data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "A mouse is sitting in a toy car on a negligibly small turntable. The car cannot turn on its own, but the mouse can control when the car is launched and when the car stops (the car has brakes). When the mouse chooses to launch, the car will immediately leave the turntable on a s... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{6}"
} | {
"index": 2300,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A moving particle starts at the point $(4,4)$ and moves until it hits one of the coordinate axes for the first time. When the particle is at the point $(a,b)$, it moves at random to one of the points $(a-1,b)$, $(a,b-1)$, or $(a-1,b-1)$, each with probability $\\frac{1}{3}$, in... | math | {
"style": "rule",
"ground_truth": "252"
} | {
"index": 2301,
"solution": "One could recursively compute the probabilities of reaching $(0,0)$ as the first axes point from any point $(x,y)$ as \\[P(x,y) = \\frac{1}{3} P(x-1,y) + \\frac{1}{3} P(x,y-1) + \\frac{1}{3} P(x-1,y-1)\\] for $x,y \\geq 1,$ and the base cases are $P(0,0) = 1, P(x,0) = P(y,0) = 0$ for a... |
DeepScaleR | [
{
"role": "user",
"content": "A moving point $A$ is on the circle $C$: $(x-1)^{2}+y^{2}=1$, and a moving point $B$ is on the line $l:x+y-4=0$. The coordinates of the fixed point $P$ are $P(-2,2)$. The minimum value of $|PB|+|AB|$ is ______."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{37}-1"
} | {
"index": 2302,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A multiple choice examination consists of $20$ questions. The scoring is $+5$ for each correct answer, $-2$ for each incorrect answer, and $0$ for each unanswered question. John's score on the examination is $48$. What is the maximum number of questions he could have answere... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 2303,
"solution": "Let $c$ be the number of questions John answered correctly, $w$ be the number of questions he answered incorrectly, and $b$ be the number of questions he left blank. We know from the problem statement that:\n1. The total number of questions is 20:\n \\[\n c + w + b = 20\n \\]\n2.... |
DeepScaleR | [
{
"role": "user",
"content": "A museum is organizing a quiz where participants must guess the year a certain artifact was created. Clues given: the year uses each of the digits: 1, 2, 2, 5, 5, 9 exactly once, and the year must start with a prime digit.\n\nHow many different possible years could be guessed b... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"index": 2304,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A national team needs to select 4 out of 6 sprinters to participate in the 4×100 m relay at the Asian Games. If sprinter A cannot run the first leg and sprinter B cannot run the fourth leg, there are a total of ______ ways to participate."
}
] | math | {
"style": "rule",
"ground_truth": "252"
} | {
"index": 2305,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A national team needs to select 4 out of 6 sprinters to participate in the 4×100m relay at the Asian Games. If one of them, A, cannot run the first leg, and another, B, cannot run the fourth leg, how many different methods are there to select the team?"
}
] | math | {
"style": "rule",
"ground_truth": "252"
} | {
"index": 2306,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A natural number $k > 1$ is called *good* if there exist natural numbers $$ a_1 < a_2 < \\cdots < a_k $$ such that $$ \\dfrac{1}{\\sqrt{a_1}} + \\dfrac{1}{\\sqrt{a_2}} + \\cdots + \\dfrac{1}{\\sqrt{a_k}} = 1 $$ .\n\nLet $f(n)$ be the sum of the first $n$ *[good* numb... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 2307,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A natural number $n$ is said to be $good$ if $n$ is the sum or $r$ consecutive positive integers, for some $r \\geq 2 $ . Find the number of good numbers in the set $\\{1,2 \\dots , 100\\}$ ."
}
] | math | {
"style": "rule",
"ground_truth": "93"
} | {
"index": 2308,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A natural number $N$ greater than 20 is a palindromic number in both base 14 and base 20 (A palindromic number is a number that reads the same forwards and backwards, such as 12321 and 3443, but not 12331). What is the smallest value of $N$ (expressed in decimal)?"
