This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 14842

2019 Balkan MO Shortlist, C1

100 couples are invited to a traditional Modolvan dance. The $200$ people stand in a line, and then in a $\textit{step}$, (not necessarily adjacent) many swap positions. Find the least $C$ such that whatever the initial order, they can arrive at an ordering where everyone is dancing next to their partner in at most $C$ steps.

1993 IMO, 3

On an infinite chessboard, a solitaire game is played as follows: at the start, we have $n^2$ pieces occupying a square of side $n.$ The only allowed move is to jump over an occupied square to an unoccupied one, and the piece which has been jumped over is removed. For which $n$ can the game end with only one piece remaining on the board?

2022 Girls in Math at Yale, R4

[b]p10 [/b]Kathy has two positive real numbers, $a$ and $b$. She mistakenly writes $$\log (a + b) = \log (a) + \log( b),$$ but miraculously, she finds that for her combination of $a$ and $b$, the equality holds. If $a = 2022b$, then $b = \frac{p}{q}$ , for positive integers $p, q$ where $gcd(p, q) = 1$. Find $p + q$. [b]p11[/b] Points $X$ and $Y$ lie on sides $AB$ and $BC$ of triangle $ABC$, respectively. Ray $\overrightarrow{XY}$ is extended to point $Z$ such that $A, C$, and $Z$ are collinear, in that order. If triangle$ ABZ$ is isosceles and triangle $CYZ$ is equilateral, then the possible values of $\angle ZXB$ lie in the interval $I = (a^o, b^o)$, such that $0 \le a, b \le 360$ and such that $a$ is as large as possible and $b$ is as small as possible. Find $a + b$. [b]p12[/b] Let $S = \{(a, b) | 0 \le a, b \le 3, a$ and $b$ are integers $\}$. In other words, $S$ is the set of points in the coordinate plane with integer coordinates between $0$ and $3$, inclusive. Prair selects four distinct points in $S$, for each selected point, she draws lines with slope $1$ and slope $-1$ passing through that point. Given that each point in $S$ lies on at least one line Prair drew, how many ways could she have selected those four points?

2019 IFYM, Sozopol, 6

There are $n$ kids. From each two at least one of them has sent an SMS to the other. For each kid $A$, among the kids on which $A$ has sent an SMS, exactly 10% of them have sent an SMS to $A$. Determine the number of possible three-digit values of $n$.

2012 Romania Team Selection Test, 4

Prove that a finite simple planar graph has an orientation so that every vertex has out-degree at most 3.

2018 Pan-African Shortlist, C2

Adamu and Afaafa choose, each in his turn, positive integers as coefficients of a polynomial of degree $n$. Adamu wins if the polynomial obtained has an integer root; otherwise, Afaafa wins. Afaafa plays first if $n$ is odd; otherwise Adamu plays first. Prove that: [list] [*] Adamu has a winning strategy if $n$ is odd. [*] Afaafa has a winning strategy if $n$ is even. [/list]

1992 Austrian-Polish Competition, 9

Given an integer $n > 1$, consider words composed of $n$ letters $A$ and $n$ letters $B$. A word $X_1...X_{2n}$ is said to belong to set $R(n)$ (respectively, $S(n)$) if no initial segment (respectively, exactly one initial segment) $X_1...X_k$ with $1 \le k < 2n$ consists of equally many letters $A$ and $B$. If $r(n)$ and $s(n)$ denote the cardinalities of $R(n)$ and $S(n)$ respectively, compute $s(n)/r(n)$.

2021 JHMT HS, 7

A number line with the integers $1$ through $20,$ from left to right, is drawn. Ten coins are placed along this number line, with one coin at each odd number on the line. A legal move consists of moving one coin from its current position to a position of strictly greater value on the number line that is not already occupied by another coin. How many ways can we perform two legal moves in sequence, starting from the initial position of the coins (different two-move sequences that result in the same position are considered distinct)?

