data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Vanessa set a school record for most points in a single basketball game when her team scored $48$ points. The six other players on her team averaged $3.5$ points each. How many points did Vanessa score to set her school record?"
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 36800,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vanya had a certain amount of cookies; he ate some, and then Tanya came to visit him and they divided the remaining cookies equally. It turned out that Vanya ate five times more cookies than Tanya did. What fraction of all the cookies did Vanya eat by the time Tanya arrived?"
... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{7}"
} | {
"index": 36801,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vasiliy came up with a new chess piece called the \"super-bishop.\" One \"super-bishop\" ($A$) attacks another ($B$) if they are on the same diagonal, there are no pieces between them, and the next cell along the diagonal after the \"super-bishop\" $B$ is empty. For example, in... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 36802,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vasya and Petya are participating in a school sports-entertainment game. They only have one pair of roller skates between the two of them, and need to traverse a distance of 3 km as quickly as possible. They start simultaneously, with one running and the other roller-skating. A... | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"index": 36803,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vasya and Petya live in the mountains and like to visit each other. They ascend the mountain at a speed of 3 km/h and descend at a speed of 6 km/h (there are no flat sections of the road). Vasya calculated that it takes him 2 hours and 30 minutes to go to Petya, and 3 hours and... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 36804,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vasya and Petya simultaneously started running from the starting point of a circular track in opposite directions at constant speeds. At some point, they met. Vasya completed a full lap and, continuing to run in the same direction, reached the point of their first meeting at th... | math | {
"style": "rule",
"ground_truth": "\\frac{1+\\sqrt{5}}{2}"
} | {
"index": 36805,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vasya has 9 different books by Arkady and Boris Strugatsky, each containing a single work by the authors. Vasya wants to arrange these books on a shelf in such a way that:\n(a) The novels \"Beetle in the Anthill\" and \"Waves Extinguish the Wind\" are next to each other (in any... | math | {
"style": "rule",
"ground_truth": "4 \\cdot 7!"
} | {
"index": 36806,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vasya has a stick that is 22 cm long. He wants to break it into three parts with integer lengths and form a triangle from the resulting pieces. How many ways can he do this? (Methods that result in identical triangles are considered the same)."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 36807,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vasya has a stick that is 22 cm long. He wants to break it into three pieces with integer lengths such that the pieces can form a triangle. In how many ways can he do this? (Ways that result in identical triangles are considered the same)."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 36808,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vasya has three cans of paint of different colors. In how many different ways can he paint a fence consisting of 10 planks so that any two adjacent planks are different colors and he uses all three colors? Provide a justification for your answer."
}
] | math | {
"style": "rule",
"ground_truth": "1530"
} | {
"index": 36809,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vasya has three cans of paint of different colors. In how many different ways can he paint a fence of 10 boards such that any two adjacent boards are of different colors, and all three colors are used?"
}
] | math | {
"style": "rule",
"ground_truth": "1530"
} | {
"index": 36810,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vasya has:\na) 2 different volumes from the collected works of A.S. Pushkin, each volume is 30 cm high;\nb) a set of works by E.V. Tarle in 4 volumes, each volume is 25 cm high;\nc) a book of lyrical poems with a height of 40 cm, published by Vasya himself.\n\nVasya wants to ar... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"index": 36811,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vasya is creating a 4-digit password for a combination lock. He dislikes the digit 2 and does not use it. Additionally, he does not like having two identical digits adjacent to each other. Moreover, he wants the first digit to be the same as the last digit. How many possible co... | math | {
"style": "rule",
"ground_truth": "576"
} | {
"index": 36812,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vasya is creating a 4-digit password for a combination lock. He dislikes the digit 2, so he does not use it. Additionally, he dislikes when two identical digits stand next to each other. Furthermore, he wants the first digit to match the last. How many possible combinations mus... | math | {
"style": "rule",
"ground_truth": "504"
} | {
"index": 36813,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vasya is inventing a 4-digit password for a combination lock. He does not like the digit 2, so he does not use it. Moreover, he doesn't like when two identical digits stand next to each other. Additionally, he wants the first digit to match the last one. How many possible combi... | math | {
"style": "rule",
"ground_truth": "504"
} | {
"index": 36814,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vasya loves picking mushrooms. He calculated that during the autumn, he collected a three-digit number of mushrooms with the sum of its digits equal to 14. Then Vasya calculated that 8% of the mushrooms were white, and 14% were boletus. How many mushrooms did Vasya collect?"
