mradermacher/Olmo-3-7B-RLZero-Math-i1-GGUF
7B
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Updated
•
1.06k
Datasets:
Tasks:
Reinforcement Learning
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English
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10K - 100K
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3.79k
| ground_truth
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In triangle $ABC$, $\sin \angle A = \frac{4}{5}$ and $\angle A < 90^\circ$. Let $D$ be a point outside triangle $ABC$ such that $\angle BAD = \angle DAC$ and $\angle BDC = 90^\circ$. Suppose that $AD = 1$ and that $\frac{BD}{CD} = \frac{3}{2}$. If $AB + AC$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $a, b, c$ are pairwise relatively prime integers, find $a + b + c$.
|
34
|
[
{
"content": "In triangle $ABC$, $\\sin \\angle A = \\frac{4}{5}$ and $\\angle A < 90^\\circ$. Let $D$ be a point outside triangle $ABC$ such that $\\angle BAD = \\angle DAC$ and $\\angle BDC = 90^\\circ$. Suppose that $AD = 1$ and that $\\frac{BD}{CD} = \\frac{3}{2}$. If $AB + AC$ can be expressed in the form $\\frac{a\\sqrt{b}}{c}$ where $a, b, c$ are pairwise relatively prime integers, find $a + b + c$.",
"role": "user"
}
] |
math
|
Let $ABCD$ be a unit square in the plane. Points $X$ and $Y$ are chosen independently and uniformly at random on the perimeter of $ABCD$. If the expected value of the area of triangle $\triangle AXY$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$.
|
113
|
[
{
"content": "Let $ABCD$ be a unit square in the plane. Points $X$ and $Y$ are chosen independently and uniformly at random on the perimeter of $ABCD$. If the expected value of the area of triangle $\\triangle AXY$ can be expressed as $\\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$.",
"role": "user"
}
] |
math
|
Let $a, b, c$ be distinct numbers such that the equations $x^2 + ax + 1 = 0$ and $x^2 + bx + c = 0$ have a common real root, and the equations $x^2 + x + a = 0$ and $x^2 + cx + b = 0$ also have a common real root. Compute the sum $a + b + c$.
|
-3
|
[
{
"content": "Let $a, b, c$ be distinct numbers such that the equations $x^2 + ax + 1 = 0$ and $x^2 + bx + c = 0$ have a common real root, and the equations $x^2 + x + a = 0$ and $x^2 + cx + b = 0$ also have a common real root. Compute the sum $a + b + c$.",
"role": "user"
}
] |
math
|
There are $7$ boxes arranged in a row and numbered $1$ through $7$. You have a stack of $2015$ cards, which you place one by one in the boxes. The first card is placed in box $1$, the second in box $2$, and so forth up to the seventh card which is placed in box $7$. You then start working back in the other direction, placing the eighth card in box $6$, the ninth in box $5$, up to the thirteenth card being placed in box $1$. The fourteenth card is then placed in box $2$, and this continues until every card is distributed. What box will the last card be placed in?
|
3
|
[
{
"content": "There are $7$ boxes arranged in a row and numbered $1$ through $7$. You have a stack of $2015$ cards, which you place one by one in the boxes. The first card is placed in box $1$, the second in box $2$, and so forth up to the seventh card which is placed in box $7$. You then start working back in the other direction, placing the eighth card in box $6$, the ninth in box $5$, up to the thirteenth card being placed in box $1$. The fourteenth card is then placed in box $2$, and this continues until every card is distributed. What box will the last card be placed in?",
"role": "user"
}
] |
math
|
Let $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. The value of the sum
\[ \sum_{(a,b,c) \in S} \frac{1}{2^a 3^b 5^c} \]
can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.
|
37
|
[
{
"content": "Let $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. The value of the sum\n\\[ \\sum_{(a,b,c) \\in S} \\frac{1}{2^a 3^b 5^c} \\]\ncan be expressed as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.",
"role": "user"
}
] |
math
|
For which $n$ is $n^4 + 6n^3 + 11n^2 + 3n + 31$ a perfect square?
|
10
|
[
{
"content": "For which $n$ is $n^4 + 6n^3 + 11n^2 + 3n + 31$ a perfect square?",
"role": "user"
}
] |
math
|
$n$ coins are arranged in a circle. If two neighboring coins both show heads or both show tails, you can flip both coins. How many distinct arrangements of the coins are there such that no arrangement can be transformed into another by performing these operations?
|
2
|
[
{
"content": "$n$ coins are arranged in a circle. If two neighboring coins both show heads or both show tails, you can flip both coins. How many distinct arrangements of the coins are there such that no arrangement can be transformed into another by performing these operations?",
"role": "user"
}
] |
math
|
Suppose you have a sphere tangent to the $xy$-plane with its center having a positive $z$-coordinate. If it is projected from a point $P=(0,b,a)$ to the $xy$-plane, it gives the conic section $y=x^2$. If we write $a=\frac{p}{q}$ where $p$ and $q$ are integers, find $p+q$.
|
3
|
[
{
"content": "Suppose you have a sphere tangent to the $xy$-plane with its center having a positive $z$-coordinate. If it is projected from a point $P=(0,b,a)$ to the $xy$-plane, it gives the conic section $y=x^2$. If we write $a=\\frac{p}{q}$ where $p$ and $q$ are integers, find $p+q$.",
"role": "user"
}
] |
math
|
There are $10^{2015}$ planets in an Intergalactic empire. Every two planets are connected by a two-way space line served by one of $2015$ travel companies. The Emperor would like to close $k$ of these companies such that it is still possible to reach any planet from any other planet. Find the maximum value of $k$ for which this is always possible.
|
1007
|
[
{
"content": "There are $10^{2015}$ planets in an Intergalactic empire. Every two planets are connected by a two-way space line served by one of $2015$ travel companies. The Emperor would like to close $k$ of these companies such that it is still possible to reach any planet from any other planet. Find the maximum value of $k$ for which this is always possible.",
"role": "user"
}
] |
math
|
Rectangle $HOMF$ has $HO = 11$ and $OM = 5$. Triangle $\triangle ABC$ has orthocenter $H$ and circumcenter $O$. $M$ is the midpoint of $BC$, and altitude $AF$ meets $BC$ at $F$. Find the length of $BC$.
|
28
|
[
{
"content": "Rectangle $HOMF$ has $HO = 11$ and $OM = 5$. Triangle $\\triangle ABC$ has orthocenter $H$ and circumcenter $O$. $M$ is the midpoint of $BC$, and altitude $AF$ meets $BC$ at $F$. Find the length of $BC$.",
"role": "user"
}
] |
math
|
Two towns, $A$ and $B$, are $100$ miles apart. Every $20$ minutes (starting at midnight), a bus traveling at $60$ mph leaves town $A$ for town $B$, and every $30$ minutes (starting at midnight), a bus traveling at $20$ mph leaves town $B$ for town $A$. Dirk starts in Town $A$ and gets on a bus leaving for town $B$ at noon. However, Dirk is always afraid he has boarded a bus going in the wrong direction, so each time the bus he is in passes another bus, he gets out and transfers to that other bus. How many hours pass before Dirk finally reaches Town $B$?
|
5
|
[
{
"content": "Two towns, $A$ and $B$, are $100$ miles apart. Every $20$ minutes (starting at midnight), a bus traveling at $60$ mph leaves town $A$ for town $B$, and every $30$ minutes (starting at midnight), a bus traveling at $20$ mph leaves town $B$ for town $A$. Dirk starts in Town $A$ and gets on a bus leaving for town $B$ at noon. However, Dirk is always afraid he has boarded a bus going in the wrong direction, so each time the bus he is in passes another bus, he gets out and transfers to that other bus. How many hours pass before Dirk finally reaches Town $B$?",
"role": "user"
}
] |
math
|
Burrito Bear has a white unit square. She inscribes a circle inside the square and paints it black. She then inscribes a square inside the black circle and paints it white. Burrito repeats this process indefinitely. The total black area can be expressed as $\frac{a\pi+b}{c}$. Find $a+b+c$.
|
1
|
[
{
"content": "Burrito Bear has a white unit square. She inscribes a circle inside the square and paints it black. She then inscribes a square inside the black circle and paints it white. Burrito repeats this process indefinitely. The total black area can be expressed as $\\frac{a\\pi+b}{c}$. Find $a+b+c$.",
"role": "user"
}
] |
math
|
Sabrina has a fair tetrahedral die with faces numbered 1, 2, 3, and 4. She rolls the die to create a sequence by recording the number on its bottom face. However, she discards any roll if appending its number to the sequence results in two consecutive terms that sum to 5. Sabrina stops once all four numbers appear in the sequence. Find the expected (average) number of terms in Sabrina's sequence.
