Found problems: 1800
1990 Brazil National Olympiad, 1
Show that a convex polyhedron with an odd number of faces has at least one face with an even number of edges.
2013 IberoAmerican, 6
A [i]beautiful configuration[/i] of points is a set of $n$ colored points, such that if a triangle with vertices in the set has an angle of at least $120$ degrees, then exactly 2 of its vertices are colored with the same color. Determine the maximum possible value of $n$.
2009 Junior Balkan Team Selection Test, 2
From the set $ \{1,2,3,\ldots,2009\}$ we choose $ 1005$ numbers, such that sum of any $ 2$ numbers isn't neither $ 2009$ nor $ 2010$. Find all ways on we can choose these $ 1005$ numbers.
1979 IMO Longlists, 14
Let $S$ be a set of $n^2 + 1$ closed intervals ($n$ a positive integer). Prove that at least one of the following assertions holds:
[b](i)[/b] There exists a subset $S'$ of $n+1$ intervals from $S$ such that the intersection of the intervals in $S'$ is nonempty.
[b](ii)[/b] There exists a subset $S''$ of $n + 1$ intervals from $S$ such that any two of the intervals in $S''$ are disjoint.
1999 Federal Competition For Advanced Students, Part 2, 1
Ninety-nine points are given on one of the diagonals of a unit square. Prove that there is at most one vertex of the square such that the average squared distance from a given point to the vertex is less than or equal to $1/2$.
2011 Iran MO (3rd Round), 5
Suppose that $n$ is a natural number. we call the sequence $(x_1,y_1,z_1,t_1),(x_2,y_2,z_2,t_2),.....,(x_s,y_s,z_s,t_s)$ of $\mathbb Z^4$ [b]good[/b] if it satisfies these three conditions:
[b]i)[/b] $x_1=y_1=z_1=t_1=0$.
[b]ii)[/b] the sequences $x_i,y_i,z_i,t_i$ be strictly increasing.
[b]iii)[/b] $x_s+y_s+z_s+t_s=n$. (note that $s$ may vary).
Find the number of good sequences.
[i]proposed by Mohammad Ghiasi[/i]
2014 China Team Selection Test, 6
Let $n\ge 2$ be a positive integer. Fill up a $n\times n$ table with the numbers $1,2,...,n^2$ exactly once each. Two cells are termed adjacent if they have a common edge. It is known that for any two adjacent cells, the numbers they contain differ by at most $n$. Show that there exist a $2\times 2$ square of adjacent cells such that the diagonally opposite pairs sum to the same number.
2008 Bosnia Herzegovina Team Selection Test, 1
$ 8$ students took part in exam that contains $ 8$ questions. If it is known that each question was solved by at least $ 5$ students, prove that we can always find $ 2$ students such that each of questions was solved by at least one of them.
2004 Pre-Preparation Course Examination, 2
Let $ H(n)$ be the number of simply connected subsets with $ n$ hexagons in an infinite hexagonal network. Also let $ P(n)$ be the number of paths starting from a fixed vertex (that do not connect itself) with lentgh $ n$ in this hexagonal network.
a) Prove that the limits \[ \alpha: \equal{}\lim_{n\rightarrow\infty}H(n)^{\frac1n}, \beta: \equal{}\lim_{n\rightarrow\infty}P(n)^{\frac1n}\]exist.
b) Prove the following inequalities:
$ \sqrt2\leq\beta\leq2$
$ \alpha\leq 12.5$
$ \alpha\geq3.5$
$ \alpha\leq\beta^4$
2007 Iran Team Selection Test, 2
Let $A$ be the largest subset of $\{1,\dots,n\}$ such that for each $x\in A$, $x$ divides at most one other element in $A$. Prove that \[\frac{2n}3\leq |A|\leq \left\lceil \frac{3n}4\right\rceil. \]
2008 Saint Petersburg Mathematical Olympiad, 2
In a kingdom, there are roads open between some cities with lanes both ways, in such a way, that you can come from one city to another using those roads. The roads are toll, and the price for taking each road is distinct. A minister made a list of all routes that go through each city exactly once. The king marked the most expensive road in each of the routes and said to close all the roads that he marked at least once. After that, it became impossible to go from city $A$ to city $B$, from city $B$ to city $C$, and from city $C$ to city $A$. Prove that the kings order was followed incorrectly.
2008 IMS, 4
A subset of $ n\times n$ table is called even if it contains even elements of each row and each column. Find the minimum $ k$ such that each subset of this table with $ k$ elements contains an even subset
1994 Romania TST for IMO, 1:
Let $ X_n\equal{}\{1,2,...,n\}$,where $ n \geq 3$.