}
] | math | {
"style": "rule",
"ground_truth": "105"
} | {
"index": 2309,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A natural number $n$ is called a \"good number\" if the column addition of $n$, $n+1$, and $n+2$ does not produce any carry-over. For example, 32 is a \"good number\" because $32+33+34$ does not result in a carry-over; however, 23 is not a \"good number\" because $23+24+25$ doe... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 2310,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A natural number \\( 1 \\leq n \\leq 221 \\) is called lucky if, when dividing 221 by \\( n \\), the remainder is wholly divisible by the incomplete quotient (the remainder can be equal to 0). How many lucky numbers are there?"
}
] | math | {
"style": "rule",
"ground_truth": "115"
} | {
"index": 2311,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A natural number \\( A \\) is written twice consecutively (for example, writing 12 as 1212) to obtain a new number \\( B \\). If \\( B \\) is a multiple of 2016, then the smallest value of \\( A \\) is ______."
}
] | math | {
"style": "rule",
"ground_truth": "288"
} | {
"index": 2312,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A natural number \\( N \\) ends with the digit 5. A ninth-grader Dima found all its divisors and discovered that the sum of the two largest proper divisors is not divisible by the sum of the two smallest proper divisors. Find the smallest possible value of the number \\( N \\).... | math | {
"style": "rule",
"ground_truth": "725"
} | {
"index": 2313,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A natural number \\( N \\) greater than 20 is a palindrome in both base 14 and base 20 (a palindrome is a number that reads the same forward and backward, such as \\( 12321 \\) and \\( 3443 \\), but \\( 12331 \\) is not a palindrome). What is the smallest value of \\( N \\) (ex... | math | {
"style": "rule",
"ground_truth": "105"
} | {
"index": 2314,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A natural number \\( n \\) is called \"cubowat\" if \\( n^{3} + 13n - 273 \\) is a cube of a natural number. Find the sum of all cubowat numbers."
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 2315,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A natural number \\( x \\) in a base \\( r \\) system (\\( r \\leq 36 \\)) is represented as \\( \\overline{ppqq} \\), where \\( 2q = 5p \\). It turns out that the base-\\( r \\) representation of \\( x^2 \\) is a seven-digit palindrome with a middle digit of zero. (A palindrom... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 2316,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A natural number greater than 1 is called \"good\" if it is equal to the product of its distinct proper divisors (excluding 1 and the number itself). Find the sum of the first ten \"good\" natural numbers."
}
] | math | {
"style": "rule",
"ground_truth": "182"
} | {
"index": 2317,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A natural number greater than 1 is called \"good\" if it is exactly equal to the product of its distinct proper divisors (excluding 1 and the number itself). Find the sum of the first ten \"good\" natural numbers."
}
] | math | {
"style": "rule",
"ground_truth": "182"
} | {
"index": 2318,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A natural number greater than 1 is defined as nice if it is equal to the product of its distinct proper divisors. A number \\( n \\) is nice if:\n1. \\( n = pq \\), where \\( p \\) and \\( q \\) are distinct prime numbers.\n2. \\( n = p^3 \\), where \\( p \\) is a prime number.... | math | {
"style": "rule",
"ground_truth": "182"
} | {
"index": 2319,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A natural number is abundant if it is less than the sum of its proper divisors. What is the smallest abundant number?"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 2320,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A natural number is called a square if it can be written as the product of two identical numbers. For example, 9 is a square because \\(9 = 3 \\times 3\\). The first squares are 1, 4, 9, 16, 25, ... A natural number is called a cube if it can be written as the product of three ... | math | {
"style": "rule",
"ground_truth": "2067"
} | {
"index": 2321,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A natural number is called lucky if all its digits are equal to 7. For example, 7 and 7777 are lucky, but 767 is not. João wrote down the first twenty lucky numbers starting from 7, and then added them. What is the remainder of that sum when divided by 1000?"
}
] | math | {
"style": "rule",
"ground_truth": "70"
} | {
"index": 2322,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A natural number is equal to the cube of the number of its thousands. Find all such numbers."
