between 1:00 PM and 3:00 PM. Each stays for 20 minutes. What is the probability that Charlie and Daisy are at the cafe at the same time?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"index": 6096,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Charlie and Dana play a game where they take turns rolling a standard die. If a player rolls $n$, she is awarded $g(n)$ points, where \n\\[g(n) = \\left\\{\n\\begin{array}{cl} \n7 & \\text{if } n \\text{ is a multiple of 3 and 5}, \\\\\n3 & \\text{if } n \\text{ is only a mult... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 6097,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Charlie folds an $\\frac{17}{2}$-inch by 11-inch piece of paper in half twice, each time along a straight line parallel to one of the paper's edges. What is the smallest possible perimeter of the piece after two such folds?"
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{39}{2}$"
} | {
"index": 6098,
"solution": "Note that a piece of paper is folded in half, one pair of opposite sides is preserved and the other pair is halved. Hence, the net effect on the perimeter is to decrease it by one of the side lengths. Hence, the original perimeter is $2\\left(\\frac{17}{2}\\right)+2 \\cdot 11=39$ and b... |
DeepScaleR | [
{
"role": "user",
"content": "Charlie is making a necklace with yellow beads and green beads. She has already used 4 green beads and 0 yellow beads. How many yellow beads will she have to add so that $\frac{4}{5}$ of the total number of beads are yellow?"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 6099,
"solution": "If $\frac{4}{5}$ of the beads are yellow, then $\frac{1}{5}$ are green. Since there are 4 green beads, the total number of beads must be $4 \times 5=20$. Thus, Charlie needs to add $20-4=16$ yellow beads."
} |
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