We define the measure $ m(X)$ of $ X\subset X_n$ as the sum of its elements.(If $ |X|\equal{}0$,then $ m(X)\equal{}0$).
A set $ X \subset X_n$ is said to be even(resp. odd) if $ m(X)$ is even(resp. odd).
(a)Show that the number of even sets equals the number of odd sets.
(b)Show that the sum of the measures of the even sets equals the sum of the measures of the odd sets.
(c)Compute the sum of the measures of the odd sets.
2005 Taiwan National Olympiad, 1
There are 94 safes and 94 keys. Each key can open only one safe, and each safe can be opened by only one key. We place randomly one key into each safe. 92 safes are then randomly chosen, and then locked. What is the probability that we can open all the safes with the two keys in the two remaining safes?
(Once a safe is opened, the key inside the safe can be used to open another safe.)
2007 Junior Balkan Team Selection Tests - Romania, 3
Consider the numbers from $1$ to $16$. The "solitar" game consists in the arbitrary grouping of the numbers in pairs and replacing each pair with the great prime divisor of the sum of the two numbers (i.e from $(1,2); (3,4); (5,6);...;(15,16)$ the numbers which result are $3,7,11,5,19,23,3,31$). The next step follows from the same procedure and the games continues untill we obtain only one number. Which is the maximum numbers with which the game ends.
1994 Balkan MO, 4
Find the smallest number $n \geq 5$ for which there can exist a set of $n$ people, such that any two people who are acquainted have no common acquaintances, and any two people who are not acquainted have exactly two common acquaintances.
[i]Bulgaria[/i]
2010 China Western Mathematical Olympiad, 7
There are $n$ $(n \ge 3)$ players in a table tennis tournament, in which any two players have a match. Player $A$ is called not out-performed by player $B$, if at least one of player $A$'s losers is not a $B$'s loser.
Determine, with proof, all possible values of $n$, such that the following case could happen: after finishing all the matches, every player is not out-performed by any other player.
2010 Switzerland - Final Round, 8
In a village with at least one inhabitant, there are several associations. Each inhabitant is a member of at least $ k$ associations, and any two associations have at most one common member.
Prove that at least $ k$ associations have the same number of members.
1983 Dutch Mathematical Olympiad, 4
Within an equilateral triangle of side $ 15$ are $ 111$ points. Prove that it is always possible to cover three of these points by a round coin of diameter $ \sqrt{3}$, part of which may lie outside the triangle.
2002 Iran Team Selection Test, 2
$n$ people (with names $1,2,\dots,n$) are around a table. Some of them are friends. At each step 2 friend can change their place. Find a necessary and sufficient condition for friendship relation between them that with these steps we can always reach to all of posiible permutations.
2000 Tuymaada Olympiad, 2
There are 2000 cities in Graphland; some of them are connected by roads.
For every city the number of roads going from it is counted. It is known that there are exactly two equal numbers among all the numbers obtained. What can be these numbers?
1998 Turkey MO (2nd round), 3
Some of the vertices of unit squares of an $n\times n$ chessboard are colored so that any $k\times k$ ( $1\le k\le n$) square consisting of these unit squares has a colored point on at least one of its sides. Let $l(n)$ denote the minimum number of colored points required to satisfy this condition. Prove that $\underset{n\to \infty }{\mathop \lim }\,\frac{l(n)}{{{n}^{2}}}=\frac{2}{7}$.
2007 Tuymaada Olympiad, 3
Several knights are arranged on an infinite chessboard. No square is attacked by more than one knight (in particular, a square occupied by a knight can be attacked by one knight but not by two). Sasha outlined a $ 14\times 16$ rectangle. What maximum number of knights can this rectangle contain?
1998 USAMO, 4
A computer screen shows a $98 \times 98$ chessboard, colored in the usual way. One can select with a mouse any rectangle with sides on the lines of the chessboard and click the mouse button: as a result, the colors in the selected rectangle switch (black becomes white, white becomes black). Find, with proof, the minimum number of mouse clicks needed to make the chessboard all one color.
1997 Turkey Team Selection Test, 3
In a football league, whenever a player is transferred from a team $X$ with $x$ players to a team $Y$ with $y$ players, the federation is paid $y-x$ billions liras by $Y$ if $y \geq x$, while the federation pays $x-y$ billions liras to $X$ if $x > y$. A player is allowed to change as many teams as he wishes during a season. Suppose that a season started with $18$ teams of $20$ players each. At the end of the season, $12$ of the teams turn out to have again $20$ players, while the remaining $6$ teams end up with $16,16, 21, 22, 22, 23$ players, respectively. What is the maximal amount the federation may have won during the season?