|
10
|
[
{
"content": "Sabrina has a fair tetrahedral die with faces numbered 1, 2, 3, and 4. She rolls the die to create a sequence by recording the number on its bottom face. However, she discards any roll if appending its number to the sequence results in two consecutive terms that sum to 5. Sabrina stops once all four numbers appear in the sequence. Find the expected (average) number of terms in Sabrina's sequence.",
"role": "user"
}
] |
math
|
For each positive integer $n$, let $S(n)$ be the sum of the digits of $n$. Determine the smallest positive integer $a$ such that there are infinitely many positive integers $n$ for which $S(n) - S(n + a) = 2018$.
|
7
|
[
{
"content": "For each positive integer $n$, let $S(n)$ be the sum of the digits of $n$. Determine the smallest positive integer $a$ such that there are infinitely many positive integers $n$ for which $S(n) - S(n + a) = 2018$.",
"role": "user"
}
] |
math
|
A class has $25$ students. The teacher wants to stock $N$ candies and hold an Olympiad, distributing all $N$ candies based on student performance. The candies should be distributed such that students who solve the same number of tasks receive the same number of candies, and those who solve fewer tasks receive fewer candies, possibly receiving none. Determine the smallest value of $N$ for which this distribution is possible, regardless of the number of tasks in the Olympiad and the students' successes.
|
300
|
[
{
"content": "A class has $25$ students. The teacher wants to stock $N$ candies and hold an Olympiad, distributing all $N$ candies based on student performance. The candies should be distributed such that students who solve the same number of tasks receive the same number of candies, and those who solve fewer tasks receive fewer candies, possibly receiving none. Determine the smallest value of $N$ for which this distribution is possible, regardless of the number of tasks in the Olympiad and the students' successes.",
"role": "user"
}
] |
math
|
If $a, b$ are real numbers such that $a^3 + 12a^2 + 49a + 69 = 0$ and $b^3 - 9b^2 + 28b - 31 = 0$, find $a + b$.
|
1
|
[
{
"content": "If $a, b$ are real numbers such that $a^3 + 12a^2 + 49a + 69 = 0$ and $b^3 - 9b^2 + 28b - 31 = 0$, find $a + b$.",
"role": "user"
}
] |
math
|
Let $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. Determine the value of the sum:
\[
\sum_{(a,b,c)\in S}\frac{1}{2^a3^b5^c}
\]
This sum can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.
|
37
|
[
{
"content": "Let $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. Determine the value of the sum:\n\\[\n\\sum_{(a,b,c)\\in S}\\frac{1}{2^a3^b5^c}\n\\]\nThis sum can be expressed as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.",
"role": "user"
}
] |
math
|
Some positive integers are initially written on a board, where each 2 of them are different. Each time we can do the following moves:
1. If there are 2 numbers (written on the board) in the form $n, n+1$, we can erase them and write down $n-2$.
2. If there are 2 numbers (written on the board) in the form $n, n+4$, we can erase them and write down $n-1$.
After some moves, there might appear negative numbers. Find the maximum value of the integer $c$ such that independently of the starting numbers, each number which appears in any move is greater than or equal to $c$.
|
-3
|
[
{
"content": "Some positive integers are initially written on a board, where each 2 of them are different. Each time we can do the following moves: \n1. If there are 2 numbers (written on the board) in the form $n, n+1$, we can erase them and write down $n-2$. \n2. If there are 2 numbers (written on the board) in the form $n, n+4$, we can erase them and write down $n-1$. \n\nAfter some moves, there might appear negative numbers. Find the maximum value of the integer $c$ such that independently of the starting numbers, each number which appears in any move is greater than or equal to $c$.",
"role": "user"
}
] |
math
|
Charlie has a fair $n$-sided die (with each face showing a positive integer between $1$ and $n$ inclusive) and a list of $n$ consecutive positive integers. He first rolls the die, and if the number showing on top is $k$, he then uniformly and randomly selects a $k$-element subset from his list and calculates the sum of the numbers in his subset. Given that the expected value of this sum is $2020$, compute the sum of all possible values of $n$.
|
8556
|
[
{
"content": "Charlie has a fair $n$-sided die (with each face showing a positive integer between $1$ and $n$ inclusive) and a list of $n$ consecutive positive integers. He first rolls the die, and if the number showing on top is $k$, he then uniformly and randomly selects a $k$-element subset from his list and calculates the sum of the numbers in his subset. Given that the expected value of this sum is $2020$, compute the sum of all possible values of $n$.",
"role": "user"
}
] |
math
|
The integers from $1$ to $n$ are written, one on each of $n$ cards. The first player removes one card. Then the second player removes two cards with consecutive integers. After that, the first player removes three cards with consecutive integers. Finally, the second player removes four cards with consecutive integers.
What is the smallest value of $n$ for which the second player can ensure that he completes both his moves?
|
14
|
[
{
"content": "The integers from $1$ to $n$ are written, one on each of $n$ cards. The first player removes one card. Then the second player removes two cards with consecutive integers. After that, the first player removes three cards with consecutive integers. Finally, the second player removes four cards with consecutive integers.\n\nWhat is the smallest value of $n$ for which the second player can ensure that he completes both his moves?",
"role": "user"
}
] |
math
|
Joe has $1729$ randomly oriented and randomly arranged unit cubes, which are initially unpainted. He makes two cubes of side lengths $9$ and $10$ or of side lengths $1$ and $12$ (randomly chosen). These cubes are dipped into white paint. Then two more cubes of side lengths $1$ and $12$ or $9$ and $10$ are formed from the same unit cubes, again randomly oriented and randomly arranged, and dipped into paint. Joe continues this process until every side of every unit cube is painted. After how many times of doing this is the expected number of painted faces closest to half of the total?
|
7
|
[
{
"content": "Joe has $1729$ randomly oriented and randomly arranged unit cubes, which are initially unpainted. He makes two cubes of side lengths $9$ and $10$ or of side lengths $1$ and $12$ (randomly chosen). These cubes are dipped into white paint. Then two more cubes of side lengths $1$ and $12$ or $9$ and $10$ are formed from the same unit cubes, again randomly oriented and randomly arranged, and dipped into paint. Joe continues this process until every side of every unit cube is painted. After how many times of doing this is the expected number of painted faces closest to half of the total?",
"role": "user"
}
] |
math
|
The circle $C_1$ is tangent to the circle $C_2$ internally at the point $A$. Let $O$ be the center of $C_2$. The point $P$ is on $C_1$, and the tangent line at $P$ passes through $O$. Let the ray $OP$ intersect with $C_2$ at $Q$, and let the tangent line of $C_1$ passing through $A$ intersect with the line $OP$ at $R$. If the radius of $C_2$ is 9 and $PQ = QR$, find the length of the segment $OP$.
|
3
|
[
{
"content": "The circle $C_1$ is tangent to the circle $C_2$ internally at the point $A$. Let $O$ be the center of $C_2$. The point $P$ is on $C_1$, and the tangent line at $P$ passes through $O$. Let the ray $OP$ intersect with $C_2$ at $Q$, and let the tangent line of $C_1$ passing through $A$ intersect with the line $OP$ at $R$. If the radius of $C_2$ is 9 and $PQ = QR$, find the length of the segment $OP$.",
"role": "user"
}
] |
math
|
Consider two distinct positive integers $a$ and $b$ that have integer arithmetic, geometric, and harmonic means. Find the minimal value of $|a-b|$.
|
30
|
[
{
"content": "Consider two distinct positive integers $a$ and $b$ that have integer arithmetic, geometric, and harmonic means. Find the minimal value of $|a-b|$.",
"role": "user"
}
] |
math
|
Dexter and Raquel are playing a game with $N$ stones. Dexter goes first and takes one stone from the pile. After that, the players alternate turns and can take anywhere from $1$ to $x + 1$ stones from the pile, where $x$ is the number of stones the other player took on the turn immediately prior. The winner is the one to take the last stone from the pile. Assuming Dexter and Raquel play optimally, compute the number of positive integers $N \le 2021$ where Dexter wins this game.
|
1010
|
[
{
"content": "Dexter and Raquel are playing a game with $N$ stones. Dexter goes first and takes one stone from the pile. After that, the players alternate turns and can take anywhere from $1$ to $x + 1$ stones from the pile, where $x$ is the number of stones the other player took on the turn immediately prior. The winner is the one to take the last stone from the pile. Assuming Dexter and Raquel play optimally, compute the number of positive integers $N \\le 2021$ where Dexter wins this game.",
"role": "user"
}
] |
math
|
Triangle $\triangle{ABC}$ is isosceles with $AB = AC$. Let the incircle of $\triangle{ABC}$ intersect $BC$ and $AC$ at $D$ and $E$ respectively. Let $F \neq A$ be the point such that $DF = DA$ and $EF = EA$. If $AF = 8$ and the circumradius of $\triangle{AED}$ is $5$, find the area of $\triangle{ABC}$.