... | math | {
"style": "rule",
"ground_truth": "950"
} | {
"index": 36815,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vasya remembers that his friend Petya lives on Kurchatovskaya street in building number 8, but he forgot the apartment number. When asked for clarification, Petya replied: \"The number of my apartment is a three-digit number. If you rearrange its digits, you get five other thre... | math | {
"style": "rule",
"ground_truth": "425"
} | {
"index": 36816,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vasya replaced the same digits in two numbers with the same letters, and different digits with different letters. It turned out that the number ZARAZA is divisible by 4, and ALMAZ is divisible by 28. Find the last two digits of the sum ZARAZA + ALMAZ."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 36817,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vasya thought of a four-digit number and wrote down the product of each pair of its adjacent digits on the board. After that, he erased one product, and the numbers 20 and 21 remained on the board. What is the smallest number Vasya could have in mind?"
}
] | math | {
"style": "rule",
"ground_truth": "3745"
} | {
"index": 36818,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vasya wrote a note on a piece of paper, folded it in four, and labeled the top with \"MAME\". Then he unfolded the note, added something else, folded it along the creases in a random manner (not necessarily the same as before), and left it on the table with a random side up. Fi... | math | {
"style": "rule",
"ground_truth": "1/8"
} | {
"index": 36819,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vasya wrote a note on a piece of paper, folded it in four, and wrote the inscription \"MAME\" on top. Then he unfolded the note, wrote something else, folded it again along the crease lines at random (not necessarily in the same way as before), and left it on the table with a r... | math | {
"style": "rule",
"ground_truth": "1/8"
} | {
"index": 36820,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vasya wrote a note on a piece of paper, folded it in quarters, and wrote \"MAME\" on top. He then unfolded the note, added something more, folded it again randomly along the crease lines (not necessarily as before), and left it on the table with a random side facing up. Find th... | math | {
"style": "rule",
"ground_truth": "1/8"
} | {
"index": 36821,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vasya wrote a note on a piece of paper, folded it into quarters, and labeled the top with \"MAME\". Then he unfolded the note, wrote something else on it, folded the note along the creases randomly (not necessarily as before), and left it on the table with a random side facing ... | math | {
"style": "rule",
"ground_truth": "1/8"
} | {
"index": 36822,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vasya wrote a set of distinct natural numbers on the board, each of which does not exceed 2023. It turned out that for any two written numbers \\(a\\) and \\(b\\), the number \\(a + b\\) is not divisible by the number \\(a - b\\). What is the maximum number of numbers Vasya mig... | math | {
"style": "rule",
"ground_truth": "675"
} | {
"index": 36823,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vasya wrote consecutive natural numbers \\( N \\), \\( N+1 \\), \\( N+2 \\), and \\( N+3 \\) in rectangles. Under each rectangle, he wrote the sum of the digits of the corresponding number in a circle.\n\nThe sum of the numbers in the first two circles turned out to be 200, and... | math | {
"style": "rule",
"ground_truth": "103"
} | {
"index": 36824,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vasya wrote consecutive natural numbers \\(N\\), \\(N+1\\), \\(N+2\\), and \\(N+3\\) in rectangular boxes. Below each rectangle, he wrote the sum of the digits of the corresponding number in a circle.\n\nThe sum of the numbers in the first and second circles equals 200, and the... | math | {
"style": "rule",
"ground_truth": "103"
} | {
"index": 36825,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vasya wrote down 11 natural numbers in one notebook. Petya wrote down the greatest common divisors of each pair of numbers recorded in Vasya's notebook in another notebook. It turned out that every number written in one of the two notebooks is also in the other notebook. What i... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 36826,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vasya wrote natural numbers on the pages of an 18-page notebook. On each page, he wrote at least 10 different numbers, and on any consecutive three pages, there are no more than 20 different numbers in total. What is the maximum number of different numbers Vasya could have writ... | math | {
"style": "rule",
"ground_truth": "190"
} | {
"index": 36827,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vasya, Petya, and Kolya are in the same class. Vasya always lies in response to any question, Petya alternates between lying and telling the truth, and Kolya lies in response to every third question but tells the truth otherwise. One day, each of them was asked six consecutive ... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"index": 36828,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vehicle A and Vehicle B start from points A and B, respectively, at the same time and travel towards each other. They meet after 3 hours, at which point Vehicle A turns back towards point A, and Vehicle B continues forward. After Vehicle A reaches point A, it turns around and h... | math | {
"style": "rule",
"ground_truth": "7.2"
} | {