2022 Nigerian Senior MO Round 2, Problem 5

For how many paths comsisting of a sequence of horizontal and/or vertical line segments, with each segment connecting a pair of adjacent letters in the diagram below, is the word $\textup{OLYMPIADS}$ spelled out as the path is traversed from beginning to end? $\begin{tabular}{ccccccccccccccccc}& & & & & & & & O & & & & & & & &\\ & & & & & & & O & L & O & & & & & & &\\ & & & & & & O & L & Y & L & O & & & & & &\\ & & & & & O & L & Y & M & Y & L & O & & & & &\\ & & & & O & L & Y & M & P & M & Y & L & O & & & &\\ & & & O & L & Y & M & P & I & P & M & Y & L & O & & &\\ & & O & L & Y & M & P & I & A & I & P & M & Y & L & O & &\\ & O & L & Y & M & P & I & A & D & A & I & P & M & Y & L & O &\\ O & L & Y & M & P & I & A & D & S & D & A & I & P & M & Y & L & O \end{tabular}$

2002 Nordic, 2

In two bowls there are in total ${N}$ balls, numbered from ${1}$ to ${N}$. One ball is moved from one of the bowls into the other. The average of the numbers in the bowls is increased in both of the bowls by the same amount, ${x}$. Determine the largest possible value of ${x}$.

1988 All Soviet Union Mathematical Olympiad, 473

Form $10A$ has $29$ students who are listed in order on its duty roster. Form $10B$ has $32$ students who are listed in order on its duty roster. Every day two students are on duty, one from form $10A$ and one from form $10B$. Each day just one of the students on duty changes and is replaced by the following student on the relevant roster (when the last student on a roster is replaced he is replaced by the first). On two particular days the same two students were on duty. Is it possible that starting on the first of these days and ending the day before the second, every pair of students (one from $10A$ and one from $10B$) shared duty exactly once?

2008 Indonesia TST, 1

Let $A$ be the subset of $\{1, 2, ..., 16\}$ that has $6$ elements. Prove that there exist $2$ subsets of $A$ that are disjoint, and the sum of their elements are the same.

2011 Bosnia And Herzegovina - Regional Olympiad, 2

At the round table there are $10$ students. Every of the students thinks of a number and says that number to its immediate neighbors (left and right) such that others do not hear him. So every student knows three numbers. After that every student publicly says arithmetic mean of two numbers he found out from his neghbors. If those arithmetic means were $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$ and $10$, respectively, which number thought student who told publicly number $6$

2017 Silk Road, 1

On an infinite white checkered sheet, a square $Q$ of size $12$ × $12$ is selected. Petya wants to paint some (not necessarily all!) cells of the square with seven colors of the rainbow (each cell is just one color) so that no two of the $288$ three-cell rectangles whose centers lie in $Q$ are the same color. Will he succeed in doing this? (Two three-celled rectangles are painted the same if one of them can be moved and possibly rotated so that each cell of it is overlaid on the cell of the second rectangle having the same color.) (Bogdanov. I)

2010 Argentina National Olympiad, 1

Given several integers, the allowed operation is to replace two of them by their non-negative difference. The operation is repeated until only one number remains. If the initial numbers are $1, 2, … , 2010$, what can be the last remaining number?

2004 Thailand Mathematical Olympiad, 10

Find the number of ways to select three distinct numbers from ${1, 2, . . . , 3n}$ with a sum divisible by $3$.

2013 Grand Duchy of Lithuania, 3

The number $1234567890$ is written on the blackboard. Two players $A$ and $B$ play the following game taking alternate moves. In one move, a player erases the number which is written on the blackboard, say, $m$, subtracts from $m$ any positive integer not exceeding the sum of the digits of $m$ and writes the obtained result instead of $m$. The first player who reduces the number written on the blackboard to $0$ wins. Determine which of the players has the winning strategy if the player $A$ makes the first move.

2022 Costa Rica - Final Round, 3

Shikaku and his son Shikamaru must climb a staircase that has $2022$ steps; the steps are listed $1$, $2$, $...$ , $2022$ and the floor is considered step $0$. This bores them both a lot, so so they decide to organize a game. They begin by tying a rope between them, so that At most they can be separated from each other by a distance of $7$ steps, that is, if they are in the steps $m$ and$ n$, then it must always be true that $|m-n| \le 7$. For the game they establish the following rules: a) They move alternately in turns. b) In his corresponding turn, the player must move to a higher step than in the one that (the same) was previously. c) If a player has just moved to the $n$-th step, then on the next turn the other player cannot be moved to any of the steps $n-1$, $n$ or $n + 1$, except when it is for reach the last step. d) Whoever reaches the last step (listed with $2022$) wins. Shikamaru is bored to start, so his father starts. Determine which of the two players has a winning strategy and describe it.

