Found problems: 1800
1991 Brazil National Olympiad, 1
At a party every woman dances with at least one man, and no man dances with every woman. Show that there are men M and M' and women W and W' such that M dances with W, M' dances with W', but M does not dance with W', and M' does not dance with W.
1997 All-Russian Olympiad, 4
On an infinite (in both directions) strip of squares, indexed by the integers, are placed several stones (more than one may be placed on a single square). We perform a sequence of moves of one of the following types:
(a) Remove one stone from each of the squares $n - 1$ and $n$ and place one stone on square $n + 1$.
(b) Remove two stones from square $n$ and place one stone on each of the squares $n + 1$, $n - 2$.
Prove that any sequence of such moves will lead to a position in which no further moves can be made, and moreover that this position is independent of the sequence of moves.
[i]D. Fon-der-Flaas[/i]
2006 Iran MO (3rd Round), 3
Let $C$ be a (probably infinite) family of subsets of $\mathbb{N}$ such that for every chain $C_{1}\subset C_{2}\subset \ldots$ of members of $C$, there is a member of $C$ containing all of them. Show that there is a member of $C$ such that no other member of $C$ contains it!
2010 Romania National Olympiad, 3
Each of the small squares of a $50\times 50$ table is coloured in red or blue. Initially all squares are red. A [i]step[/i] means changing the colour of all squares on a row or on a column.
a) Prove that there exists no sequence of steps, such that at the end there are exactly $2011$ blue squares.
b) Describe a sequence of steps, such that at the end exactly $2010$ squares are blue.
[i]Adriana & Lucian Dragomir[/i]
2001 Baltic Way, 5
Let $2001$ given points on a circle be coloured either red or green. In one step all points are recoloured simultaneously in the following way: If both direct neighbours of a point $P$ have the same colour as $P$, then the colour of $P$ remains unchanged, otherwise $P$ obtains the other colour. Starting with the first colouring $F_1$, we obtain the colourings $F_2,F_3 ,\ldots .$ after several recolouring steps. Prove that there is a number $n_0\le 1000$ such that $F_{n_0}=F_{n_0 +2}$. Is the assertion also true if $1000$ is replaced by $999$?
2024 Bundeswettbewerb Mathematik, 4
In Sikinia, there are $2024$ cities. Between some of them there are flight connections, which can be used in either direction. No city has a direct flight to all $2023$ other cities. It is known, however, that there is a positive integer $n$ with the following property: For any $n$ cities in Sikinia, there is another city which is directly connected to all these cities.
Determine the largest possible value of $n$.
2002 India IMO Training Camp, 9
On each day of their tour of the West Indies, Sourav and Srinath have either an apple or an orange for breakfast. Sourav has oranges for the first $m$ days, apples for the next $m$ days, followed by oranges for the next $m$ days, and so on. Srinath has oranges for the first $n$ days, apples for the next $n$ days, followed by oranges for the next $n$ days, and so on.
If $\gcd(m,n)=1$, and if the tour lasted for $mn$ days, on how many days did they eat the same kind of fruit?
2010 Contests, 1
The integer number $n > 1$ is given and a set $S \subset \{0, 1, 2, \ldots, n-1\}$ with $|S| > \frac{3}{4} n$. Prove that there exist integer numbers $a, b, c$ such that the remainders after the division by $n$ of the numbers:
\[a, b, c, a+b, b+c, c+a, a+b+c\]
belong to $S$.
2014 Korea - Final Round, 6
In an island there are $n$ castles, and each castle is in country $A$ or $B$. There is one commander per castle, and each commander belongs to the same country as the castle he's initially in. There are some (two-way) roads between castles (there may be roads between castles of different countries), and call two castles adjacent if there is a road between them.
Prove that the following two statements are equivalent:
(1) If some commanders from country $B$ move to attack an adjacent castle in country $A$, some commanders from country $A$ could appropriately move in defense to adjacent castles in country $A$ so that in every castle of country $A$, the number of country $A$'s commanders defending that castle is not less than the number of country $B$'s commanders attacking that castle. (Each commander can defend or attack only one castle at a time.)
(2) For any arbitrary set $X$ of castles in country $A$, the number of country $A$'s castles that are in $X$ or adjacent to at least one of the castle in $X$ is not less than the number of country $B$'s castles that are adjacent to at least one of the castles in $X$.
2004 Pre-Preparation Course Examination, 4
Let $ G$ be a simple graph. Suppose that size of largest independent set in $ G$ is $ \alpha$. Prove that:
a) Vertices of $ G$ can be partitioned to at most $ \alpha$ paths.
b) Suppose that a vertex and an edge are also cycles. Prove that vertices of $ G$ can be partitioned to at most $ \alpha$ cycles.
2008 Romania Team Selection Test, 1
Let $ n$ be an integer, $ n\geq 2$. Find all sets $ A$ with $ n$ integer elements such that the sum of any nonempty subset of $ A$ is not divisible by $ n\plus{}1$.
2009 Indonesia TST, 4
2008 boys and 2008 girls sit on 4016 chairs around a round table. Each boy brings a garland and each girl brings a chocolate. In an "activity", each person gives his/her goods to the nearest person on the left. After some activities, it turns out that all boys get chocolates and all girls get garlands. Find the number of possible arrangements.