"index": 36829,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vendelín lives between two bus stops, at three-eighths of their distance. Today he left home and discovered that whether he ran to one or the other stop, he would arrive at the stop at the same time as the bus. The average speed of the bus is $60 \\mathrm{~km} / \\mathrm{h}$.\n... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 36830,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vera has a set of weights, each of which has a distinct mass and weighs an integer number of grams. It is known that the lightest weight in the set weighs 71 times less than the sum of the weights of all other weights in the set. It is also known that the two lightest weights t... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 36831,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vera has several identical matches, from which she makes a triangle. Vera wants any two sides of this triangle to differ in length by at least $10$ matches, but it turned out that it is impossible to add such a triangle from the available matches (it is impossible to leave ex... | math | {
"style": "rule",
"ground_truth": "62"
} | {
"index": 36832,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Veronica has 6 marks on her report card.\n\nThe mean of the 6 marks is 74.\n\nThe mode of the 6 marks is 76.\n\nThe median of the 6 marks is 76.\n\nThe lowest mark is 50.\n\nThe highest mark is 94.\n\nOnly one mark appears twice, and no mark appears more than twice.\n\nAssuming... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"index": 36833,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Veronica put on five rings: one on her little finger, one on her middle finger, and three on her ring finger. In how many different orders can she take them all off one by one?"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 36834,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Veronica put on five rings: one on her little finger, one on her middle finger, and three on her ring finger. In how many different orders can she take them all off one by one? \n\nA) 16\nB) 20\nC) 24\nD) 30\nE) 45"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 36835,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vertex $E$ of equilateral $\\triangle ABE$ is in the interior of square $ABCD$, and $F$ is the point of intersection of diagonal $BD$ and line segment $AE$. If length $AB$ is $\\sqrt{1+\\sqrt{3}}$ then the area of $\\triangle ABF$ is"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"index": 36836,
"solution": "1. **Positioning the Square and Triangle on the Coordinate Plane**:\n Place square $ABCD$ on the coordinate plane with $A$ at the origin $(0,0)$, $B$ at $(\\sqrt{1+\\sqrt{3}}, 0)$, $C$ at $(\\sqrt{1+\\sqrt{3}}, \\sqrt{1+\\sqrt{3}})$, and $D$ at $(0, \\sqrt{1+\\sqrt{3}})$.\n\n2. **Eq... |
DeepScaleR | [
{
"role": "user",
"content": "Vertex $E$ of equilateral $\\triangle{ABE}$ is in the interior of unit square $ABCD$. Let $R$ be the region consisting of all points inside $ABCD$ and outside $\\triangle{ABE}$ whose distance from $AD$ is between $\\frac{1}{3}$ and $\\frac{2}{3}$. What is the area of $R$?"
}
... | math | {
"style": "rule",
"ground_truth": "\\frac{3-\\sqrt{3}}{9}"
} | {
"index": 36837,
"solution": "1. **Identify the region $R$:** The region $R$ consists of all points inside the unit square $ABCD$ but outside the equilateral triangle $\\triangle{ABE}$, and whose distance from side $AD$ is between $\\frac{1}{3}$ and $\\frac{2}{3}$.\n\n2. **Calculate the area of the middle third of... |
DeepScaleR | [
{
"role": "user",
"content": "Vertex E of equilateral triangle ∆ABE is inside square ABCD. F is the intersection point of diagonal BD and line segment AE. If AB has length √(1+√3), calculate the area of ∆ABF."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"index": 36838,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Victor has $3$ piles of $3$ cards each. He draws all of the cards, but cannot draw a card until all the cards above it have been drawn. (For example, for his first card, Victor must draw the top card from one of the $3$ piles.) In how many orders can Victor draw the cards... | math | {
"style": "rule",
"ground_truth": "1680"
} | {
"index": 36839,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Victor has a drawer with 6 socks of 3 different types: 2 complex socks, 2 synthetic socks, and 2 trigonometric socks. He repeatedly draws 2 socks at a time from the drawer at random, and stops if the socks are of the same type. However, Victor is 'synthetic-complex type-blind',... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{7}"
} | {
"index": 36840,
"solution": "Let the socks be $C_{1}, C_{2}, S_{1}, S_{2}, T_{1}, T_{2}$, where $C, S$ and $T$ stand for complex, synthetic and trigonometric respectively. The possible stopping points consist of three pairs of socks of the same type plus four different complex-synthetic $(C-S)$ pairs, for a total... |
DeepScaleR | [
{
"role": "user",
"content": "Victor has a drawer with two red socks, two green socks, two blue socks, two magenta socks, two lavender socks, two neon socks, two mauve socks, two wisteria socks, and 2000 copper socks, for a total of 2016 socks. He repeatedly draws two socks at a time from the drawer at rand... | math | {
"style": "rule",
"ground_truth": "\\frac{1999008}{1999012}"
} | {
"index": 36841,
"solution": "There are $\\binom{2000}{2}+8\\binom{2}{2}=1999008$ ways to get socks which are matching colors, and four extra ways to get a red-green pair, hence the answer."
} |
DeepScaleR | [
{
"role": "user",
"content": "Victor was driving to the airport in a neighboring city. Half an hour into the drive at a speed of 60 km/h, he realized that if he did not change his speed, he would be 15 minutes late. So he increased his speed, covering the remaining distance at an average speed of 80 km/h, a... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"index": 36842,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Victoria wants to order at least 550 donuts from Dunkin' Donuts for the HMMT 2014 November contest. However, donuts only come in multiples of twelve. Assuming every twelve donuts cost \\$7.49, what is the minimum amount Victoria needs to pay, in dollars?"
