Found problems: 1800
2005 QEDMO 1st, 10 (C3)
Let $n\geq 3$ be an integer. Let also $P_1,P_2,...,P_n$ be different two-element-subsets of $M=\{1,2,...,n\}$, such that when for $i,j \in M , i\neq j$ the sets $P_i,P_j$ are not totally disjoint, then there is a $k \in M$ with $P_k = \{ i,j\}$.
Prove that every element of $M$ occurse in exactly $2$ of these subsets.
2009 China Second Round Olympiad, 4
Let $P=[a_{ij}]_{3\times 9}$ be a $3\times 9$ matrix where $a_{ij}\ge 0$ for all $i,j$. The following conditions are given:
[list][*]Every row consists of distinct numbers;
[*]$\sum_{i=1}^{3}x_{ij}=1$ for $1\le j\le 6$;
[*]$x_{17}=x_{28}=x_{39}=0$;
[*]$x_{ij}>1$ for all $1\le i\le 3$ and $7\le j\le 9$ such that $j-i\not= 6$.
[*]The first three columns of $P$ satisfy the following property $(R)$: for an arbitrary column $[x_{1k},x_{2k},x_{3k}]^T$, $1\le k\le 9$, there exists an $i\in\{1,2,3\}$ such that $x_{ik}\le u_i=\min (x_{i1},x_{i2},x_{i3})$.[/list]
Prove that:
a) the elements $u_1,u_2,u_3$ come from three different columns;
b) if a column $[x_{1l},x_{2l},x_{3l}]^T$ of $P$, where $l\ge 4$, satisfies the condition that after replacing the third column of $P$ by it, the first three columns of the newly obtained matrix $P'$ still have property $(R)$, then this column uniquely exists.
2002 Iran Team Selection Test, 11
A $10\times10\times10$ cube has $1000$ unit cubes. $500$ of them are coloured black and $500$ of them are coloured white. Show that there are at least $100$ unit squares, being the common face of a white and a black unit cube.
2010 Contests, 4
A $9\times 7$ rectangle is tiled with tiles of the two types: L-shaped tiles composed by three unit squares (can be rotated repeatedly with $90^\circ$) and square tiles composed by four unit squares.
Let $n\ge 0$ be the number of the $2 \times 2 $ tiles which can be used in such a tiling. Find all the values of $n$.
2012 Bosnia Herzegovina Team Selection Test, 6
A unit square is divided into polygons, so that all sides of a polygon are parallel to sides of the given square. If the total length of the segments inside the square (without the square) is $2n$ (where $n$ is a positive real number), prove that there exists a polygon whose area is greater than $\frac{1}{(n+1)^2}$.
2009 Romania Team Selection Test, 2
Prove that the edges of a finite simple planar graph (with no loops, multiple edges) may be oriented in such a way that at most three fourths of the total number of dges of any cycle share the same orientation. Moreover, show that this is the best global bound possible.
Comment: The actual problem in the TST asked to prove that the edges can be $2$-colored so that the same conclusion holds. Under this circumstances, the problem is wrong and a counterexample was found in the contest by Marius Tiba.
2014 Argentina Cono Sur TST, 2
The numbers $1$ through $9$ are written on a $3 \times 3$ board, without repetitions. A valid operation is to choose a row or a column of the board, and replace its three numbers $a, b, c$ (in order, i.e., the first number of the row/column is $a$, the second number of the row/column is $b$, the third number of the row/column is $c$) with either the three non-negative numbers $a-x, b-x, c+x$ (in order) or with the three non-negative numbers $a+x, b-x, c-x$ (in order), where $x$ is a real positive number which may vary in each operation .
a) Determine if there is a way of getting all $9$ numbers on the board to be the same, starting with the following board:
$\begin{array}{|c|c|c|c|c|c|c|c|} \hline 1 & 2 & 3 \\ \hline 4 & 5 & 6 \\ \hline 7 & 8 & 9 \\ \hline \end{array}$
b) For all posible configurations such that it is possible to get all $9$ numbers to be equal to a number $m$ using the valid operations, determine the maximum value of $m$.
2003 Tournament Of Towns, 5
Is it possible to tile $2003 \times 2003$ board by $1 \times 2$ dominoes placed horizontally and $1 \times 3$ rectangles placed vertically?