|
40
|
[
{
"content": "Triangle $\\triangle{ABC}$ is isosceles with $AB = AC$. Let the incircle of $\\triangle{ABC}$ intersect $BC$ and $AC$ at $D$ and $E$ respectively. Let $F \\neq A$ be the point such that $DF = DA$ and $EF = EA$. If $AF = 8$ and the circumradius of $\\triangle{AED}$ is $5$, find the area of $\\triangle{ABC}$.",
"role": "user"
}
] |
math
|
Petya and Vasya play the following game. Petya conceives a polynomial $P(x)$ with integer coefficients. On each move, Vasya pays him a ruble and calls an integer $a$ of his choice, which has not yet been called by him. Petya has to reply with the number of distinct integer solutions of the equation $P(x) = a$. The game continues until Petya is forced to repeat an answer. What is the minimal amount of rubles Vasya must pay in order to win?
|
4
|
[
{
"content": "Petya and Vasya play the following game. Petya conceives a polynomial $P(x)$ with integer coefficients. On each move, Vasya pays him a ruble and calls an integer $a$ of his choice, which has not yet been called by him. Petya has to reply with the number of distinct integer solutions of the equation $P(x) = a$. The game continues until Petya is forced to repeat an answer. What is the minimal amount of rubles Vasya must pay in order to win?",
"role": "user"
}
] |
math
|
Determine the greatest power of $2$ that is a factor of $3^{15} + 3^{11} + 3^{6} + 1$.
|
64
|
[
{
"content": "Determine the greatest power of $2$ that is a factor of $3^{15} + 3^{11} + 3^{6} + 1$.",
"role": "user"
}
] |
math
|
For positive integers $n$, let $s(n)$ be the sum of the digits of $n$. Over all four-digit positive integers $n$, which value of $n$ maximizes the ratio $\frac{s(n)}{n}$?
|
1099
|
[
{
"content": "For positive integers $n$, let $s(n)$ be the sum of the digits of $n$. Over all four-digit positive integers $n$, which value of $n$ maximizes the ratio $\\frac{s(n)}{n}$?",
"role": "user"
}
] |
math
|
Suppose that $n$ persons meet in a meeting, and that each of the persons is acquainted with exactly $8$ others. Any two acquainted persons have exactly $4$ common acquaintances, and any two non-acquainted persons have exactly $2$ common acquaintances. Find all possible values of $n$.
|
21
|
[
{
"content": "Suppose that $n$ persons meet in a meeting, and that each of the persons is acquainted with exactly $8$ others. Any two acquainted persons have exactly $4$ common acquaintances, and any two non-acquainted persons have exactly $2$ common acquaintances. Find all possible values of $n$.",
"role": "user"
}
] |
math
|
How many ways can $8$ mutually non-attacking rooks be placed on a $9 \times 9$ chessboard so that all $8$ rooks are on squares of the same color?
(Two rooks are said to be attacking each other if they are placed in the same row or column of the board.)
|
40320
|
[
{
"content": "How many ways can $8$ mutually non-attacking rooks be placed on a $9 \\times 9$ chessboard so that all $8$ rooks are on squares of the same color?\n\n(Two rooks are said to be attacking each other if they are placed in the same row or column of the board.)",
"role": "user"
}
] |
math
|
Let $ABCDEFGHIJ$ be a regular 10-sided polygon inscribed in a circle with center $O$ and radius $5$. The diagonals $AD$ and $BE$ intersect at point $P$, and the diagonals $AH$ and $BI$ intersect at point $Q$. Calculate the length of segment $PQ$.
|
5
|
[
{
"content": "Let $ABCDEFGHIJ$ be a regular 10-sided polygon inscribed in a circle with center $O$ and radius $5$. The diagonals $AD$ and $BE$ intersect at point $P$, and the diagonals $AH$ and $BI$ intersect at point $Q$. Calculate the length of segment $PQ$.",
"role": "user"
}
] |
math
|
Three boxes contain 600 balls each. The first box contains 600 identical red balls, the second box contains 600 identical white balls, and the third box contains 600 identical blue balls. From these three boxes, 900 balls are chosen. In how many ways can the balls be chosen? For example, one can choose 250 red balls, 187 white balls, and 463 blue balls, or one can choose 360 red balls and 540 blue balls.
|
270901
|
[
{
"content": "Three boxes contain 600 balls each. The first box contains 600 identical red balls, the second box contains 600 identical white balls, and the third box contains 600 identical blue balls. From these three boxes, 900 balls are chosen. In how many ways can the balls be chosen? For example, one can choose 250 red balls, 187 white balls, and 463 blue balls, or one can choose 360 red balls and 540 blue balls.",
"role": "user"
}
] |
math
|
Let $A$, $B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $\triangle ABC$ is an equilateral triangle with each side measuring $86$ meters. A boy swam from $A$ directly towards $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after covering a distance of $y$ meters. If $x$ and $y$ are both positive integers, determine $y$.
|
12
|
[
{
"content": "Let $A$, $B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $\\triangle ABC$ is an equilateral triangle with each side measuring $86$ meters. A boy swam from $A$ directly towards $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after covering a distance of $y$ meters. If $x$ and $y$ are both positive integers, determine $y$.",
"role": "user"
}
] |
math
|
Let $A$, $B$, $C$, and $D$ be points on a circle such that $AB = 11$ and $CD = 19$. Point $P$ is on segment $AB$ with $AP = 6$, and $Q$ is on segment $CD$ with $CQ = 7$. The line through $P$ and $Q$ intersects the circle at $X$ and $Y$. If $PQ = 27$, find $XY$.
|
31
|
[
{
"content": "Let $A$, $B$, $C$, and $D$ be points on a circle such that $AB = 11$ and $CD = 19$. Point $P$ is on segment $AB$ with $AP = 6$, and $Q$ is on segment $CD$ with $CQ = 7$. The line through $P$ and $Q$ intersects the circle at $X$ and $Y$. If $PQ = 27$, find $XY$.",
"role": "user"
}
] |
math
|
There are $2023$ distinct points on a plane, which are colored either white or red. For each white point, there are exactly two red points such that the distance from the white point to each of these red points is $2023$. Find the minimum number of red points.
|
45
|
[
{
"content": "There are $2023$ distinct points on a plane, which are colored either white or red. For each white point, there are exactly two red points such that the distance from the white point to each of these red points is $2023$. Find the minimum number of red points.",
"role": "user"
}
] |
math
|
Determine the largest $k$ such that for all competitive graphs with $2019$ vertices, if the difference between the in-degree and out-degree of any vertex is less than or equal to $k$, then this graph is strongly connected.
|
1
|
[
{
"content": "Determine the largest $k$ such that for all competitive graphs with $2019$ vertices, if the difference between the in-degree and out-degree of any vertex is less than or equal to $k$, then this graph is strongly connected.",
"role": "user"
}
] |
math
|
Given a regular tetrahedron with edge $a$, its edges are divided into $n$ equal segments, thus obtaining $n + 1$ points: $2$ at the ends and $n - 1$ inside. The following set of planes is considered:
- Those that contain the faces of the tetrahedron, and
- Each of the planes parallel to a face of the tetrahedron and containing at least one of the points determined above.
Now all those points $P$ that belong (simultaneously) to four planes of that set are considered. Determine the smallest positive natural $n$ so that among those points $P$ the eight vertices of a square-based rectangular parallelepiped can be chosen.
|
3
|
[
{
"content": "Given a regular tetrahedron with edge $a$, its edges are divided into $n$ equal segments, thus obtaining $n + 1$ points: $2$ at the ends and $n - 1$ inside. The following set of planes is considered:\n- Those that contain the faces of the tetrahedron, and\n- Each of the planes parallel to a face of the tetrahedron and containing at least one of the points determined above.\n\nNow all those points $P$ that belong (simultaneously) to four planes of that set are considered. Determine the smallest positive natural $n$ so that among those points $P$ the eight vertices of a square-based rectangular parallelepiped can be chosen.",
"role": "user"
}
] |
math
|
For some positive integer $n > m$, it turns out that $n$ can be represented as a sum of $2021$ non-negative integer powers of $m$, and also as a sum of $2021$ non-negative integer powers of $m+1$. Find the maximal value of the positive integer $m$.
|
2021
|
[
{
"content": "For some positive integer $n > m$, it turns out that $n$ can be represented as a sum of $2021$ non-negative integer powers of $m$, and also as a sum of $2021$ non-negative integer powers of $m+1$. Find the maximal value of the positive integer $m$.",
"role": "user"
}
] |
math
|
Suppose integers $a < b < c$ satisfy the following conditions:
\[
a + b + c = 95 \\
a^2 + b^2 + c^2 = 3083
\]
Find the value of $c$.