}
] | math | {
"style": "rule",
"ground_truth": "344.54"
} | {
"index": 36843,
"solution": "The smallest multiple of 12 larger than 550 is $552=12 \\cdot 46$. So the answer is $46 \\cdot \\$7.49$. To make the multiplication easier, we can write this as $46 \\cdot(\\$7.5-\\$0.01)=\\$345-\\$0.46=\\$344.54$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Vijay chooses three distinct integers \\(a, b, c\\) from the set \\(\\{1,2,3,4,5,6,7,8,9,10,11\\}\\). If \\(k\\) is the minimum value taken on by the polynomial \\(a(x-b)(x-c)\\) over all real numbers \\(x\\), and \\(l\\) is the minimum value taken on by the polynomial \\(a(x-b... | math | {
"style": "rule",
"ground_truth": "990"
} | {
"index": 36844,
"solution": "Quadratics are minimized at the average of their roots, so \\(k=a\\left(\\frac{b+c}{2}-b\\right)\\left(\\frac{b+c}{2}-c\\right)=-\\frac{a(b-c)^{2}}{4}\\), and \\(l=a\\left(\\frac{b-c}{2}-b\\right)\\left(\\frac{b-c}{2}+c\\right)=-\\frac{a(b+c)^{2}}{4}\\). Therefore, \\(k-l=-\\frac{a}{4... |
DeepScaleR | [
{
"role": "user",
"content": "Vinny wrote down all the single-digit base-$b$ numbers and added them in base $b$, getting $34_b$.\n\nWhat is $b$?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 36845,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Violet has one-half of the money she needs to buy her mother a necklace. After her sister gives her $\\$30$, she has three-quarters of the amount she needs. How much will Violet's father give her?"
}
] | math | {
"style": "rule",
"ground_truth": "$30"
} | {
"index": 36846,
"solution": "Violet starts with one-half of the money that she needed to buy the necklace. After her sister gives her money, she has three-quarters of the amount that she needs. This means that her sister gave her $\\frac{3}{4}-\\frac{1}{2}=\\frac{1}{4}$ of the total amount that she needs. Since s... |
DeepScaleR | [
{
"role": "user",
"content": "Vitya Perestukin always incorrectly calculates percentages during surveys: he divides the number of respondents who answered a certain way by the number of all remaining respondents. For instance, in the survey \"What is your name?\" conducted among 7 Annas, 9 Olgas, 8 Julias, ... | math | {
"style": "rule",
"ground_truth": "110"
} | {
"index": 36847,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vitya and Masha were born in the same year in June. Find the probability that Vitya is at least one day older than Masha."
}
] | math | {
"style": "rule",
"ground_truth": "29/60"
} | {
"index": 36848,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vitya collects toy cars from the \"Retro\" series. The problem is that the total number of different models in the series is unknown — it's a big commercial secret. However, it is known that different cars are produced in equal quantities, so it can be assumed that all models a... | math | {
"style": "rule",
"ground_truth": "58"
} | {
"index": 36849,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vitya has five math lessons a week, one on each day from Monday to Friday. Vitya knows that with a probability of \\( \\frac{1}{2} \\) the teacher will not check his homework at all during the week, and with a probability of \\( \\frac{1}{2} \\) the teacher will check it exactl... | math | {
"style": "rule",
"ground_truth": "1/6"
} | {
"index": 36850,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vitya has five math lessons per week, one on each day from Monday to Friday. Vitya knows that with a probability of \\(1 / 2\\), the teacher will not check his homework at all during the week, and with a probability of \\(1 / 2\\), the teacher will check it, but only once on an... | math | {
"style": "rule",
"ground_truth": "1/6"
} | {
"index": 36851,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vivek is painting three doors numbered 1, 2, and 3. Each door is to be painted either black or gold. How many different ways can the three doors be painted?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 36852,
"solution": "Since there are 3 doors and 2 colour choices for each door, there are $2^{3}=8$ ways of painting the three doors. Using 'B' to represent black and 'G' to represent gold, these ways are BBB, BBG, BGB, BGG, GBB, GBG, GGB, and GGG."
} |
DeepScaleR | [
{
"role": "user",
"content": "Voldemort bought a book for $\\$5$. It was one-tenth of its original price. What was the original price in dollars?"