2001 Turkey MO (2nd round), 3

One wants to distribute $n$ same sized cakes between $k$ people equally by cutting every cake at most once. If the number of positive divisors of $n$ is denoted as $d(n)$, show that the number of different values of $k$ which makes such distribution possible is $n+d(n)$

1995 Singapore Team Selection Test, 3

In a dance, a group $S$ of $1994$ students stand in a big circle. Each student claps the hands of each of his two neighbours a number of times. For each student $x,$ let $f(x)$ be the total number of times $x$ claps the hands of his neighbours. As an example, suppose there are $3$ students $A, B$ and $C$. A claps hand with $B$ two times, $B$ claps hand with $C$ three times and $C$ claps hand with $A$ five times. Then $f(A) = 7, f(B) = 5$ and $f(C) = 8.$ (i) Prove that $\{f(x) | x \in S\}\ne\{n | n$ is an integer, $2 \le n \le 1995\}$. (ii) Find an example in which $\{f(x) | x \in S\} = \{n | n$ is an integer, $n \ne 3, 2 \le n \le 1996\}$

2022 Switzerland - Final Round, 6

Let $n\ge 3$ be an integer. Annalena has infinitely many cowbells in each of $n$ different colours. Given an integer $m \ge n + 1$ and a group of $m$ cows standing in a circle, she is tasked with tying one cowbell around the neck of every cow so that every group of $n + 1$ consecutive cows have cowbells of all the possible $n$ colours. Prove that there are only finitely many values of $m$ for which this is not possible and determine the largest such $m$ in terms of $n$.

2017 Korea National Olympiad, problem 1

Denote $U$ as the set of $20$ diagonals of the regular polygon $P_1P_2P_3P_4P_5P_6P_7P_8$. Find the number of sets $S$ which satisfies the following conditions. 1. $S$ is a subset of $U$. 2. If $P_iP_j \in S$ and $P_j P_k \in S$, and $i \neq k$, $P_iP_k \in S$.

2007 IMO, 3

In a mathematical competition some competitors are friends. Friendship is always mutual. Call a group of competitors a [i]clique[/i] if each two of them are friends. (In particular, any group of fewer than two competitiors is a clique.) The number of members of a clique is called its [i]size[/i]. Given that, in this competition, the largest size of a clique is even, prove that the competitors can be arranged into two rooms such that the largest size of a clique contained in one room is the same as the largest size of a clique contained in the other room. [i]Author: Vasily Astakhov, Russia[/i]

2024 Thailand TST, 3

Let $N$ be a positive integer, and consider an $N \times N$ grid. A [i]right-down path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell below the previous cell in the sequence. A [i]right-up path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell above the previous cell in the sequence. Prove that the cells of the $N \times N$ grid cannot be partitioned into less than $N$ right-down or right-up paths. For example, the following partition of the $5 \times 5$ grid uses $5$ paths. [asy] size(4cm); draw((5,-1)--(0,-1)--(0,-2)--(5,-2)--(5,-3)--(0,-3)--(0,-4)--(5,-4),gray+linewidth(0.5)+miterjoin); draw((1,-5)--(1,0)--(2,0)--(2,-5)--(3,-5)--(3,0)--(4,0)--(4,-5),gray+linewidth(0.5)+miterjoin); draw((0,0)--(5,0)--(5,-5)--(0,-5)--cycle,black+linewidth(2.5)+miterjoin); draw((0,-1)--(3,-1)--(3,-2)--(1,-2)--(1,-4)--(4,-4)--(4,-3)--(2,-3)--(2,-2),black+linewidth(2.5)+miterjoin); draw((3,0)--(3,-1),black+linewidth(2.5)+miterjoin); draw((1,-4)--(1,-5),black+linewidth(2.5)+miterjoin); draw((4,-3)--(4,-1)--(5,-1),black+linewidth(2.5)+miterjoin); [/asy] [i]Proposed by Zixiang Zhou, Canada[/i]

2018 NZMOC Camp Selection Problems, 3

Show that amongst any $ 8$ points in the interior of a $7 \times 12$ rectangle, there exists a pair whose distance is less than $5$. Note: The interior of a rectangle excludes points lying on the sides of the rectangle.