|
37
|
[
{
"content": "Suppose integers $a < b < c$ satisfy the following conditions:\n\\[\n a + b + c = 95 \\\\\n a^2 + b^2 + c^2 = 3083\n\\]\nFind the value of $c$.",
"role": "user"
}
] |
math
|
Let $a$, $b$, $c$ be real numbers such that $a^2 - 2 = 3b - c$, $b^2 + 4 = 3 + a$, and $c^2 + 4 = 3a - b$. Find $a^4 + b^4 + c^4$.
|
18
|
[
{
"content": "Let $a$, $b$, $c$ be real numbers such that $a^2 - 2 = 3b - c$, $b^2 + 4 = 3 + a$, and $c^2 + 4 = 3a - b$. Find $a^4 + b^4 + c^4$.",
"role": "user"
}
] |
math
|
Find the smallest natural number $n$ such that the expression $n^2 - n + 11$ has exactly four prime factors, which may not be distinct.
|
132
|
[
{
"content": "Find the smallest natural number $n$ such that the expression $n^2 - n + 11$ has exactly four prime factors, which may not be distinct.",
"role": "user"
}
] |
math
|
Let $P$ be a regular $2006$-gon. A diagonal is called *good* if its endpoints divide the boundary of $P$ into two parts, each composed of an odd number of sides of $P$. The sides of $P$ are also called *good*.
Suppose $P$ has been dissected into triangles by $2003$ diagonals, no two of which have a common point in the interior of $P$. Find the maximum number of isosceles triangles having two good sides that could appear in such a configuration.
|
1003
|
[
{
"content": "Let $P$ be a regular $2006$-gon. A diagonal is called *good* if its endpoints divide the boundary of $P$ into two parts, each composed of an odd number of sides of $P$. The sides of $P$ are also called *good*.\n\nSuppose $P$ has been dissected into triangles by $2003$ diagonals, no two of which have a common point in the interior of $P$. Find the maximum number of isosceles triangles having two good sides that could appear in such a configuration.",
"role": "user"
}
] |
math
|
Byan is playing a game called "raven, raven, falcon" with his three friends. His friends sit in a circle, and Byan walks clockwise around them, tapping each friend he passes on the head and saying either "raven" or "falcon," each with a probability of $\frac{1}{2}$. The game ends when Byan has said "falcon" twice. The probability that one of his friends will be called a "falcon" twice can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $100m + n$.
|
107
|
[
{
"content": "Byan is playing a game called \"raven, raven, falcon\" with his three friends. His friends sit in a circle, and Byan walks clockwise around them, tapping each friend he passes on the head and saying either \"raven\" or \"falcon,\" each with a probability of $\\frac{1}{2}$. The game ends when Byan has said \"falcon\" twice. The probability that one of his friends will be called a \"falcon\" twice can be expressed as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $100m + n$.",
"role": "user"
}
] |
math
|
The numbers $1, 2, \ldots, 10000$ are placed in the cells of a $100 \times 100$ square grid, each number appearing exactly once. In this arrangement, numbers that differ by $1$ are located in adjacent cells. We then calculate the distances between the centers of every pair of cells where the numbers differ by exactly $5000$. Let $S$ be the minimum of these distances. What is the largest value $S$ can take?
|
50
|
[
{
"content": "The numbers $1, 2, \\ldots, 10000$ are placed in the cells of a $100 \\times 100$ square grid, each number appearing exactly once. In this arrangement, numbers that differ by $1$ are located in adjacent cells. We then calculate the distances between the centers of every pair of cells where the numbers differ by exactly $5000$. Let $S$ be the minimum of these distances. What is the largest value $S$ can take?",
"role": "user"
}
] |
math
|
An arrangement of 12 real numbers in a row is called *good* if for any four consecutive numbers, the arithmetic mean of the first and last numbers is equal to the product of the two middle numbers. Determine how many good arrangements exist in which the first and last numbers are 1, and the second number is the same as the third.
|
45
|
[
{
"content": "An arrangement of 12 real numbers in a row is called *good* if for any four consecutive numbers, the arithmetic mean of the first and last numbers is equal to the product of the two middle numbers. Determine how many good arrangements exist in which the first and last numbers are 1, and the second number is the same as the third.",
"role": "user"
}
] |
math
|
Find all odd natural numbers $n$ such that $d(n)$ is the largest divisor of the number $n$ different from $n$. Here, $d(n)$ represents the number of divisors of the number $n$, including $1$ and $n$.
|
9
|
[
{
"content": "Find all odd natural numbers $n$ such that $d(n)$ is the largest divisor of the number $n$ different from $n$. Here, $d(n)$ represents the number of divisors of the number $n$, including $1$ and $n$.",
"role": "user"
}
] |
math
|
Let's consider a set of distinct positive integers with a sum equal to 2023. Among these integers, there are a total of $d$ even numbers and $m$ odd numbers. Determine the maximum possible value of $2d + 4m$.
|
200
|
[
{
"content": "Let's consider a set of distinct positive integers with a sum equal to 2023. Among these integers, there are a total of $d$ even numbers and $m$ odd numbers. Determine the maximum possible value of $2d + 4m$.",
"role": "user"
}
] |
math
|
It is given that $x = -2272$, $y = 10^3 + 10^2c + 10b + a$, and $z = 1$ satisfy the equation $ax + by + cz = 1$, where $a, b, c$ are positive integers with $a < b < c$. Find $y$.
|
1987
|
[
{
"content": "It is given that $x = -2272$, $y = 10^3 + 10^2c + 10b + a$, and $z = 1$ satisfy the equation $ax + by + cz = 1$, where $a, b, c$ are positive integers with $a < b < c$. Find $y$.",
"role": "user"
}
] |
math
|
Let $f(x)$ be a quadratic polynomial such that $f(f(1)) = f(-f(-1)) = 0$ and $f(1) \neq -f(-1)$. Suppose furthermore that the quadratic $2f(x)$ has coefficients that are nonzero integers. Find $f(0)$.
|
3
|
[
{
"content": "Let $f(x)$ be a quadratic polynomial such that $f(f(1)) = f(-f(-1)) = 0$ and $f(1) \\neq -f(-1)$. Suppose furthermore that the quadratic $2f(x)$ has coefficients that are nonzero integers. Find $f(0)$.",
"role": "user"
}
] |
math
|
Find the smallest positive integer $k$ such that $k + 25973$ is a palindrome, a number that reads the same forward and backward.
|
89
|
[
{
"content": "Find the smallest positive integer $k$ such that $k + 25973$ is a palindrome, a number that reads the same forward and backward.",
"role": "user"
}
] |
math
|
Find all values of the positive integer $m$ such that there exist polynomials $P(x)$, $Q(x)$, and $R(x,y)$ with real coefficients satisfying the condition: For every pair of real numbers $a$ and $b$ satisfying $a^m - b^2 = 0$, we always have $P(R(a,b)) = a$ and $Q(R(a,b)) = b$.
|
1
|
[
{
"content": "Find all values of the positive integer $m$ such that there exist polynomials $P(x)$, $Q(x)$, and $R(x,y)$ with real coefficients satisfying the condition: For every pair of real numbers $a$ and $b$ satisfying $a^m - b^2 = 0$, we always have $P(R(a,b)) = a$ and $Q(R(a,b)) = b$.",
"role": "user"
}
] |
math
|
Find the smallest positive integer $n$ such that the decimal representation of $n!(n+1)!(2n+1)! - 1$ has its last 30 digits all equal to 9.
|
34
|
[
{
"content": "Find the smallest positive integer $n$ such that the decimal representation of $n!(n+1)!(2n+1)! - 1$ has its last 30 digits all equal to 9.",
"role": "user"
}
] |
math
|
Let the number of ways for a rook to return to its original square on a $4 \times 4$ chessboard in 8 moves, starting from a corner, be $k$. Find the number of positive integers that are divisors of $k$. A "move" counts as shifting the rook by a positive number of squares on the board along a row or column. Note that the rook may return back to its original square during an intermediate step within its 8-move path.
|
4
|
[
{
"content": "Let the number of ways for a rook to return to its original square on a $4 \\times 4$ chessboard in 8 moves, starting from a corner, be $k$. Find the number of positive integers that are divisors of $k$. A \"move\" counts as shifting the rook by a positive number of squares on the board along a row or column. Note that the rook may return back to its original square during an intermediate step within its 8-move path.",
"role": "user"
}
] |
math
|
\( \triangle ABC \) is an isosceles triangle with \( AB = AC \). Point \( P \) is located inside \( \triangle ABC \) such that \( \angle BCP = 30^\circ \), \( \angle APB = 150^\circ \), and \( \angle CAP = 39^\circ \). Find \( \angle BAP \).