}
] | math | {
"style": "rule",
"ground_truth": "\\$50"
} | {
"index": 36853,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Von Neumann's Poker: The first step in Von Neumann's game is selecting a random number on $[0,1]$. To generate this number, Chebby uses the factorial base: the number $0 . A_{1} A_{2} A_{3} A_{4} \\ldots$ stands for $\\sum_{n=0}^{\\infty} \\frac{A_{n}}{(n+1)!}$, where each $A_{... | math | {
"style": "rule",
"ground_truth": "0.57196"
} | {
"index": 36854,
"solution": "The expected value of a number generated in the factorial base system is given by the sum of the expected values of each digit divided by its factorial weight. For this specific setup, the expected value is approximately 0.57196."
} |
DeepScaleR | [
{
"role": "user",
"content": "Vovochka adds three-digit numbers in a column in the following way: he does not carry over tens, but writes the sum of pairs of digits in the same place value position under each pair of digits, even if the result is a two-digit number. For example, for the sum \\(248 + 208\\),... | math | {
"style": "rule",
"ground_truth": "1800"
} | {
"index": 36855,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Vovochka approached an arcade machine which displayed the number 0 on the screen. The rules of the game stated: \"The screen shows the number of points. If you insert a 1 ruble coin, the number of points increases by 1. If you insert a 2 ruble coin, the number of points doubles... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 36856,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 36857,
"solution": "We are given the conditions:\n1. Ralph passed the orange house before the red house.\n2. Ralph passed the blue house before the yellow house.\n3. The blue house was not next to the yellow house.\n\nWe need to determine the possible orderings of the houses under these constraints.\n\n#... |
DeepScaleR | [
{
"role": "user",
"content": "Walking is a form of exercise that falls between walking and racewalking. It is a simple and safe aerobic exercise that can enhance lung capacity and promote heart health. A sports physiologist conducted a large number of surveys on the body fat percentage ($X$) of people engag... | math | {
"style": "rule",
"ground_truth": "0.03"
} | {
"index": 36858,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Walnuts and hazelnuts were delivered to a store in $1 \\mathrm{~kg}$ packages. The delivery note only mentioned that the shipment's value is $1978 \\mathrm{Ft}$, and its weight is $55 \\mathrm{~kg}$. The deliverers remembered the following:\n\n- Walnuts are more expensive;\n- T... | math | {
"style": "rule",
"ground_truth": "43"
} | {
"index": 36859,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Walter gets up at 6:30 a.m., catches the school bus at 7:30 a.m., has 6 classes that last 50 minutes each, has 30 minutes for lunch, and has 2 hours additional time at school. He takes the bus home and arrives at 4:00 p.m. How many minutes has he spent on the bus?"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 36860,
"solution": "1. **Calculate the total time Walter was away from home**: Walter leaves home at 7:30 a.m. and returns at 4:00 p.m. The time from 7:30 a.m. to 4:00 p.m. is calculated as follows:\n - From 7:30 a.m. to 12:00 p.m. is $4$ hours and $30$ minutes, which is $4.5$ hours.\n - From 12:00 p... |
DeepScaleR | [
{
"role": "user",
"content": "Walter gets up at 6:30 a.m., catches the school bus at 7:30 a.m., has 6 classes that last 50 minutes each, has 30 minutes for lunch, and has 2 hours additional time at school. He takes the bus home and arrives at 4:00 p.m. Calculate the total number of minutes Walter spent on t... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 36861,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Walter has exactly one penny, one nickel, one dime and one quarter in his pocket. What percent of one dollar is in his pocket?"
}
] | math | {
"style": "rule",
"ground_truth": "41\\%"
} | {
"index": 36862,
"solution": "1. **Identify the coins and their values**: Walter has the following coins:\n - One penny = $1$ cent\n - One nickel = $5$ cents\n - One dime = $10$ cents\n - One quarter = $25$ cents\n\n2. **Calculate the total value of the coins**:\n \\[\n 1 \\text{ cent} + 5 \\text{ cent... |
DeepScaleR | [
{
"role": "user",
"content": "Walter wakes up at 6:30 a.m., catches the school bus at 7:30 a.m., has 7 classes that last 45 minutes each, enjoys a 30-minute lunch break, and spends an additional 3 hours at school for various activities. He takes the bus home and arrives back at 5:00 p.m. Calculate the total... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"index": 36863,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Wanda is trying to locate the Fermat point $P$ of $\\triangle ABC$, where $A$ is at the origin, $B$ is at $(10,0)$, and $C$ is at $(3,5)$ (the Fermat point is the point such that the sum of its distances from the vertices of a triangle is minimized). She guesses that the point ... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 36864,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Wanda is trying to locate the Fermat point $P$ of $\\triangle ABC$, where $A$ is at the origin, $B$ is at $(8,-1)$, and $C$ is at $(5,4)$ (the Fermat point is the point such that the sum of its distances from the vertices of a triangle is minimized). She guesses that the point ... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 36865,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Wanda, Darren, Beatrice, and Chi are tutors in the school math lab. Their schedule is as follows: Darren works every third school day, Wanda works every fourth school day, Beatrice works every sixth school day, and Chi works every seventh school day. Today they are all working ... | math | {
"style": "rule",
"ground_truth": "84"
} | {
"index": 36866,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Wang Hong's father deposited 20,000 yuan in the bank for a fixed term of three years, with an annual interest rate of 3.33%. How much money, including the principal and interest, can Wang Hong's father withdraw at the end of the term?"