}
] | math | {
"style": "rule",
"ground_truth": "32768"
} | {
"index": 2323,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A natural number is written on the board. If its last digit (in the units place) is erased, the remaining non-zero number is divisible by 20. If the first digit is erased, the remaining number is divisible by 21. What is the smallest number that could be on the board if its sec... | math | {
"style": "rule",
"ground_truth": "1609"
} | {
"index": 2324,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A natural number of five digits is called *Ecuadorian*if it satisfies the following conditions: $\\bullet$ All its digits are different. $\\bullet$ The digit on the far left is equal to the sum of the other four digits. Example: $91350$ is an Ecuadorian number since $9 = ... | math | {
"style": "rule",
"ground_truth": "168"
} | {
"index": 2325,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A natural number plus 13 is a multiple of 5, and its difference with 13 is a multiple of 6. What is the smallest natural number that satisfies these conditions?"
}
] | math | {
"style": "rule",
"ground_truth": "37"
} | {
"index": 2326,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A natural number undergoes the following operation: the rightmost digit of its decimal representation is discarded, and then the number obtained after discarding is added to twice the discarded digit. For example, $157 \\mapsto 15 + 2 \\times 7 = 29$, $5 \\mapsto 0 + 2 \\times ... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 2327,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A natural number's proper divisors are defined as positive integer divisors other than 1 and the number itself. A natural number greater than 1 is called \"good\" if it is equal to the product of all its distinct proper divisors. What is the sum of the first 10 \"good\" natural... | math | {
"style": "rule",
"ground_truth": "182"
} | {
"index": 2328,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A natural number, when raised to the sixth power, has digits which, when arranged in ascending order, are:\n\n$$\n0,2,3,4,4,7,8,8,9\n$$\n\nWhat is this number?"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 2329,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A necklace consists of 100 beads of red, blue, and green colors. It is known that among any five consecutive beads, there is at least one blue bead, and among any seven consecutive beads, there is at least one red bead. What is the maximum number of green beads that can be in t... | math | {
"style": "rule",
"ground_truth": "65"
} | {
"index": 2330,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A necklace consists of 50 blue beads and some quantity of red beads. It is known that on any segment of the necklace containing 8 blue beads, there are at least 4 red beads. What is the minimum number of red beads that can be in this necklace? (The beads in the necklace are arr... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 2331,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A necklace consists of 80 beads of red, blue, and green colors. It is known that in any segment of the necklace between two blue beads, there is at least one red bead, and in any segment of the necklace between two red beads, there is at least one green bead. What is the minimu... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 2332,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A necklace has a total of 99 beads. Among them, the first bead is white, the 2nd and 3rd beads are red, the 4th bead is white, the 5th, 6th, 7th, and 8th beads are red, the 9th bead is white, and so on. Determine the total number of red beads on this necklace."
}
] | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 2333,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A necklace is strung with gems in the order of A, B, C, D, E, F, G, H. Now, we want to select 8 gems from it in two rounds, with the requirement that only 4 gems can be taken each time, and at most two adjacent gems can be taken (such as A, B, E, F). How many ways are there to ... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 2334,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A neighbor bought a certain quantity of beef at two shillings a pound, and the same quantity of sausages at eighteenpence a pound. If she had divided the same money equally between beef and sausages, she would have gained two pounds in the total weight. Determine the exact amou... | math | {
"style": "rule",
"ground_truth": "168"
} | {
"index": 2335,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A net for hexagonal pyramid is constructed by placing a triangle with side lengths $x$ , $x$ , and $y$ on each side of a regular hexagon with side length $y$ . What is the maximum volume of the pyramid formed by the net if $x+y=20$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "128\\sqrt{15}"
} | {