|
13
|
[
{
"content": "\\( \\triangle ABC \\) is an isosceles triangle with \\( AB = AC \\). Point \\( P \\) is located inside \\( \\triangle ABC \\) such that \\( \\angle BCP = 30^\\circ \\), \\( \\angle APB = 150^\\circ \\), and \\( \\angle CAP = 39^\\circ \\). Find \\( \\angle BAP \\).",
"role": "user"
}
] |
math
|
Five congruent circles have centers at the vertices of a regular pentagon so that each of the circles is tangent to its two neighbors. A sixth circle, congruent to the other five, is placed tangent to two of the five. If this sixth circle is allowed to roll without slipping around the exterior of the figure formed by the other five circles, then it will turn through an angle of $k$ degrees before it returns to its starting position. Find $k$.
|
720
|
[
{
"content": "Five congruent circles have centers at the vertices of a regular pentagon so that each of the circles is tangent to its two neighbors. A sixth circle, congruent to the other five, is placed tangent to two of the five. If this sixth circle is allowed to roll without slipping around the exterior of the figure formed by the other five circles, then it will turn through an angle of $k$ degrees before it returns to its starting position. Find $k$.",
"role": "user"
}
] |
math
|
The thousands digit of a five-digit number which is divisible by $37$ and $173$ is $3$. What is the hundreds digit of this number?
\[
\textbf{a)}\ 0 \quad \textbf{b)}\ 2 \quad \textbf{c)}\ 4 \quad \textbf{d)}\ 6 \quad \textbf{e)}\ 8
\]
|
2
|
[
{
"content": "The thousands digit of a five-digit number which is divisible by $37$ and $173$ is $3$. What is the hundreds digit of this number?\n\n\\[\n\\textbf{a)}\\ 0 \\quad \\textbf{b)}\\ 2 \\quad \\textbf{c)}\\ 4 \\quad \\textbf{d)}\\ 6 \\quad \\textbf{e)}\\ 8\n\\]",
"role": "user"
}
] |
math
|
Let $f(n)$ be a function satisfying the following three conditions for all positive integers $n$:
1. $f(n)$ is a positive integer.
2. $f(n + 1) > f(n)$.
3. $f(f(n)) = 3n$.
Find $f(2001)$.
|
3816
|
[
{
"content": "Let $f(n)$ be a function satisfying the following three conditions for all positive integers $n$:\n\n1. $f(n)$ is a positive integer.\n2. $f(n + 1) > f(n)$.\n3. $f(f(n)) = 3n$.\n\nFind $f(2001)$.",
"role": "user"
}
] |
math
|
Let $ABC$ be a triangle in the $xy$ plane, where $B$ is at the origin $(0,0)$. Extend $BC$ to $D$ such that $BC: CD = 1:1$, extend $CA$ to $E$ such that $CA: AE = 1:2$, and extend $AB$ to $F$ such that $AB: BF = 1:3$. Let $G(32,24)$ be the centroid of the triangle $ABC$ and $K$ be the centroid of the triangle $DEF$. Find the length $GK$.
|
40
|
[
{
"content": "Let $ABC$ be a triangle in the $xy$ plane, where $B$ is at the origin $(0,0)$. Extend $BC$ to $D$ such that $BC: CD = 1:1$, extend $CA$ to $E$ such that $CA: AE = 1:2$, and extend $AB$ to $F$ such that $AB: BF = 1:3$. Let $G(32,24)$ be the centroid of the triangle $ABC$ and $K$ be the centroid of the triangle $DEF$. Find the length $GK$.",
"role": "user"
}
] |
math
|
Real nonzero numbers $x, y, z$ satisfy $x + y + z = 0$. Furthermore, it is given that:
\[
A = \frac{x}{y} + \frac{y}{z} + \frac{z}{x} = \frac{x}{z} + \frac{z}{y} + \frac{y}{x} + 1
\]
Determine the value of $A$.
|
-1
|
[
{
"content": "Real nonzero numbers $x, y, z$ satisfy $x + y + z = 0$. Furthermore, it is given that:\n\\[\nA = \\frac{x}{y} + \\frac{y}{z} + \\frac{z}{x} = \\frac{x}{z} + \\frac{z}{y} + \\frac{y}{x} + 1\n\\]\nDetermine the value of $A$.",
"role": "user"
}
] |
math
|
Find the sum of all positive integers $x$ such that $3 \times 2^x = n^2 - 1$ for some positive integer $n$.
|
7
|
[
{
"content": "Find the sum of all positive integers $x$ such that $3 \\times 2^x = n^2 - 1$ for some positive integer $n$.",
"role": "user"
}
] |
math
|
Let $x$, $y$, $z$ be positive real numbers satisfying the simultaneous equations:
\[
\begin{align*}
x(y^2+yz+z^2)&=3y+10z,\\
y(z^2+zx+x^2)&=21z+24x,\\
z(x^2+xy+y^2)&=7x+28y.
\end{align*}
\]
Find $xy+yz+zx$.
|
31
|
[
{
"content": "Let $x$, $y$, $z$ be positive real numbers satisfying the simultaneous equations:\n\\[\n\\begin{align*}\nx(y^2+yz+z^2)&=3y+10z,\\\\\ny(z^2+zx+x^2)&=21z+24x,\\\\\nz(x^2+xy+y^2)&=7x+28y.\n\\end{align*}\n\\]\nFind $xy+yz+zx$.",
"role": "user"
}
] |
math
|
Given a pentagon $ABCDE$ with the following properties:
1. Angles $\angle A$ and $\angle C$ are right angles.
2. The side lengths are $|AB| = 4$, $|BC| = 5$, $|CD| = 10$, and $|DE| = 6$.
3. Point $C'$ is the reflection of point $C$ across line $BD$ and lies on line segment $AE$.
Find the measure of angle $\angle E$.
|
90
|
[
{
"content": "Given a pentagon $ABCDE$ with the following properties:\n\n1. Angles $\\angle A$ and $\\angle C$ are right angles.\n2. The side lengths are $|AB| = 4$, $|BC| = 5$, $|CD| = 10$, and $|DE| = 6$.\n3. Point $C'$ is the reflection of point $C$ across line $BD$ and lies on line segment $AE$.\n\nFind the measure of angle $\\angle E$.",
"role": "user"
}
] |
math
|
Let $\triangle ABC$ have $AB=9$ and $AC=10$. A semicircle is inscribed in $\triangle ABC$ with its center on segment $BC$ such that it is tangent to $AB$ at point $D$ and $AC$ at point $E$. If $AD=2DB$ and $r$ is the radius of the semicircle, $r^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
|
415
|
[
{
"content": "Let $\\triangle ABC$ have $AB=9$ and $AC=10$. A semicircle is inscribed in $\\triangle ABC$ with its center on segment $BC$ such that it is tangent to $AB$ at point $D$ and $AC$ at point $E$. If $AD=2DB$ and $r$ is the radius of the semicircle, $r^2$ can be expressed as $\\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.",
"role": "user"
}
] |
math
|
A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.
Ada has been told to write down five haikus plus two more every hour. This means she needs to write down five in the first hour, seven in the second hour, nine in the third hour, and so on.
Ada has written forty haikus so far and writes down seven haikus every hour. At which hour after she begins will she not have enough haikus done?
|
4
|
[
{
"content": "A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.\n\nAda has been told to write down five haikus plus two more every hour. This means she needs to write down five in the first hour, seven in the second hour, nine in the third hour, and so on.\n\nAda has written forty haikus so far and writes down seven haikus every hour. At which hour after she begins will she not have enough haikus done?",
"role": "user"
}
] |
math
|
The quadrilateral $ABCD$ satisfies the following conditions: \( \angle ABC = \angle BCD = 150^{\circ} \). Additionally, \( AB = 18 \) and \( BC = 24 \). Equilateral triangles \( \triangle APB \), \( \triangle BQC \), and \( \triangle CRD \) are constructed outside the quadrilateral. If \( P(X) \) represents the perimeter of polygon \( X \), then the equation \( P(APQRD) = P(ABCD) + 32 \) holds. Determine the length of side \( CD \).
|
10
|
[
{
"content": "The quadrilateral $ABCD$ satisfies the following conditions: \\( \\angle ABC = \\angle BCD = 150^{\\circ} \\). Additionally, \\( AB = 18 \\) and \\( BC = 24 \\). Equilateral triangles \\( \\triangle APB \\), \\( \\triangle BQC \\), and \\( \\triangle CRD \\) are constructed outside the quadrilateral. If \\( P(X) \\) represents the perimeter of polygon \\( X \\), then the equation \\( P(APQRD) = P(ABCD) + 32 \\) holds. Determine the length of side \\( CD \\).",
"role": "user"
}
] |
math
|
We are given a coin of diameter $\frac{1}{2}$ and a checkerboard of $1 \times 1$ squares with an area of $2010 \times 2010$. We toss the coin such that it lands completely on the checkerboard. If the probability that the coin doesn't touch any of the lattice lines is $\frac{a^2}{b^2}$, where $\frac{a}{b}$ is a reduced fraction, find $a + b$.