2012 Iran MO (3rd Round), 4
[b]a)[/b] Prove that for all $m,n\in \mathbb N$ there exists a natural number $a$ such that if we color every $3$-element subset of the set $\mathcal A=\{1,2,3,...,a\}$ using $2$ colors red and green, there exists an $m$-element subset of $\mathcal A$ such that all $3$-element subsets of it are red or there exists an $n$-element subset of $\mathcal A$ such that all $3$-element subsets of it are green.
[b]b)[/b] Prove that for all $m,n\in \mathbb N$ there exists a natural number $a$ such that if we color every $k$-element subset ($k>3$) of the set $\mathcal A=\{1,2,3,...,a\}$ using $2$ colors red and green, there exists an $m$-element subset of $\mathcal A$ such that all $k$-element subsets of it are red or there exists an $n$-element subset of $\mathcal A$ such that all $k$-element subsets of it are green.
2004 Iran MO (2nd round), 6
We have a $m\times n$ table and $m\geq{4}$ and we call a $1\times 1$ square a room. When we put an alligator coin in a room, it menaces all the rooms in his column and his adjacent rooms in his row. What's the minimum number of alligator coins required, such that each room is menaced at least by one alligator coin? (Notice that all alligator coins are vertical.)
2008 Tournament Of Towns, 5
On a straight track are several runners, each running at a di fferent constant speed. They start at one end of the track at the same time. When a runner reaches any end of the track, he immediately turns around and runs back with the same speed (then he reaches the other end and turns back again, and so on). Some time after the start, all runners meet at the same point. Prove that this will happen again.
2010 ELMO Shortlist, 5
Let $n > 1$ be a positive integer. A 2-dimensional grid, infinite in all directions, is given. Each 1 by 1 square in a given $n$ by $n$ square has a counter on it. A [i]move[/i] consists of taking $n$ adjacent counters in a row or column and sliding them each by one space along that row or column. A [i]returning sequence[/i] is a finite sequence of moves such that all counters again fill the original $n$ by $n$ square at the end of the sequence.
[list]
[*] Assume that all counters are distinguishable except two, which are indistinguishable from each other. Prove that any distinguishable arrangement of counters in the $n$ by $n$ square can be reached by a returning sequence.
[*] Assume all counters are distinguishable. Prove that there is no returning sequence that switches two counters and returns the rest to their original positions.[/list]
[i]Mitchell Lee and Benjamin Gunby.[/i]
1986 Iran MO (2nd round), 5
We have erasers, four pencils, two note books and three pens and we want to divide them between two persons so that every one receives at least one of the above stationery. In how many ways is this possible? [Note that the are not distinct.]
2008 Grigore Moisil Intercounty, 1
On a circle there are given $ n\plus{}3$ distinct points,from which $ n$ are colored red, two yellow, and one blue. Determine the number of polygons which have
a) the vertices of the same color
b) the vertices of two colors
c) the vertices of three colors.
2004 Silk Road, 4
Natural $n \geq 2$ is given. Group of people calls $n-compact$, if for any men from group, we can found $n$ people (without he), each two of there are familiar.
Find maximum $N$ such that for any $n-compact$ group, consisting $N$ people contains subgroup from $n+1$ people, each of two of there are familiar.
2011 Vietnam National Olympiad, 4
A convex pentagon $ABCDE$ satisfies that the sidelengths and $AC,AD\leq \sqrt 3.$ Let us choose $2011$ distinct points inside this pentagon. Prove that there exists an unit circle with centre on one edge of the pentagon, and which contains at least $403$ points out of the $2011$ given points.
{Edited}
{I posted it correctly before but because of a little confusion deleted the sidelength part, sorry.}
1978 Canada National Olympiad, 5
Eve and Odette play a game on a $3\times 3$ checkerboard, with black checkers and white checkers. The rules are as follows:
$\text{I.}$ They play alternately.
$\text{II.}$ A turn consists of placing one checker on an unoccupied square of the board.
$\text{III.}$ In her turn, a player may select either a white checker or a black checker and need not always use the same colour.
$\text{IV.}$ When the board is full, Eve obtains one point for every row, column or diagonal that has an even number of black checkers, and Odette obtains one point for very row, column or diagonal that has an odd number of black checkers.