"index": 2336,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A new city is constructed in the form of 12 rectangular blocks of houses, divided by streets; at each corner of the block and at every street intersection, there is a mailbox. The blocks either touch each other along entire sides or share only a single corner.\n\nAssuming that ... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"index": 2337,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A new definition: $\\left[a,b,c\\right]$ represents the \"graph number\" of a quadratic function $y=ax^{2}+bx+c$ (where $a\\neq 0$, and $a$, $b$, $c$ are real numbers). For example, the \"graph number\" of $y=-x^{2}+2x+3$ is $\\left[-1,2,3\\right]$.\n\n$(1)$ The \"graph number\... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 2338,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A new dump truck delivered sand to a construction site, forming a conical pile with a diameter of $12$ feet. The height of the cone was $50\\%$ of its diameter. However, the pile was too large, causing some sand to spill, forming a cylindrical layer directly around the base of ... | math | {
"style": "rule",
"ground_truth": "98\\pi"
} | {
"index": 2339,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A new model car travels 4.2 kilometers more per liter of gasoline than an old model car. Additionally, the fuel consumption for the new model is 2 liters less per 100 km. How many liters of gasoline per 100 km does the new car consume? Round your answer to the nearest hundredth... | math | {
"style": "rule",
"ground_truth": "5.97"
} | {
"index": 2340,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A new model car travels 4.4 kilometers more per liter of gasoline than an old model car. Additionally, the fuel consumption per 100 km for the new model is 2 liters less than that of the old model. How many liters of gasoline does the new model car consume per 100 km? Round the... | math | {
"style": "rule",
"ground_truth": "5.82"
} | {
"index": 2341,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A new pyramid is added on one of the pentagonal faces of a pentagonal prism. Calculate the total number of exterior faces, vertices, and edges of the composite shape formed by the fusion of the pentagonal prism and the pyramid. What is the maximum value of this sum?"
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 2342,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A new solid $T$ is defined by the set of all points $(x, y, z)$ in space such that $|x| + |y| + |z| \\leq 2$. Find the volume of solid $T$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{32}{3}"
} | {
"index": 2343,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A new train goes $20\\%$ farther than an older train in the same amount of time. During the time it takes the older train to go 200 miles, how many miles can the newer train complete?"
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 2344,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A new website registered $2000$ people. Each of them invited $1000$ other registered people to be their friends. Two people are considered to be friends if and only if they have invited each other. What is the minimum number of pairs of friends on this website?"
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"index": 2345,
"solution": "\nConsider a new website with \\(2000\\) registered people. Each person invites \\(1000\\) other people from the group to be their friends. According to the rules, two people are actually considered friends if and only if they have invited each other.\n\nWe need to determine the minimu... |
DeepScaleR | [
{
"role": "user",
"content": "A new window design consists of a rectangle topped with a semi-circle at both ends. The ratio of the length AD of the rectangle to its width AB is 4:3. If AB is 36 inches, calculate the ratio of the area of the rectangle to the combined area of the semicircles."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{16}{3\\pi}"
} | {
"index": 2346,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A newly designed car travels 4.2 kilometers further per liter of gasoline than an older model. The fuel consumption for the new car is 2 liters less per 100 kilometers. How many liters of gasoline does the new car consume per 100 kilometers? If necessary, round your answer to t... | math | {
"style": "rule",
"ground_truth": "5.97"
} | {
"index": 2347,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A nickel is placed on a table. The number of nickels which can be placed around it, each tangent to it and to two others is:"
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 2348,
"solution": "To solve this problem, we need to determine how many nickels can be placed around a central nickel such that each is tangent to the central nickel and to its two neighboring nickels.\n\n1. **Understanding the Problem:**\n - Each nickel is a circle, and we are asked to place other nic... |
DeepScaleR | [
{
"role": "user",
"content": "A nine-digit integer is formed by repeating a positive three-digit integer three times. For example, 123,123,123 or 307,307,307 are integers of this form. What is the greatest common divisor of all nine-digit integers of this form?"
}
] | math | {
"style": "rule",
"ground_truth": "1001001"
} | {
"index": 2349,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A nine-digit integer is formed by repeating a positive three-digit integer three times. For example, 123,123,123 or 456,456,456 are integers of this form. What is the greatest common divisor of all nine-digit integers of this form?"
}
] | math | {
"style": "rule",
"ground_truth": "1001001"
} | {
"index": 2350,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A nine-digit integer is formed by repeating a three-digit integer three times. For example, 123,123,123 or 456,456,456 are integers of this form. What is the greatest common divisor of all nine-digit integers of this form?"