|
6029
|
[
{
"content": "We are given a coin of diameter $\\frac{1}{2}$ and a checkerboard of $1 \\times 1$ squares with an area of $2010 \\times 2010$. We toss the coin such that it lands completely on the checkerboard. If the probability that the coin doesn't touch any of the lattice lines is $\\frac{a^2}{b^2}$, where $\\frac{a}{b}$ is a reduced fraction, find $a + b$.",
"role": "user"
}
] |
math
|
Let $ABC$ be an acute-angled triangle. Construct points $A'$, $B'$, $C'$ on its sides $BC$, $CA$, $AB$ such that:
- $A'B' \parallel AB$,
- $C'C$ is the bisector of angle $A'C'B'$,
- $A'C' + B'C' = AB$.
|
1
|
[
{
"content": "Let $ABC$ be an acute-angled triangle. Construct points $A'$, $B'$, $C'$ on its sides $BC$, $CA$, $AB$ such that:\n- $A'B' \\parallel AB$, \n- $C'C$ is the bisector of angle $A'C'B'$, \n- $A'C' + B'C' = AB$.",
"role": "user"
}
] |
math
|
Using each of the digits $1, 2, 3, \ldots, 8, 9$ exactly once, we form nine, not necessarily distinct, nine-digit numbers. Their sum ends in $n$ zeroes, where $n$ is a non-negative integer. Determine the maximum possible value of $n$.
|
8
|
[
{
"content": "Using each of the digits $1, 2, 3, \\ldots, 8, 9$ exactly once, we form nine, not necessarily distinct, nine-digit numbers. Their sum ends in $n$ zeroes, where $n$ is a non-negative integer. Determine the maximum possible value of $n$.",
"role": "user"
}
] |
math
|
Determine the largest positive integer $n$ such that the following statement holds:
If $a_1, a_2, a_3, a_4, a_5, a_6$ are six distinct positive integers less than or equal to $n$, then there exist three distinct positive integers from these six, say $a$, $b$, and $c$, such that $ab > c$, $bc > a$, and $ca > b$.
|
107
|
[
{
"content": "Determine the largest positive integer $n$ such that the following statement holds:\n\nIf $a_1, a_2, a_3, a_4, a_5, a_6$ are six distinct positive integers less than or equal to $n$, then there exist three distinct positive integers from these six, say $a$, $b$, and $c$, such that $ab > c$, $bc > a$, and $ca > b$.",
"role": "user"
}
] |
math
|
Solve for $x$:
\[
v - w + x - y + z = 79 \\
v + w + x + y + z = -1 \\
v + 2w + 4x + 8y + 16z = -2 \\
v + 3w + 9x + 27y + 81z = -1 \\
v + 5w + 25x + 125y + 625z = 79.
\]
|
24
|
[
{
"content": "Solve for $x$:\n\\[\nv - w + x - y + z = 79 \\\\\nv + w + x + y + z = -1 \\\\\nv + 2w + 4x + 8y + 16z = -2 \\\\\nv + 3w + 9x + 27y + 81z = -1 \\\\\nv + 5w + 25x + 125y + 625z = 79.\n\\]",
"role": "user"
}
] |
math
|
You have a 4-digit whole number that is a perfect square. Another number is constructed by making the following changes to the digits of the original number:
1. Add 1 to the unit's digit.
2. Subtract 1 from the ten's digit.
3. Add 1 to the hundred's digit.
4. Subtract 1 from the thousand's digit.
If the resulting number is also a perfect square, find the original number. Is it unique?
|
3025
|
[
{
"content": "You have a 4-digit whole number that is a perfect square. Another number is constructed by making the following changes to the digits of the original number:\n\n1. Add 1 to the unit's digit.\n2. Subtract 1 from the ten's digit.\n3. Add 1 to the hundred's digit.\n4. Subtract 1 from the thousand's digit.\n\nIf the resulting number is also a perfect square, find the original number. Is it unique?",
"role": "user"
}
] |
math
|
Three 12 cm $\times$ 12 cm squares are each cut into two pieces $A$ and $B$, as shown in the first figure below, by joining the midpoints of two adjacent sides. These six pieces are then attached to a regular hexagon, as shown in the second figure, so as to fold into a polyhedron. What is the volume (in $\text{cm}^3$) of this polyhedron?
|
864
|
[
{
"content": "Three 12 cm $\\times$ 12 cm squares are each cut into two pieces $A$ and $B$, as shown in the first figure below, by joining the midpoints of two adjacent sides. These six pieces are then attached to a regular hexagon, as shown in the second figure, so as to fold into a polyhedron. What is the volume (in $\\text{cm}^3$) of this polyhedron?",
"role": "user"
}
] |
math
|
Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following $100$ $9$-digit integers:
\[
\begin{align*}
700 \cdot 712 \cdot 718 + 320,\\
701 \cdot 713 \cdot 719 + 320,\\
702 \cdot 714 \cdot 720 + 320,\\
&\vdots\\
798 \cdot 810 \cdot 816 + 320,\\
799 \cdot 811 \cdot 817 + 320.
\end{align*}
\]
She notes that two of them have exactly $8$ positive divisors each. Find the common prime divisor of those two integers.
|
7
|
[
{
"content": "Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following $100$ $9$-digit integers:\n\n\\[\n\\begin{align*}\n700 \\cdot 712 \\cdot 718 + 320,\\\\\n701 \\cdot 713 \\cdot 719 + 320,\\\\\n702 \\cdot 714 \\cdot 720 + 320,\\\\\n&\\vdots\\\\\n798 \\cdot 810 \\cdot 816 + 320,\\\\\n799 \\cdot 811 \\cdot 817 + 320.\n\\end{align*}\n\\]\n\nShe notes that two of them have exactly $8$ positive divisors each. Find the common prime divisor of those two integers.",
"role": "user"
}
] |
math
|
Given a regular octagon $ABCDEFGH$ with side length $3$. By drawing the four diagonals $AF$, $BE$, $CH$, and $DG$, the octagon is divided into a square, four triangles, and four rectangles. Find the sum of the areas of the square and the four triangles.
|
18
|
[
{
"content": "Given a regular octagon $ABCDEFGH$ with side length $3$. By drawing the four diagonals $AF$, $BE$, $CH$, and $DG$, the octagon is divided into a square, four triangles, and four rectangles. Find the sum of the areas of the square and the four triangles.",
"role": "user"
}
] |
math
|
In a bag, there are $1007$ black and $1007$ white balls, each randomly numbered from $1$ to $2014$. At each step, one ball is drawn and placed on the table. Additionally, if desired, two balls of different colors can be selected from the table and placed in a separate bag. This action earns points equal to the absolute difference of their numbers. Determine the maximum number of points that can be guaranteed after $2014$ steps.
|
1007
|
[
{
"content": "In a bag, there are $1007$ black and $1007$ white balls, each randomly numbered from $1$ to $2014$. At each step, one ball is drawn and placed on the table. Additionally, if desired, two balls of different colors can be selected from the table and placed in a separate bag. This action earns points equal to the absolute difference of their numbers. Determine the maximum number of points that can be guaranteed after $2014$ steps.",
"role": "user"
}
] |
math
|
In the country Máxico, there are two islands: the island "Mayor" and the island "Menor". The island "Mayor" has $k > 3$ states, with exactly $n > 3$ cities in each state. The island "Menor" has only one state with $31$ cities. "Aeropapantla" and "Aerocenzontle" are the airlines that offer flights in Máxico. "Aeropapantla" offers direct flights from every city in Máxico to any other city in Máxico. "Aerocenzontle" only offers direct flights from every city on the island "Mayor" to any other city on the island "Mayor".
