Found problems: 1800
2012 Canadian Mathematical Olympiad Qualification Repechage, 7
Six tennis players gather to play in a tournament where each pair of persons play one game, with one person declared the winner and the other person the loser. A triplet of three players $\{\mathit{A}, \mathit{B}, \mathit{C}\}$ is said to be [i]cyclic[/i] if $\mathit{A}$ wins against $\mathit{B}$, $\mathit{B}$ wins against $\mathit{C}$ and $\mathit{C}$ wins against $\mathit{A}$.
[list]
[*] (a) After the tournament, the six people are to be separated in two rooms such that none of the two rooms contains a cyclic triplet. Prove that this is always possible.
[*] (b) Suppose there are instead seven people in the tournament. Is it always possible that the seven people can be separated in two rooms such that none of the two rooms contains a cyclic triplet?[/list]
2007 Junior Balkan MO, 3
Given are $50$ points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least $130$ scalene triangles with vertices of that color.
2007 Junior Balkan Team Selection Tests - Romania, 3
At a party there are eight guests, and each participant can't talk with at most three persons. Prove that we can group the persons in four pairs such that in every pair a conversation can take place.
2002 Baltic Way, 7
We draw $n$ convex quadrilaterals in the plane. They divide the plane into regions (one of the regions is infinite). Determine the maximal possible number of these regions.
2013 Mexico National Olympiad, 4
A $n \times n \times n$ cube is constructed using $1 \times 1 \times 1$ cubes, some of them black and others white, such that in each $n \times 1 \times 1$, $1 \times n \times 1$, and $1 \times 1 \times n$ subprism there are exactly two black cubes, and they are separated by an even number of white cubes (possibly 0).
Show it is possible to replace half of the black cubes with white cubes such that each $n \times 1 \times 1$, $1 \times n \times 1$ and $1 \times 1 \times n$ subprism contains exactly one black cube.
2019 Turkey MO (2nd round), 3
There are 2019 students in a school, and some of these students are members of different student clubs. Each student club has an advisory board consisting of 12 students who are members of that particular club. An {\em advisory meeting} (for a particular club) can be realized only when each participant is a member of that club, and moreover, each of the 12 students forming the advisory board are present among the participants. It is known that each subset of at least 12 students in this school can realize an advisory meeting for exactly one student club. Determine all possible numbers of different student clubs with exactly 27 members.
2008 IberoAmerican, 6
[i]Biribol[/i] is a game played between two teams of 4 people each (teams are not fixed). Find all the possible values of $ n$ for which it is possible to arrange a tournament with $ n$ players in such a way that every couple of people plays a match in opposite teams exactly once.
2013 Middle European Mathematical Olympiad, 4
Consider finitely many points in the plane with no three points on a line. All these points can be coloured red or green such that any triangle with vertices of the same colour contains at least one point of the other colour in its interior.
What is the maximal possible number of points with this property?
2000 Turkey MO (2nd round), 3
Let $f(x,y)$ and $g(x,y)$ be real valued functions defined for every $x,y \in \{1,2,..,2000\}$. If there exist $X,Y \subset \{1,2,..,2000\}$ such that $s(X)=s(Y)=1000$ and $x\notin X$ and $y\notin Y$ implies that $f(x,y)=g(x,y)$ than, what is the maximum number of $(x,y)$ couples where $f(x,y)\neq g(x,y)$.
2012 Iran Team Selection Test, 3
Let $n$ be a positive integer. Let $S$ be a subset of points on the plane with these conditions:
$i)$ There does not exist $n$ lines in the plane such that every element of $S$ be on at least one of them.
$ii)$ for all $X \in S$ there exists $n$ lines in the plane such that every element of $S - {X} $ be on at least one of them.
Find maximum of $\mid S\mid$.
[i]Proposed by Erfan Salavati[/i]
2009 All-Russian Olympiad, 4
Given a set $ M$ of points $ (x,y)$ with integral coordinates satisfying $ x^2 + y^2\leq 10^{10}$. Two players play a game. One of them marks a point on his first move. After this, on each move the moving player marks a point, which is not yet marked and joins it with the previous marked point. Players are not allowed to mark a point symmetrical to the one just chosen. So, they draw a broken line. The requirement is that lengths of edges of this broken line must strictly increase. The player, which can not make a move, loses. Who have a winning strategy?
1983 IMO Longlists, 45
Let two glasses, numbered $1$ and $2$, contain an equal quantity of liquid, milk in glass $1$ and coffee in glass $2$. One does the following: Take one spoon of mixture from glass $1$ and pour it into glass $2$, and then take the same spoon of the new mixture from glass $2$ and pour it back into the first glass. What happens after this operation is repeated $n$ times, and what as $n$ tends to infinity?