}
] | math | {
"style": "rule",
"ground_truth": "21998"
} | {
"index": 36867,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Wang Lei and her older sister walk from home to the gym to play badminton. It is known that the older sister walks 20 meters more per minute than Wang Lei. After 25 minutes, the older sister reaches the gym, and then realizes she forgot the racket. She immediately returns along... | math | {
"style": "rule",
"ground_truth": "1500"
} | {
"index": 36868,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Warehouse A and Warehouse B originally stored whole bags of grain. If 90 bags are transferred from Warehouse A to Warehouse B, then the grain in Warehouse B will be twice that in Warehouse A. If a certain number of bags are transferred from Warehouse B to Warehouse A, then the ... | math | {
"style": "rule",
"ground_truth": "153"
} | {
"index": 36869,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Water is the source of life and one of the indispensable important material resources for human survival and development. In order to better manage water quality and protect the environment, the Municipal Sewage Treatment Office plans to purchase 10 sewage treatment equipment i... | math | {
"style": "rule",
"ground_truth": "102"
} | {
"index": 36870,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Wayne has 3 green buckets, 3 red buckets, 3 blue buckets, and 3 yellow buckets. He randomly distributes 4 hockey pucks among the green buckets, with each puck equally likely to be put in each bucket. Similarly, he distributes 3 pucks among the red buckets, 2 pucks among the blu... | math | {
"style": "rule",
"ground_truth": "\\frac{89}{243}"
} | {
"index": 36871,
"solution": "We use the notation ' $a / b / c$ ' to mean $a$ pucks in one bucket, $b$ pucks in a second bucket, and $c$ pucks in the third bucket, ignoring the order of the buckets. Yellow buckets: 1/0/0: With 1 puck to distribute, the distribution will always be $1 / 0 / 0$. Blue buckets: Since t... |
DeepScaleR | [
{
"role": "user",
"content": "We are allowed to remove exactly one integer from the list $$-1,0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,11,$$and then we choose two distinct integers at random from the remaining list. What number should we remove if we wish to maximize the probability that the sum of the two chosen ... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 36872,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We are approaching a 120-meter high skyscraper on a horizontal road. After traveling 300 meters, we see the building at an angle of elevation that is $45^\\circ$ greater than at the start of our journey. How close have we approached the skyscraper?"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 36873,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We are given $2n$ natural numbers\n\\[1, 1, 2, 2, 3, 3, \\ldots, n - 1, n - 1, n, n.\\]\nFind all $n$ for which these numbers can be arranged in a row such that for each $k \\leq n$, there are exactly $k$ numbers between the two numbers $k$."
}
] | math | {
"style": "rule",
"ground_truth": "$n=3,4,7,8$"
} | {
"index": 36874,
"solution": "\nWe are given \\(2n\\) natural numbers: \n\\[\n1, 1, 2, 2, 3, 3, \\ldots, n-1, n-1, n, n.\n\\]\nand we need to find all values of \\(n\\) for which these numbers can be arranged such that there are exactly \\(k\\) numbers between the two occurrences of the number \\(k\\).\n\nFirst, c... |
DeepScaleR | [
{
"role": "user",
"content": "We are given $n$ coins of different weights and $n$ balances, $n>2$. On each turn one can choose one balance, put one coin on the right pan and one on the left pan, and then delete these coins out of the balance. It's known that one balance is wrong (but it's not known ehich ex... | math | {
"style": "rule",
"ground_truth": "2n - 1"
} | {
"index": 36875,
"solution": "\nGiven are \\( n \\) coins of different weights and \\( n \\) balances, where \\( n > 2 \\). One of these balances is faulty and provides arbitrary results on each turn. Our goal is to determine the smallest number of turns required to find the heaviest coin.\n\n### Strategy\n\n1. **... |
DeepScaleR | [
{
"role": "user",
"content": "We are given 5771 weights weighing 1,2,3,...,5770,5771. We partition the weights into $n$ sets of equal weight. What is the maximal $n$ for which this is possible?"
}
] | math | {
"style": "rule",
"ground_truth": "2886"
} | {
"index": 36876,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We are given a cone with height 6, whose base is a circle with radius $\\sqrt{2}$ . Inside the cone, there is an inscribed cube: Its bottom face on the base of the cone, and all of its top vertices lie on the cone. What is the length of the cube's edge?\n\n![Image](https://i.i... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"index": 36877,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We are given a triangle $ABC$ . Points $D$ and $E$ on the line $AB$ are such that $AD=AC$ and $BE=BC$ , with the arrangment of points $D - A - B - E$ . The circumscribed circles of the triangles $DBC$ and $EAC$ meet again at the point $X\\neq C$ , and the circum... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 36878,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We are given some similar triangles. Their areas are $1^{2}, 3^{2}, 5^{2} \\ldots$, and $49^{2}$. If the smallest triangle has a perimeter of 4, what is the sum of all the triangles' perimeters?"