If the percentage of flights that connect two cities in the same state is the same for the flights of each airline, what is the least number of cities that can be on the island "Mayor"?
|
65
|
[
{
"content": "In the country Máxico, there are two islands: the island \"Mayor\" and the island \"Menor\". The island \"Mayor\" has $k > 3$ states, with exactly $n > 3$ cities in each state. The island \"Menor\" has only one state with $31$ cities. \"Aeropapantla\" and \"Aerocenzontle\" are the airlines that offer flights in Máxico. \"Aeropapantla\" offers direct flights from every city in Máxico to any other city in Máxico. \"Aerocenzontle\" only offers direct flights from every city on the island \"Mayor\" to any other city on the island \"Mayor\".\n\nIf the percentage of flights that connect two cities in the same state is the same for the flights of each airline, what is the least number of cities that can be on the island \"Mayor\"?",
"role": "user"
}
] |
math
|
Several cells are marked in a $100 \times 100$ table. Vasya wants to split the square into several rectangles such that each rectangle does not contain more than two marked cells, and there are at most $k$ rectangles containing less than two cells. What is the smallest $k$ such that Vasya will certainly be able to do this?
|
100
|
[
{
"content": "Several cells are marked in a $100 \\times 100$ table. Vasya wants to split the square into several rectangles such that each rectangle does not contain more than two marked cells, and there are at most $k$ rectangles containing less than two cells. What is the smallest $k$ such that Vasya will certainly be able to do this?",
"role": "user"
}
] |
math
|
Kermit the frog enjoys hopping around the infinite square grid in his backyard. It takes him $1$ Joule of energy to hop one step north or one step south, and $1$ Joule of energy to hop one step east or one step west. He wakes up one morning on the grid with $100$ Joules of energy, and hops till he falls asleep with $0$ energy. How many different places could he have gone to sleep?
|
10201
|
[
{
"content": "Kermit the frog enjoys hopping around the infinite square grid in his backyard. It takes him $1$ Joule of energy to hop one step north or one step south, and $1$ Joule of energy to hop one step east or one step west. He wakes up one morning on the grid with $100$ Joules of energy, and hops till he falls asleep with $0$ energy. How many different places could he have gone to sleep?",
"role": "user"
}
] |
math
|
In a chemistry experiment, a tube contains 100 particles, 68 on the right and 32 on the left. Each second, if there are $a$ particles on the left side of the tube, some number $n$ of these particles move to the right side, where $n \in \{0,1,\dots,a\}$ is chosen uniformly at random. Similarly, some number of the particles from the right side of the tube move to the left at the same time. The experiment ends at the moment when all particles are on the same side of the tube. The probability that all particles end on the left side is $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a+b$.
|
102
|
[
{
"content": "In a chemistry experiment, a tube contains 100 particles, 68 on the right and 32 on the left. Each second, if there are $a$ particles on the left side of the tube, some number $n$ of these particles move to the right side, where $n \\in \\{0,1,\\dots,a\\}$ is chosen uniformly at random. Similarly, some number of the particles from the right side of the tube move to the left at the same time. The experiment ends at the moment when all particles are on the same side of the tube. The probability that all particles end on the left side is $\\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a+b$.",
"role": "user"
}
] |
math
|
30 volleyball teams have participated in a league. Any two teams have played a match with each other exactly once. At the end of the league, a match is called **unusual** if the winner of the match has fewer wins than the loser. A team is called **astonishing** if all its matches are **unusual** matches. Find the maximum number of **astonishing** teams.
|
1
|
[
{
"content": "30 volleyball teams have participated in a league. Any two teams have played a match with each other exactly once. At the end of the league, a match is called **unusual** if the winner of the match has fewer wins than the loser. A team is called **astonishing** if all its matches are **unusual** matches. Find the maximum number of **astonishing** teams.",
"role": "user"
}
] |
math
|
In the arrow-shaped polygon (refer to the provided figure), the angles at vertices $A$, $C$, $D$, $E$, and $F$ are right angles. The side lengths are given as follows: $BC = FG = 5$, $CD = FE = 20$, $DE = 10$, and $AB = AG$. Determine the area of the polygon that is closest to one of the following options:
\[ \textbf{(A)}\ 288 \quad \textbf{(B)}\ 291 \quad \textbf{(C)}\ 294 \quad \textbf{(D)}\ 297 \quad \textbf{(E)}\ 300 \]
|
300
|
[
{
"content": "In the arrow-shaped polygon (refer to the provided figure), the angles at vertices $A$, $C$, $D$, $E$, and $F$ are right angles. The side lengths are given as follows: $BC = FG = 5$, $CD = FE = 20$, $DE = 10$, and $AB = AG$. Determine the area of the polygon that is closest to one of the following options:\n\n\\[ \\textbf{(A)}\\ 288 \\quad \\textbf{(B)}\\ 291 \\quad \\textbf{(C)}\\ 294 \\quad \\textbf{(D)}\\ 297 \\quad \\textbf{(E)}\\ 300 \\]",
"role": "user"
}
] |
math
|
For a positive integer $n$, let $p(n)$ denote the smallest prime divisor of $n$. Find the maximum number of divisors $m$ can have if $p(m)^4 > m$.
|
6
|
[
{
"content": "For a positive integer $n$, let $p(n)$ denote the smallest prime divisor of $n$. Find the maximum number of divisors $m$ can have if $p(m)^4 > m$.",
"role": "user"
}
] |
math
|
Find the smallest positive integer $n$ such that there are at least three distinct ordered pairs $(x,y)$ of positive integers satisfying the equation $x^2 - y^2 = n$.
|
45
|
[
{
"content": "Find the smallest positive integer $n$ such that there are at least three distinct ordered pairs $(x,y)$ of positive integers satisfying the equation $x^2 - y^2 = n$.",
"role": "user"
}
] |
math
|
Let $a, b, c$ be positive integers such that $a, b, c, a+b-c, a+c-b, b+c-a, a+b+c$ are 7 distinct primes. The sum of two of $a, b, c$ is 800. If $d$ is the difference between the largest prime and the smallest prime among these 7 primes, find the maximum value of $d$.
|
1594
|
[
{
"content": "Let $a, b, c$ be positive integers such that $a, b, c, a+b-c, a+c-b, b+c-a, a+b+c$ are 7 distinct primes. The sum of two of $a, b, c$ is 800. If $d$ is the difference between the largest prime and the smallest prime among these 7 primes, find the maximum value of $d$.",
"role": "user"
}
] |
math
|
10 students are arranged in a row. Every minute, a new student is inserted in the row (which can occur in the front and in the back as well, hence $11$ possible places) with a uniform $\frac{1}{11}$ probability of each location. Then, either the frontmost or the backmost student is removed from the row (each with a $\frac{1}{2}$ probability).
Suppose you are the eighth in the line from the front. The probability that you exit the row from the front rather than the back is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $100m+n$.
|
828
|
[
{
"content": "10 students are arranged in a row. Every minute, a new student is inserted in the row (which can occur in the front and in the back as well, hence $11$ possible places) with a uniform $\\frac{1}{11}$ probability of each location. Then, either the frontmost or the backmost student is removed from the row (each with a $\\frac{1}{2}$ probability). \n\nSuppose you are the eighth in the line from the front. The probability that you exit the row from the front rather than the back is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $100m+n$.",
"role": "user"
}
] |
math
|
On a semicircle with diameter $|AB|=d$, there are points $C$ and $D$ such that $|BC|=|CD|=a$ and $|DA|=b$, where $a$, $b$, and $d$ are different positive integers. Find the minimum possible value of $d$.
|
8
|
[
{
"content": "On a semicircle with diameter $|AB|=d$, there are points $C$ and $D$ such that $|BC|=|CD|=a$ and $|DA|=b$, where $a$, $b$, and $d$ are different positive integers. Find the minimum possible value of $d$.",
"role": "user"
}
] |
math
|
Suppose in the plane, there are 10 pairwise nonparallel lines intersecting one another. What is the maximum possible number of polygons (with finite areas) that can be formed?
|
36
|
[
{
"content": "Suppose in the plane, there are 10 pairwise nonparallel lines intersecting one another. What is the maximum possible number of polygons (with finite areas) that can be formed?",
"role": "user"
}
] |
math
|
The average age of the participants in a mathematics competition (gymnasts and high school students) increases by exactly one month if three high school age students $18$ years each are included in the competition or if three gymnasts aged $12$ years each are excluded from the competition. How many participants were initially in the contest?
|
108
|
[
{
"content": "The average age of the participants in a mathematics competition (gymnasts and high school students) increases by exactly one month if three high school age students $18$ years each are included in the competition or if three gymnasts aged $12$ years each are excluded from the competition. How many participants were initially in the contest?",
"role": "user"
}
] |
math
|
Let $\triangle ABC$ be a triangle with $AB=65$, $BC=70$, and $CA=75$. A semicircle $\Gamma$ with diameter $\overline{BC}$ is constructed outside the triangle. Suppose there exists a circle $\omega$ tangent to $AB$ and $AC$ and furthermore internally tangent to $\Gamma$ at a point $X$. The length $AX$ can be written in the form $m\sqrt{n}$ where $m$ and $n$ are positive integers with $n$ not divisible by the square of any prime. Find $m+n$.
|
76
|
[
{
"content": "Let $\\triangle ABC$ be a triangle with $AB=65$, $BC=70$, and $CA=75$. A semicircle $\\Gamma$ with diameter $\\overline{BC}$ is constructed outside the triangle. Suppose there exists a circle $\\omega$ tangent to $AB$ and $AC$ and furthermore internally tangent to $\\Gamma$ at a point $X$. The length $AX$ can be written in the form $m\\sqrt{n}$ where $m$ and $n$ are positive integers with $n$ not divisible by the square of any prime. Find $m+n$.",
"role": "user"
}
] |
math
|
A herder has forgotten the number of cows she has and does not want to count all of them. She remembers these four facts about the number of cows:
- It has $3$ digits.