}
] | math | {
"style": "rule",
"ground_truth": "1001001"
} | {
"index": 2351,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A nine-digit number is formed by repeating a three-digit number three times. For example, 123,123,123 or 456,456,456 are numbers of this form. What is the greatest common divisor of all nine-digit numbers of this form?"
}
] | math | {
"style": "rule",
"ground_truth": "1001001"
} | {
"index": 2352,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A nine-digit number is formed by repeating a three-digit number three times; for example, $256256256$. Determine the common factor that divides any number of this form exactly."
}
] | math | {
"style": "rule",
"ground_truth": "1001001"
} | {
"index": 2353,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A nine-joint bamboo tube has rice capacities of 4.5 *Sheng* in the lower three joints and 3.8 *Sheng* in the upper four joints. Find the capacity of the middle two joints."
}
] | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"index": 2354,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A non-increasing sequence of 100 non-negative reals has the sum of the first two terms at most 100 and the sum of the remaining terms at most 100. What is the largest possible value for the sum of the squares of the terms?"
}
] | math | {
"style": "rule",
"ground_truth": "10000"
} | {
"index": 2355,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A non-square rectangle has integer dimensions. The number of square units in its area is numerically equal to the number of units in its perimeter. What is the number of units in the perimeter of this rectangle?"
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 2356,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A non-zero digit is chosen in such a way that the probability of choosing digit $d$ is $\\log_{10}{(d+1)}-\\log_{10}{d}$. The probability that the digit $2$ is chosen is exactly $\\frac{1}{2}$ the probability that the digit chosen is in the set"
}
] | math | {
"style": "rule",
"ground_truth": "{4, 5, 6, 7, 8}"
} | {
"index": 2357,
"solution": "1. **Understanding the Probability Function**: The probability of choosing a digit $d$ is given by $\\log_{10}(d+1) - \\log_{10}(d)$. Using the properties of logarithms, this can be rewritten as:\n \\[\n \\log_{10}\\left(\\frac{d+1}{d}\\right)\n \\]\n\n2. **Calculating the Probab... |
DeepScaleR | [
{
"role": "user",
"content": "A nonempty set $S$ is called well-filled if for every $m \\in S$, there are fewer than $\\frac{1}{2}m$ elements of $S$ which are less than $m$. Determine the number of well-filled subsets of $\\{1,2, \\ldots, 42\\}$."
}
] | math | {
"style": "rule",
"ground_truth": "\\binom{43}{21}-1"
} | {
"index": 2358,
"solution": "Let $a_{n}$ be the number of well-filled subsets whose maximum element is $n$ (setting $a_{0}=1$). Then it's easy to see that $a_{2k+1}=a_{2k}+a_{2k-1}+\\cdots+a_{0}$ and $a_{2k+2}=(a_{2k+1}-C_{k})+a_{2k}+\\cdots+a_{0}$ where $C_{k}$ is the number of well-filled subsets of size $k+1$ w... |
DeepScaleR | [
{
"role": "user",
"content": "A nonzero polynomial with rational coefficients has all of the numbers \\[1+\\sqrt{2}, \\; 2+\\sqrt{3}, \\;3+\\sqrt{4},\\; \\dots, \\;1000+\\sqrt{1001}\\]as roots. What is the smallest possible degree of such a polynomial?"