}
] | math | {
"style": "rule",
"ground_truth": "2500"
} | {
"index": 36879,
"solution": "Because the triangles are all similar, they all have the same ratio of perimeter squared to area, or, equivalently, the same ratio of perimeter to the square root of area. Because the latter ratio is 4 for the smallest triangle, it is 4 for all the triangles, and thus their perimeters... |
DeepScaleR | [
{
"role": "user",
"content": "We are given that $$54+(98\\div14)+(23\\cdot 17)-200-(312\\div 6)=200.$$Now, let's remove the parentheses: $$54+98\\div14+23\\cdot 17-200-312\\div 6.$$What does this expression equal?"
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 36880,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We are given triangle $A B C$, with $A B=9, A C=10$, and $B C=12$, and a point $D$ on $B C . B$ and $C$ are reflected in $A D$ to $B^{\\prime}$ and $C^{\\prime}$, respectively. Suppose that lines $B C^{\\prime}$ and $B^{\\prime} C$ never meet (i.e., are parallel and distinct). ... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 36881,
"solution": "The lengths of $A B$ and $A C$ are irrelevant. Because the figure is symmetric about $A D$, lines $B C^{\\prime}$ and $B^{\\prime} C$ meet if and only if they meet at a point on line $A D$. So, if they never meet, they must be parallel to $A D$. Because $A D$ and $B C^{\\prime}$ are p... |
DeepScaleR | [
{
"role": "user",
"content": "We are laying railway tracks that are $15 \\text{ meters }$ long at a temperature of $t = -8^{\\circ} \\text{C}$. What gap should we leave between each rail if the maximum expected temperature is $t = 60^{\\circ} \\text{C}$? The coefficient of expansion for the rail is $\\lambd... | math | {
"style": "rule",
"ground_truth": "12.24"
} | {
"index": 36882,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We build a $4 \\times 4 \\times 4$ cube out of sugar cubes. How many different rectangular parallelepipeds can the sugar cubes determine, if the rectangular parallelepipeds differ in at least one sugar cube?"
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"index": 36883,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We call a four-digit number with the following property a \"centered four-digit number\": Arrange the four digits of this four-digit number in any order, and when all the resulting four-digit numbers (at least 2) are sorted from smallest to largest, the original four-digit numb... | math | {
"style": "rule",
"ground_truth": "90"
} | {
"index": 36884,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We call a natural number \\( b \\) lucky if for any natural \\( a \\) such that \\( a^{5} \\) is divisible by \\( b^{2} \\), the number \\( a^{2} \\) is divisible by \\( b \\).\n\nFind the number of lucky natural numbers less than 2010."
}
] | math | {
"style": "rule",
"ground_truth": "1961"
} | {
"index": 36885,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We call a number a mountain number if its middle digit is larger than any other digit. For example, 284 is a mountain number. How many 3-digit mountain numbers are there?"
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 36886,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We call a number antitriangular if it can be expressed in the form \\(\\frac{2}{n(n+1)}\\) for some natural number \\(n\\). For how many numbers \\(k\\) (where \\(1000 \\leq k \\leq 2000\\)) can the number 1 be expressed as the sum of \\(k\\) antitriangular numbers (not necessa... | math | {
"style": "rule",
"ground_truth": "1001"
} | {
"index": 36887,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We call a number greater than $25$, [i] semi-prime[/i] if it is the sum of some two different prime numbers. What is the greatest number of consecutive natural numbers that can be [i]semi-prime[/i]?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 36888,
"solution": "\nTo solve this problem, we need to find the greatest number of consecutive natural numbers greater than 25 that can each be represented as the sum of two different prime numbers, which we refer to as semi-prime.\n\nFirst, let's identify a few prime numbers as potential candidates: 2,... |
DeepScaleR | [
{
"role": "user",
"content": "We call a pair $(a,b)$ of positive integers, $a<391$ , *pupusa* if $$ \\textup{lcm}(a,b)>\\textup{lcm}(a,391) $$ Find the minimum value of $b$ across all *pupusa* pairs.\n\nFun Fact: OMCC 2017 was held in El Salvador. *Pupusa* is their national dish. It is a corn tortil... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 36889,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We call a pair of natural numbers $(a, p)$ good if the number $a^{3} + p^{3}$ is divisible by $a^{2} - p^{2}$, and $a > p$.\n\n(a) (1 point) Give any possible value of $a$ for which the pair $(a, 11)$ is good.\n\n(b) (3 points) Find the number of good pairs, where $p$ is a prim... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 36890,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We call a pair of natural numbers \\((a, p)\\) good if the number \\(a^3 + p^3\\) is divisible by \\(a^2 - p^2\\), with \\(a > p\\).\n\n(a) (1 point) Specify any possible value of \\(a\\) for which the pair \\((a, 13)\\) is good.\n\n(b) (3 points) Find the number of good pairs ... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 36891,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We call a positive integer $N$ [i]contagious[/i] if there are $1000$ consecutive non-negative integers such that the sum of all their digits is $N$. Find all contagious positive integers."