- It is a palindrome.
- The middle digit is a multiple of $4$.
- It is divisible by $11$.
What is the sum of all possible numbers of cows that the herder has?
$\textbf{(A) }343 \ \textbf{(B) }494 \ \textbf{(C) }615 \ \textbf{(D) }635 \ \textbf{(E) }726$
|
726
|
[
{
"content": "A herder has forgotten the number of cows she has and does not want to count all of them. She remembers these four facts about the number of cows:\n\n- It has $3$ digits.\n- It is a palindrome.\n- The middle digit is a multiple of $4$.\n- It is divisible by $11$.\n\nWhat is the sum of all possible numbers of cows that the herder has?\n\n$\\textbf{(A) }343 \\ \\textbf{(B) }494 \\ \\textbf{(C) }615 \\ \\textbf{(D) }635 \\ \\textbf{(E) }726$",
"role": "user"
}
] |
math
|
All six-digit natural numbers from $100000$ to $999999$ are written on the page in ascending order without spaces. What is the largest value of $k$ for which the same $k$-digit number can be found in at least two different places in this string?
|
6
|
[
{
"content": "All six-digit natural numbers from $100000$ to $999999$ are written on the page in ascending order without spaces. What is the largest value of $k$ for which the same $k$-digit number can be found in at least two different places in this string?",
"role": "user"
}
] |
math
|
Let $f(n)$ be a function that fulfills the following properties:
- For each natural $n$, $f(n)$ is an integer greater than or equal to $0$.
- $f(n) = 2010$, if $n$ ends in $7$. For example, $f(137) = 2010$.
- If $a$ is a divisor of $b$, then: $f\left(\frac{b}{a}\right) = |f(b) - f(a)|$.
Find $\displaystyle f(2009^{2009^{2009}})$ and justify your answer.
|
0
|
[
{
"content": "Let $f(n)$ be a function that fulfills the following properties:\n\n- For each natural $n$, $f(n)$ is an integer greater than or equal to $0$.\n- $f(n) = 2010$, if $n$ ends in $7$. For example, $f(137) = 2010$.\n- If $a$ is a divisor of $b$, then: $f\\left(\\frac{b}{a}\\right) = |f(b) - f(a)|$.\n\nFind $\\displaystyle f(2009^{2009^{2009}})$ and justify your answer.",
"role": "user"
}
] |
math
|
Let $n$ be a positive integer. Find as many zeros as possible for the last digits of the following expression: $1^n + 2^n + 3^n + 4^n$.
|
2
|
[
{
"content": "Let $n$ be a positive integer. Find as many zeros as possible for the last digits of the following expression: $1^n + 2^n + 3^n + 4^n$.",
"role": "user"
}
] |
math
|
Find the number of ordered triplets $(a, b, c)$ of positive integers such that $a < b < c$ and $abc = 2008$.
|
4
|
[
{
"content": "Find the number of ordered triplets $(a, b, c)$ of positive integers such that $a < b < c$ and $abc = 2008$.",
"role": "user"
}
] |
math
|
A large gathering of people stand in a triangular array with $2020$ rows, such that the first row has $1$ person, the second row has $2$ people, and so on. Every day, the people in each row infect all of the people adjacent to them in their own row. Additionally, the people at the ends of each row infect the people at the ends of the rows in front of and behind them that are at the same side of the row as they are. Given that two people are chosen at random to be infected with COVID at the beginning of day 1, what is the earliest possible day that the last uninfected person will be infected with COVID?
|
2019
|
[
{
"content": "A large gathering of people stand in a triangular array with $2020$ rows, such that the first row has $1$ person, the second row has $2$ people, and so on. Every day, the people in each row infect all of the people adjacent to them in their own row. Additionally, the people at the ends of each row infect the people at the ends of the rows in front of and behind them that are at the same side of the row as they are. Given that two people are chosen at random to be infected with COVID at the beginning of day 1, what is the earliest possible day that the last uninfected person will be infected with COVID?",
"role": "user"
}
] |
math
|
Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get?
|
32
|
[
{
"content": "Vitya cut the chessboard along the borders of the cells into pieces of the same perimeter. It turned out that not all of the received parts are equal. What is the largest possible number of parts that Vitya could get?",
"role": "user"
}
] |
math
|
Willy Wonka has $n$ distinguishable pieces of candy that he wants to split into groups. If the number of ways for him to do this is $p(n)$, then we have the following table:
\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
p(n) & 1 & 2 & 5 & 15 & 52 & 203 & 877 & 4140 & 21147 & 115975 \\
\hline
\end{array}
\]
Define a "splitting" of the $n$ distinguishable pieces of candy to be a way of splitting them into groups. If Willy Wonka has $8$ candies, what is the sum of the number of groups over all splittings he can use?
|
17007
|
[
{
"content": "Willy Wonka has $n$ distinguishable pieces of candy that he wants to split into groups. If the number of ways for him to do this is $p(n)$, then we have the following table:\n\n\\[\n\\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}\n\\hline\nn & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\\n\\hline\np(n) & 1 & 2 & 5 & 15 & 52 & 203 & 877 & 4140 & 21147 & 115975 \\\\\n\\hline\n\\end{array}\n\\]\n\nDefine a \"splitting\" of the $n$ distinguishable pieces of candy to be a way of splitting them into groups. If Willy Wonka has $8$ candies, what is the sum of the number of groups over all splittings he can use?",
"role": "user"
}
] |
math
|
In rectangle $ABCD$, diagonal $AC$ is intersected by the angle bisector from $B$ at point $B'$ and by the angle bisector from $D$ at point $D'$. Similarly, diagonal $BD$ is intersected by the angle bisector from $A$ at point $A'$ and by the angle bisector from $C$ at point $C'$. The area of quadrilateral $A'B'C'D'$ is $\frac{9}{16}$ of the area of rectangle $ABCD$. Determine the ratio of the longer side to the shorter side of rectangle $ABCD$.
|
7
|
[
{
"content": "In rectangle $ABCD$, diagonal $AC$ is intersected by the angle bisector from $B$ at point $B'$ and by the angle bisector from $D$ at point $D'$. Similarly, diagonal $BD$ is intersected by the angle bisector from $A$ at point $A'$ and by the angle bisector from $C$ at point $C'$. The area of quadrilateral $A'B'C'D'$ is $\\frac{9}{16}$ of the area of rectangle $ABCD$. Determine the ratio of the longer side to the shorter side of rectangle $ABCD$.",
"role": "user"
}
] |
math
|
Find the smallest positive $m$ such that there exist at least 11 even and 11 odd positive integers $n$ for which $\frac{n^3 + m}{n + 2}$ is an integer.
|
8
|
[
{
"content": "Find the smallest positive $m$ such that there exist at least 11 even and 11 odd positive integers $n$ for which $\\frac{n^3 + m}{n + 2}$ is an integer.",
"role": "user"
}
] |
math
|
A $8 \times 8$ board is given. Seven out of $64$ unit squares are painted black. Suppose that there exists a positive $k$ such that no matter which squares are black, there exists a rectangle (with sides parallel to the sides of the board) with area $k$ containing no black squares. Find the maximum value of $k$.
|
8
|
[
{
"content": "A $8 \\times 8$ board is given. Seven out of $64$ unit squares are painted black. Suppose that there exists a positive $k$ such that no matter which squares are black, there exists a rectangle (with sides parallel to the sides of the board) with area $k$ containing no black squares. Find the maximum value of $k$.",
"role": "user"
}
] |
math
|
End of preview. Expand
in Data Studio
Dolci RL-Zero Math
Dolci RL-Zero Math is a dataset of 13.3k math questions and answers for RLVR training of Olmo 3 RL-Zero Math 7B
This dataset was collected from a subset of DAPO Math and Klear Reasoner Math, see the Olmo 3 paper for details.
Downloading
You can download and load this data using HuggingFace's datasets library with the following code:
from datasets import load_dataset
dataset = load_dataset("allenai/dolci-rlzero-math-7b", split="train",)
Licensing Information
This dataset is licensed under ODC-BY. It is intended for research and educational use in accordance with Ai2's Responsible Use Guidelines.
Citation
Technical manuscript coming soon!
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Size of downloaded dataset files:
4.29 MB
Size of the auto-converted Parquet files:
4.29 MB
Number of rows:
13,314
Models trained or fine-tuned on allenai/Dolci-RLZero-Math-7B