$\text{V.}$ The player obtaining at least five of the eight points WINS.
$\text{(a)}$ Is a $4-4$ tie possible? Explain.
$\text{(b)}$ Describe a winning strategy for the girl who is first to play.
2009 China Team Selection Test, 2
Let $ n,k$ be given positive integers satisfying $ k\le 2n \minus{} 1$. On a table tennis tournament $ 2n$ players take part, they play a total of $ k$ rounds match, each round is divided into $ n$ groups, each group two players match. The two players in different rounds can match on many occasions. Find the greatest positive integer $ m \equal{} f(n,k)$ such that no matter how the tournament processes, we always find $ m$ players each of pair of which didn't match each other.
1999 CentroAmerican, 6
Denote $S$ as the subset of $\{1,2,3,\dots,1000\}$ with the property that none of the sums of two different elements in $S$ is in $S$. Find the maximum number of elements in $S$.
2006 Kyiv Mathematical Festival, 1
See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url]
The number $123456789$ is written on the blackboard. At each step it is allowed to choose its digits $a$ and $b$ of the same parity and to replace each of them by $\frac{a+b}{2}.$ Is it possible to obtain a number larger then
a)$800000000$; b)$880000000$ by such replacements?
2007 China Western Mathematical Olympiad, 4
A circular disk is partitioned into $ 2n$ equal sectors by $ n$ straight lines through its center. Then, these $ 2n$ sectors are colored in such a way that exactly $ n$ of the sectors are colored in blue, and the other $ n$ sectors are colored in red. We number the red sectors with numbers from $ 1$ to $ n$ in counter-clockwise direction (starting at some of these red sectors), and then we number the blue sectors with numbers from $ 1$ to $ n$ in clockwise direction (starting at some of these blue sectors).
Prove that one can find a half-disk which contains sectors numbered with all the numbers from $ 1$ to $ n$ (in some order). (In other words, prove that one can find $ n$ consecutive sectors which are numbered by all numbers $ 1$, $ 2$, ..., $ n$ in some order.)
[hide="Problem 8 from CWMO 2007"]$ n$ white and $ n$ black balls are placed at random on the circumference of a circle.Starting from a certain white ball,number all white balls in a clockwise direction by $ 1,2,\dots,n$. Likewise number all black balls by $ 1,2,\dots,n$ in anti-clockwise direction starting from a certain black ball.Prove that there exists a chain of $ n$ balls whose collection of numbering forms the set $ \{1,2,3\dots,n\}$.[/hide]
2007 Czech and Slovak Olympiad III A, 1
A stone is placed in a square of a chessboard with $n$ rows and $n$ columns. We can alternately undertake two operations:
[b](a)[/b] move the stone to a square that shares a common side with the square in which it stands;
[b](b)[/b] move it to a square sharing only one common vertex with the square in which it stands.
In addition, we are required that the first step must be [b](b)[/b]. Find all integers $n$ such that the stone can go through a certain path visiting every square exactly once.
2012 Iran MO (3rd Round), 2
Suppose $W(k,2)$ is the smallest number such that if $n\ge W(k,2)$, for each coloring of the set $\{1,2,...,n\}$ with two colors there exists a monochromatic arithmetic progression of length $k$. Prove that
$W(k,2)=\Omega (2^{\frac{k}{2}})$.
2005 Iran Team Selection Test, 3
Suppose $S= \{1,2,\dots,n\}$ and $n \geq 3$. There is $f:S^k \longmapsto S$ that if $a,b \in S^k$ and $a$ and $b$ differ in all of elements then $f(a) \neq f(b)$. Prove that $f$ is a function of one of its elements.
2014 All-Russian Olympiad, 4
Two players play a card game. They have a deck of $n$ distinct cards. About any two cards from the deck know which of them has a different (in this case, if $A$ beats $B$, and $B$ beats $C$, then it may be that $C$ beats $A$). The deck is split between players in an arbitrary manner. In each turn the players over the top card from his deck and one whose card has a card from another player takes both cards and puts them to the bottom of your deck in any order of their discretion. Prove that for any initial distribution of cards, the players can with knowing the location agree and act so that one of the players left without a card.
[i]E. Lakshtanov[/i]