}
] | math | {
"style": "rule",
"ground_truth": "\\{13500, 13501, 13502, \\ldots\\}"
} | {
"index": 36892,
"solution": "\nTo determine which positive integers \\( N \\) are contagious, we consider 1000 consecutive non-negative integers and the sum of all their digits equating to \\( N \\).\n\nLet the consecutive integers be \\( x, x+1, x+2, \\ldots, x+999 \\). We need to calculate the sum of the digits... |
DeepScaleR | [
{
"role": "user",
"content": "We call a positive integer $t$ good if there is a sequence $a_{0}, a_{1}, \\ldots$ of positive integers satisfying $a_{0}=15, a_{1}=t$, and $a_{n-1} a_{n+1}=\\left(a_{n}-1\\right)\\left(a_{n}+1\\right)$ for all positive integers $n$. Find the sum of all good numbers."
}
] | math | {
"style": "rule",
"ground_truth": "296"
} | {
"index": 36893,
"solution": "By the condition of the problem statement, we have $a_{n}^{2}-a_{n-1} a_{n+1}=1=a_{n-1}^{2}-a_{n-2} a_{n}$. This is equivalent to $\\frac{a_{n-2}+a_{n}}{a_{n-1}}=\\frac{a_{n-1}+a_{n+1}}{a_{n}}$. Let $k=\\frac{a_{0}+a_{2}}{a_{1}}$. Then we have $\\frac{a_{n-1}+a_{n+1}}{a_{n}}=\\frac{a_... |
DeepScaleR | [
{
"role": "user",
"content": "We call any eight squares in a diagonal of a chessboard as a fence. The rook is moved on the chessboard in such way that he stands neither on each square over one time nor on the squares of the fences (the squares which the rook passes is not considered ones it has stood on). T... | math | {
"style": "rule",
"ground_truth": "47"
} | {
"index": 36894,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We call the pair $(m, n)$ of positive integers a happy pair if the greatest common divisor of $m$ and $n$ is a perfect square. For example, $(20, 24)$ is a happy pair because the greatest common divisor of 20 and 24 is 4. Suppose that $k$ is a positive integer such that $(20580... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 36895,
"solution": "Suppose that $(205800, 35k)$ is a happy pair. We find the prime factorization of 205800: $205800 = 2^3 \\times 3^1 \\times 5^2 \\times 7^3$. Note also that $35k = 5^1 \\times 7^1 \\times k$. Let $d$ be the greatest common divisor of 205800 and $35k$. We want to find the number of poss... |
DeepScaleR | [
{
"role": "user",
"content": "We can view these conditions as a geometry diagram as seen below. So, we know that $\\frac{e}{f}=\\frac{3}{4}$ (since $e=a-b=\\frac{3}{4} c-\\frac{3}{4} d=\\frac{3}{4} f$ and we know that $\\sqrt{e^{2}+f^{2}}=15$ (since this is $\\left.\\sqrt{a^{2}+c^{2}}-\\sqrt{b^{2}+d^{2}}\\r... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"index": 36896,
"solution": "The value of $ef$ is 108."
} |
DeepScaleR | [
{
"role": "user",
"content": "We can write\n\\[\\sum_{k = 1}^{100} (-1)^k \\cdot \\frac{k^2 + k + 1}{k!} = \\frac{a}{b!} - c,\\]where $a,$ $b,$ and $c$ are positive integers. Find the smallest possible value of $a + b + c.$"
}
] | math | {
"style": "rule",
"ground_truth": "202"
} | {
"index": 36897,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We choose 100 points in the coordinate plane. Let $N$ be the number of triples $(A,B,C)$ of distinct chosen points such that $A$ and $B$ have the same $y$ -coordinate, and $B$ and $C$ have the same $x$ -coordinate. Find the greatest value that $N$ can attain con... | math | {
"style": "rule",
"ground_truth": "8100"
} | {
"index": 36898,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "We color certain squares of an $8 \\times 8$ chessboard red. How many squares can we color at most if we want no red trimino? How many squares can we color at least if we want every trimino to have at least one red square?"
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"index": 36899,
"solution": ""
} |
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