2015 Brazil National Olympiad, 2
Consider $S=\{1, 2, 3, \cdots, 6n\}$, $n>1$. Find the largest $k$ such that the following statement is true: every subset $A$ of $S$ with $4n$ elements has at least $k$ pairs $(a,b)$, $a<b$ and $b$ is divisible by $a$.
2001 Junior Balkan Team Selection Tests - Romania, 3
In the interior of a circle centred at $O$ consider the $1200$ points $A_1,A_2,\ldots ,A_{1200}$, where for every $i,j$ with $1\le i\le j\le 1200$, the points $O,A_i$ and $A_j$ are not collinear. Prove that there exist the points $M$ and $N$ on the circle, with $\angle MON=30^{\circ}$, such that in the interior of the angle $\angle MON$ lie exactly $100$ points.
1983 Federal Competition For Advanced Students, P2, 6
Planes $ \pi _1$ and $ \pi _2$ in Euclidean space $ \mathbb{R} ^3$ partition $ S\equal{}\mathbb{R} ^3 \setminus (\pi _1 \cup \pi _2)$ into several components. Show that for any cube in $ \mathbb{R} ^3$, at least one of the components of $ S$ meets at least three faces of the cube.
2010 Tournament Of Towns, 4
In a school, more than $90\% $ of the students know both English and German, and more than $90\%$ percent of the students know both English and French. Prove that more than $90\%$ percent of the students who know both German and French also know English.
2013 All-Russian Olympiad, 4
A square with horizontal and vertical sides is drawn on the plane. It held several segments parallel to the sides, and there are no two segments which lie on one line or intersect at an interior point for both segments. It turned out that the segments cuts square into rectangles, and any vertical line intersecting the square and not containing segments of the partition intersects exactly $ k $ rectangles of the partition, and any horizontal line intersecting the square and not containing segments of the partition intersects exactly $\ell$ rectangles. How much the number of rectangles can be?
[i]I. Bogdanov, D. Fon-Der-Flaass[/i]
2005 Indonesia MO, 8
There are $ 90$ contestants in a mathematics competition. Each contestant gets acquainted with at least $ 60$ other contestants. One of the contestants, Amin, state that at least four contestants have the same number of new friends. Prove or disprove his statement.
1990 Turkey Team Selection Test, 3
Let $n$ be an odd integer greater than $11$; $k\in \mathbb{N}$, $k \geq 6$, $n=2k-1$.
We define \[d(x,y) = \left | \{ i\in \{1,2,\dots, n \} \bigm | x_i \neq y_i \} \right |\] for $T=\{ (x_1, x_2, \dots, x_n) \bigm | x_i \in \{0,1\}, i=1,2,\dots, n \}$ and $x=(x_1,x_2,\dots, x_n), y=(y_1, y_2, \dots, y_n) \in T$.
Show that $n=23$ if $T$ has a subset $S$ satisfying
[list=i]
[*]$|S|=2^k$
[*]For each $x \in T$, there exists exacly one $y\in S$ such that $d(x,y)\leq 3$[/list]
2011 Turkey Team Selection Test, 2
Graphistan has $2011$ cities and Graph Air (GA) is running one-way flights between all pairs of these cities. Determine the maximum possible value of the integer $k$ such that no matter how these flights are arranged it is possible to travel between any two cities in Graphistan riding only GA flights as long as the absolute values of the difference between the number of flights originating and terminating at any city is not more than $k.$
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.
2012 Indonesia TST, 2
A TV station holds a math talent competition, where each participant will be scored by 8 people. The scores are F (failed), G (good), or E (exceptional). The competition is participated by three people, A, B, and C. In the competition, A and B get the same score from exactly 4 people. C states that he has differing scores with A from at least 4 people, and also differing scores with B from at least 4 people. Assuming C tells the truth, how many scoring schemes can occur?
2021 Bulgaria EGMO TST, 4
In a convex $n$-gon, several diagonals are drawn. Among these diagonals, a diagonal is called [i]good[/i] if it intersects exactly one other diagonal drawn (in the interior of the $n$-gon). Find the maximum number of good diagonals.
2020 Bundeswettbewerb Mathematik, 1
Leo and Smilla find $2020$ gold nuggets with masses $1,2,\dots,2020$ gram, which they distribute to a red and a blue treasure chest according to the following rule:
First, Leo chooses one of the chests and tells its colour to Smilla. Then Smilla chooses one of the not yet distributed nuggets and puts it into this chest.
This is repeated until all the nuggets are distributed. Finally, Smilla chooses one of the chests and wins all the nuggets from this chest.
How many gram of gold can Smilla make sure to win?
2006 Cono Sur Olympiad, 2
Two players, A and B, play the following game: they retire coins of a pile which contains initially 2006 coins. The players play removing alternatingly, in each move, from 1 to 7 coins, each player keeps the coins that retires. If a player wishes he can pass(he doesn't retire any coin), but to do that he must pay 7 coins from the ones he retired from the pile in past moves. These 7 coins are taken to a separated box and don't interfere in the game any more. The winner is the one who retires the last coin, and A starts the game. Determine which player can win for sure, it doesn't matter how the other one plays. Show the winning strategy and explain why it works.