1975 Miklós Schweitzer, 2
Let $ \mathcal{A}_n$ denote the set of all mappings $ f: \{1,2,\ldots ,n \} \rightarrow \{1,2,\ldots, n \}$ such that $ f^{-1}(i) :=\{ k \colon f(k)=i\ \} \neq \varnothing$ implies $ f^{-1}(j) \neq \varnothing, j \in \{1,2,\ldots, i \} .$ Prove \[ |\mathcal{A}_n| =
\sum_{k=0}^{\infty} \frac{k^n}{2^{k+1}}.\]
[i]L. Lovasz[/i]
2014 Baltic Way, 6
In how many ways can we paint $16$ seats in a row, each red or green, in such a way that the number of consecutive seats painted in the same colour is always odd?
2010 Middle European Mathematical Olympiad, 7
In each vertex of a regular $n$-gon, there is a fortress. At the same moment, each fortress shoots one of the two nearest fortresses and hits it. The [i]result of the shooting[/i] is the set of the hit fortresses; we do not distinguish whether a fortress was hit once or twice. Let $P(n)$ be the number of possible results of the shooting. Prove that for every positive integer $k\geqslant 3$, $P(k)$ and $P(k+1)$ are relatively prime.
[i](4th Middle European Mathematical Olympiad, Team Competition, Problem 3)[/i]
2005 Georgia Team Selection Test, 6
Let $ A$ be the subset of the set of positive integers, having the following $ 2$ properties:
1) If $ a$ belong to $ A$,than all of the divisors of $ a$ also belong to $ A$;
2) If $ a$ and $ b$, $ 1 < a < b$, belong to $ A$, than $ 1 \plus{} ab$ is also in $ A$;
Prove that if $ A$ contains at least $ 3$ positive integers, than $ A$ contains all positive integers.
2010 Contests, 4
Find all positive integers $N$ such that an $N\times N$ board can be tiled using tiles of size $5\times 5$ or $1\times 3$.
Note: The tiles must completely cover all the board, with no overlappings.
2013 China Team Selection Test, 1
For a positive integer $k\ge 2$ define $\mathcal{T}_k=\{(x,y)\mid x,y=0,1,\ldots, k-1\}$ to be a collection of $k^2$ lattice points on the cartesian coordinate plane. Let $d_1(k)>d_2(k)>\cdots$ be the decreasing sequence of the distinct distances between any two points in $T_k$. Suppose $S_i(k)$ be the number of distances equal to $d_i(k)$.
Prove that for any three positive integers $m>n>i$ we have $S_i(m)=S_i(n)$.
1983 IMO Longlists, 15
Find all possible finite sequences $\{n_0, n_1, n_2, \ldots, n_k \}$ of integers such that for each $i, i$ appears in the sequence $n_i$ times $(0 \leq i \leq k).$
2014 ELMO Shortlist, 5
Let $n$ be a positive integer. For any $k$, denote by $a_k$ the number of permutations of $\{1,2,\dots,n\}$ with exactly $k$ disjoint cycles. (For example, if $n=3$ then $a_2=3$ since $(1)(23)$, $(2)(31)$, $(3)(12)$ are the only such permutations.) Evaluate
\[ a_n n^n + a_{n-1} n^{n-1} + \dots + a_1 n. \][i]Proposed by Sammy Luo[/i]
2013 India IMO Training Camp, 3
A marker is placed at the origin of an integer lattice. Calvin and Hobbes play the following game. Calvin starts the game and each of them takes turns alternatively. At each turn, one can choose two (not necessarily distinct) integers $a, b$, neither of which was chosen earlier by any player and move the marker by $a$ units in the horizontal direction and $b$ units in the vertical direction. Hobbes wins if the marker is back at the origin any time after the first turn. Prove or disprove that Calvin can prevent Hobbes from winning.
Note: A move in the horizontal direction by a positive quantity will be towards the right, and by a negative quantity will be towards the left (and similar directions in the vertical case as well).
2002 Tournament Of Towns, 5
A convex $N\text{-gon}$ is divided by diagonals into triangles so that no two diagonals intersect inside the polygon. The triangles are painted in black and white so that any two triangles are painted in black and white so that any two triangles with a common side are painted in different colors. For each $N$ find the maximal difference between the numbers of black and white triangles.
2013 China National Olympiad, 1
Let $n \geqslant 2$ be an integer. There are $n$ finite sets ${A_1},{A_2},\ldots,{A_n}$ which satisfy the condition
\[\left| {{A_i}\Delta {A_j}} \right| = \left| {i - j} \right| \quad \forall i,j \in \left\{ {1,2,...,n} \right\}.\]
Find the minimum of $\sum\limits_{i = 1}^n {\left| {{A_i}} \right|} $.
1985 Balkan MO, 3
Let $S$ be the set of all positive integers of the form $19a+85b$, where $a,b$ are arbitrary positive integers. On the real axis, the points of $S$ are colored in red and the remaining integer numbers are colored in green. Find, with proof, whether or not there exists a point $A$ on the real axis such that any two points with integer coordinates which are symmetrical with respect to $A$ have necessarily distinct colors.
1985 Iran MO (2nd round), 5
In the Archery with an especial gun, the probability of goal is $90 \%.$ If we continue our work until we goal.
[b]i)[/b] What is the probability which exactly $3$ balls consumed.
[b]ii)[/b] What is the probability which at least $3$ balls consumed.