}
] | math | {
"style": "rule",
"ground_truth": "1970"
} | {
"index": 2359,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A noodle shop offers classic specialty noodles to customers, who can either dine in at the shop (referred to as \"dine-in\" noodles) or purchase packaged fresh noodles with condiments (referred to as \"fresh\" noodles). It is known that the total price of 3 portions of \"dine-i... | math | {
"style": "rule",
"ground_truth": "\\frac{40}{9}"
} | {
"index": 2360,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A normal line (a line that passes through a point of tangency and is perpendicular to the tangent line) is drawn to the parabola \\( y = x^2 \\) at point \\( A \\). The normal line intersects the parabola at another point \\( B \\). Let \\( O \\) be the origin of coordinates. W... | math | {
"style": "rule",
"ground_truth": "\\frac{-3+\\sqrt{33}}{24}"
} | {
"index": 2361,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A note contains three two-digit numbers that are said to form a sequence with a fourth number under a cryptic condition. The numbers provided are 46, 19, and 63, but the fourth number is unreadable. You know that the sum of the digits of all four numbers is $\\frac{1}{4}$ of th... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"index": 2362,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A notebook containing 67 pages, numbered from 1 to 67, is renumbered such that the last page becomes the first one, the second-last becomes the second one, and so on. Determine how many pages have the same units digit in their old and new numbering."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 2363,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A notebook sheet is painted in 23 colors, with each cell in the sheet painted in one of these colors. A pair of colors is called \"good\" if there exist two adjacent cells painted in these colors. What is the minimum number of good pairs?"
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 2364,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A notebook with 75 pages numbered from 1 to 75 is renumbered in reverse, from 75 to 1. Determine how many pages have the same units digit in both the old and new numbering systems."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 2365,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A novel is recorded onto compact discs, taking a total of 505 minutes to read aloud. Each disc can hold up to 53 minutes of reading. Assuming the smallest possible number of discs is used and each disc contains the same length of reading, calculate the number of minutes of read... | math | {
"style": "rule",
"ground_truth": "50.5"
} | {
"index": 2366,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A novel is to be recorded onto compact discs and takes 528 minutes to read aloud. Each disc can now hold up to 45 minutes of reading. Assuming that the smallest possible number of discs is used and that each disc contains the same length of reading, calculate the number of minu... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"index": 2367,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A novice economist-cryptographer received a cryptogram from a ruler which contained a secret decree about implementing an itemized tax on a certain market. The cryptogram specified the amount of tax revenue that needed to be collected, emphasizing that a greater amount could no... | math | {
"style": "rule",
"ground_truth": "6480"
} | {
"index": 2368,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number $N$ is defined as follows: \n\\[N=2+22+202+2002+20002+\\cdots+2\\overbrace{00\\ldots000}^{19~0\\text{'s}}2\\] \nWhen the value of $N$ is simplified, what is the sum of its digits?"
}
] | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 2369,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number $N$ has three digits when expressed in base $7$. When $N$ is expressed in base $9$ the digits are reversed. Then the middle digit is:"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 2370,
"solution": "1. **Expressing $N$ in different bases**: Let $N$ be represented as $\\overline{abc}_7$ in base $7$ and as $\\overline{cba}_9$ in base $9$. This means:\n - In base $7$: $N = 49a + 7b + c$\n - In base $9$: $N = 81c + 9b + a$\n\n2. **Setting up the equation**: Since both expressions ... |
DeepScaleR | [
{
"role": "user",
"content": "A number $m$ is randomly selected from the set $\\{11,13,15,17,19\\}$, and a number $n$ is randomly selected from $\\{1999,2000,2001,\\ldots,2018\\}$. What is the probability that $m^n$ has a units digit of $1$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{20}"
} | {
"index": 2371,
"solution": "1. **Reduction to Units Digits**:\n Since we are interested in the units digit of $m^n$, we can reduce the set $\\{11, 13, 15, 17, 19\\}$ to their respective units digits, resulting in the set $A = \\{1, 3, 5, 7, 9\\}$.\n\n2. **Case Analysis**:\n - **Case $m = 1$**:\n The units... |
DeepScaleR | [
{
"role": "user",
"content": "A number $m$ is randomly selected from the set $\\{12, 14, 16, 18, 20\\}$, and a number $n$ is randomly selected from $\\{2005, 2006, 2007, \\ldots, 2024\\}$. What is the probability that $m^n$ has a units digit of $6$?\nA) $\\frac{1}{5}$\nB) $\\frac{1}{4}$\nC) $\\frac{2}{5}$\n... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"index": 2372,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number $n$ has $3$ divisors. How many divisors does $n^2$ have?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 2373,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number $n$ is $b a d$ if there exists some integer $c$ for which $x^{x} \\equiv c(\\bmod n)$ has no integer solutions for $x$. Find the number of bad integers between 2 and 42 inclusive."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"index": 2374,
"solution": "Call a number good if it is not bad. We claim all good numbers are products of distinct primes, none of which are equivalent to 1 modulo another. We first show that all such numbers are good. Consider $n=p_{1} p_{2} \\ldots p_{k}$, and let $x$ be a number satisfying $x \\equiv c\\left(... |
DeepScaleR | [
{
"role": "user",
"content": "A number $p$ is $perfect$ if the sum of its divisors, except $p$ is $p$. Let $f$ be a function such that:\n\n$f(n)=0$, if n is perfect\n$f(n)=0$, if the last digit of n is 4\n$f(a.b)=f(a)+f(b)$\n\nFind $f(1998)$"
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 2375,
"solution": "\nTo determine \\( f(1998) \\), we start by analyzing the given function \\( f \\) and the properties it holds.\n\n1. **Perfect Number Property**: \n If \\( n \\) is a perfect number, then \\( f(n) = 0 \\).\n\n2. **Ending with Digit 4 Property**: \n If the last digit of \\( n \\) i... |
DeepScaleR | [
{
"role": "user",
"content": "A number $x$ is $2$ more than the product of its reciprocal and its additive inverse. In which interval does the number lie?"
}
] | math | {
"style": "rule",
"ground_truth": "0 < x \\le 2"
} | {
"index": 2376,
"solution": "Let's analyze the given problem step by step.\n\n1. **Understanding the terms:**\n - The **reciprocal** of a number $x$ is $\\frac{1}{x}$.\n - The **additive inverse** of a number $x$ is $-x$.\n\n2. **Setting up the equation:**\n The problem states that $x$ is $2$ more than the p... |
DeepScaleR | [
{
"role": "user",
"content": "A number $x$ is equal to $6 \\cdot 18 \\cdot 42$. What is the smallest positive integer $y$ such that the product $xy$ is a perfect cube?"
}
] | math | {
"style": "rule",
"ground_truth": "441"
} | {
"index": 2377,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number $x$ is equal to $7\\cdot24\\cdot48$. What is the smallest positive integer $y$ such that the product $xy$ is a perfect cube?"
}
] | math | {
"style": "rule",
"ground_truth": "588"
} | {
"index": 2378,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number $x$ is randomly chosen from the interval $[-1, 1]$. What is the probability that the value of $\\cos \\frac{\\pi x}{2}$ lies between $0$ and $\\frac{1}{2}$?"
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{1}{3}$"
} | {
"index": 2379,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number $x$ is randomly selected from the interval $\\left[ -\\frac{\\pi}{6}, \\frac{\\pi}{2} \\right]$. Calculate the probability that $\\sin x + \\cos x \\in [1, \\sqrt{2}]$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"index": 2380,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number \\( a \\) is randomly chosen from \\( 1, 2, 3, \\cdots, 10 \\), and a number \\( b \\) is randomly chosen from \\( -1, -2, -3, \\cdots, -10 \\). What is the probability that \\( a^{2} + b \\) is divisible by 3?"
}
] | math | {
"style": "rule",
"ground_truth": "37/100"
} | {
"index": 2381,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number \\( n \\) has a sum of digits equal to 100, while \\( 44n \\) has a sum of digits equal to 800. Find the sum of the digits of \\( 3n \\)."
}
] | math | {
"style": "rule",
"ground_truth": "300"
} | {
"index": 2382,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number composed of ten million, three hundred thousand, and fifty is written as \\_\\_\\_\\_\\_\\_, and this number is read as \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "10300050"
} | {
"index": 2383,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number contains only two kinds of digits: 3 or 4, and both 3 and 4 appear at least once. The number is a multiple of both 3 and 4. What is the smallest such number?"
}
] | math | {
"style": "rule",
"ground_truth": "3444"
} | {
"index": 2384,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number from the set $\\{30, 31, 32, \\ldots, 500\\}$ is randomly selected. What is the probability that the number is greater than 100 but less than or equal to 200? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{100}{471}"
} | {
"index": 2385,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number has 6 on both its tens and hundredths places, and 0 on both its ones and tenths places. This number is written as \\_\\_\\_\\_\\_\\_."
}
] | math | {
"style": "rule",
"ground_truth": "60.06"
} | {
"index": 2386,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number in the set $\\{50, 51, 52, 53, ... , 500\\}$ is randomly selected. What is the probability that it is a two-digit number divisible by 3? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{451}"
} | {
"index": 2387,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number in the set $\\{50, 51, 52, 53, ... , 999\\}$ is randomly selected. What is the probability that it is a two-digit number? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{19}"
} | {
"index": 2388,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number is called *6-composite* if it has exactly 6 composite factors. What is the 6th smallest 6-composite number? (A number is *composite* if it has a factor not equal to 1 or itself. In particular, 1 is not composite.)\n\n*Ray Li.*"
}
] | math | {
"style": "rule",
"ground_truth": "441"
} | {
"index": 2389,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number is called a visible factor number if it is divisible by each of its non-zero digits. For example, 102 is divisible by 1 and 2, so it is a visible factor number. How many visible factor numbers are there from 100 through 150, inclusive?"
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 2390,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number is called ascending if each of its digits is greater than the digit to its left. For example, 2568 is ascending, and 175 is not. How many ascending numbers are there between 400 and 600?"
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 2391,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number is called flippy if its digits alternate between two distinct digits. For example, $2020$ and $37373$ are flippy, but $3883$ and $123123$ are not. How many five-digit flippy numbers are divisible by $15?$"
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 2392,
"solution": "1. **Identify the conditions for a number to be flippy and divisible by 15**:\n - A flippy number alternates between two distinct digits.\n - A number is divisible by 15 if it is divisible by both 3 and 5.\n\n2. **Condition for divisibility by 5**:\n - The last digit must be eith... |
DeepScaleR | [
{
"role": "user",
"content": "A number is chosen at random from the set of consecutive natural numbers $\\{1, 2, 3, \\ldots, 24\\}$. What is the probability that the number chosen is a factor of $4!$? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 2393,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number is chosen uniformly at random from the set of all positive integers with at least two digits, none of which are repeated. Find the probability that the number is even."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{41}{81}"
} | {
"index": 2394,
"solution": "Since the number has at least two digits, all possible combinations of first and last digits have the same number of possibilities, which is \\(\\sum_{i=0}^{8} \\frac{8!}{i!}\\). Since the first digit cannot be zero, all of the last digits have 8 possible first digits, except for 0, wh... |
DeepScaleR | [
{
"role": "user",
"content": "A number is composed of 10 ones, 9 tenths (0.1), and 6 hundredths (0.01). This number is written as ____, and when rounded to one decimal place, it is approximately ____."
}
] | math | {
"style": "rule",
"ground_truth": "11.0"
} | {
"index": 2395,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number is composed of 6 millions, 3 tens of thousands, and 4 thousands. This number is written as ____, and when rewritten in terms of \"ten thousands\" as the unit, it becomes ____ ten thousands."
}
] | math | {
"style": "rule",
"ground_truth": "603.4"
} | {
"index": 2396,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number is considered a visible factor number if it is divisible by each of its non-zero digits. For example, 204 is divisible by 2 and 4 and is therefore a visible factor number. Determine how many visible factor numbers exist from 200 to 250, inclusive."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"index": 2397,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number is divided by \\(7, 11, 13\\). The sum of the quotients is 21, and the sum of the remainders is 21. What is the number?"
}
] | math | {
"style": "rule",
"ground_truth": "74"
} | {
"index": 2398,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "A number is divisible by $8$ if the number formed by its last $3$ digits is divisible by $8.$ For example, the number $47\\,389\\,248$ is divisible by $8$ because $248$ is divisible by $8.$ However, $47\\,389\\,284$ is not divisible by $8$ because $284$ is not divisible by $8.$... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 2399,
"solution": ""
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.