Found problems: 1800
2003 Iran MO (2nd round), 3
$n$ volleyball teams have competed to each other (each $2$ teams have competed exactly $1$ time.). For every $2$ distinct teams like $A,B$, there exist exactly $t$ teams which have lost their match with $A,B$. Prove that $n=4t+3$. (Notabene that in volleyball, there doesn’t exist tie!)
2008 IberoAmerican, 1
The integers from 1 to $ 2008^2$ are written on each square of a $ 2008 \times 2008$ board. For every row and column the difference between the maximum and minimum numbers is computed. Let $ S$ be the sum of these 4016 numbers. Find the greatest possible value of $ S$.
2014 Olympic Revenge, 3
Let $n$ a positive integer. In a $2n\times 2n$ board, $1\times n$ and $n\times 1$ pieces are arranged without overlap.
Call an arrangement [b]maximal[/b] if it is impossible to put a new piece in the board without overlapping the previous ones.
Find the least $k$ such that there is a [b]maximal[/b] arrangement that uses $k$ pieces.
2006 Moldova Team Selection Test, 4
Let $f(n)$ denote the number of permutations $(a_{1}, a_{2}, \ldots ,a_{n})$ of the set $\{1,2,\ldots,n\}$, which satisfy the conditions: $a_{1}=1$ and $|a_{i}-a_{i+1}|\leq2$, for any $i=1,2,\ldots,n-1$. Prove that $f(2006)$ is divisible by 3.
2003 Tuymaada Olympiad, 1
A $2003\times 2004$ rectangle consists of unit squares. We consider rhombi formed by four diagonals of unit squares.
What maximum number of such rhombi can be arranged in this rectangle so that no two of them have any common points except vertices?
[i]Proposed by A. Golovanov[/i]
2010 All-Russian Olympiad, 4
In the county some pairs of towns connected by two-way non-stop flight. From any town we can flight to any other (may be not on one flight). Gives, that if we consider any cyclic (i.e. beginning and finish towns match) route, consisting odd number of flights, and close all flights of this route, then we can found two towns, such that we can't fly from one to other.
Proved, that we can divided all country on $4$ regions, such that any flight connected towns from other regions.
2010 Contests, 1
There are $24$ different pencils, $4$ different colors, and $6$ pencils of each color. They were given to $6$ children in such a way that each got $4$ pencils. What is the least number of children that you can randomly choose so that you can guarantee that you have pencils of all colors.
P.S. for 10 grade gives same problem with $40$ pencils, $10$ of each color and $10$ children.
2006 Baltic Way, 9
To every vertex of a regular pentagon a real number is assigned. We may perform the following operation repeatedly: we choose two adjacent vertices of the pentagon and replace each of the two numbers assigned to these vertices by their arithmetic mean. Is it always possible to obtain the position in which all five numbers are zeroes, given that in the initial position the sum of all five numbers is equal to zero?
2007 IberoAmerican, 3
Two teams, $ A$ and $ B$, fight for a territory limited by a circumference.
$ A$ has $ n$ blue flags and $ B$ has $ n$ white flags ($ n\geq 2$, fixed). They play alternatively and $ A$ begins the game. Each team, in its turn, places one of his flags in a point of the circumference that has not been used in a previous play. Each flag, once placed, cannot be moved.
Once all $ 2n$ flags have been placed, territory is divided between the two teams. A point of the territory belongs to $ A$ if the closest flag to it is blue, and it belongs to $ B$ if the closest flag to it is white. If the closest blue flag to a point is at the same distance than the closest white flag to that point, the point is neutral (not from $ A$ nor from $ B$). A team wins the game is their points cover a greater area that that covered by the points of the other team. There is a draw if both cover equal areas.
Prove that, for every $ n$, team $ B$ has a winning strategy.
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.
2019 Peru Cono Sur TST, P4
Positive integers 1 to 9 are written in each square of a $ 3 \times 3 $ table. Let us define an operation as follows: Take an arbitrary row or column and replace these numbers $ a, b, c$ with either non-negative numbers $ a-x, b-x, c+x $ or $ a+x, b-x, c-x$, where $ x $ is a positive number and can vary in each operation.
(1) Does there exist a series of operations such that all 9 numbers turn out to be equal from the following initial arrangement a)? b)?
\[ a) \begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array} )\]
\[ b) \begin{array}{ccc} 2 & 8 & 5 \\ 9 & 3 & 4 \\ 6 & 7 & 1 \end{array} )\]
(2) Determine the maximum value which all 9 numbers turn out to be equal to after some steps.
2003 Bulgaria Team Selection Test, 3
Some of the vertices of a convex $n$-gon are connected by segments, such that any two of them have no common interior point. Prove that, for any $n$ points in general position, there exists a one-to-one correspondence between the points and the vertices of the $n$ gon, such that any two segments between the points, corresponding to the respective segments from the $n$ gon, have no common interior point.
2004 Junior Tuymaada Olympiad, 8
Zeroes and ones are arranged in all the squares of $n\times n$ table.
All the squares of the left column are filled by ones, and the sum of numbers in every figure of the form
[asy]size(50); draw((2,1)--(0,1)--(0,2)--(2,2)--(2,0)--(1,0)--(1,2));[/asy]
(consisting of a square and its neighbours from left and from below)
is even.
Prove that no two rows of the table are identical.
[i]Proposed by O. Vanyushina[/i]
2010 Contests, 3
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]
2007 Tournament Of Towns, 1
A $9 \times 9$ chessboard with the standard checkered pattern has white squares at its four corners. What is the least number of rooks that can be placed on this board so that all the white squares are attacked? (A rook also attacks the square it is on, in addition to every other square in the same row or column.)
2012 Indonesia MO, 1
Given positive integers $m$ and $n$. Let $P$ and $Q$ be two collections of $m \times n$ numbers of $0$ and $1$, arranged in $m$ rows and $n$ columns. An example of such collections for $m=3$ and $n=4$ is
\[\left[ \begin{array}{cccc}
1 & 1 & 1 & 0 \\
1 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 \end{array} \right].\]
Let those two collections satisfy the following properties:
(i) On each row of $P$, from left to right, the numbers are non-increasing,
(ii) On each column of $Q$, from top to bottom, the numbers are non-increasing,
(iii) The sum of numbers on the row in $P$ equals to the same row in $Q$,
(iv) The sum of numbers on the column in $P$ equals to the same column in $Q$.
Show that the number on row $i$ and column $j$ of $P$ equals to the number on row $i$ and column $j$ of $Q$ for $i=1,2,\dots,m$ and $j=1,2,\dots,n$.
[i]Proposer: Stefanus Lie[/i]
1979 Miklós Schweitzer, 3
Let $ g(n,k)$ denote the number of strongly connected, $ \textit{simple}$ directed graphs with $ n$ vertices and $ k$ edges. ($ \textit{Simple}$ means no loops or multiple edges.) Show that \[ \sum_{k=n}^{n^2-n}(-1)^kg(n,k)=(n-1)!.\]
[i]A. A. Schrijver[/i]
2005 Taiwan TST Round 3, 1
A club provides 30 snacks to 18 members, and each member orders 3 different snacks. It is known that every snack is ordered by at least one member, and that any two members order at most one same snack. Is it possible to find 12 snacks, such that the snacks ordered by any member is not completely in these 12 snacks?
2011 Canadian Mathematical Olympiad Qualification Repechage, 5
Each vertex of a regular $11$-gon is colored black or gold. All possible triangles are formed using these vertices. Prove that there are either two congruent triangles with three black vertices or two congruent triangles with three gold vertices.
VMEO III 2006, 10.4
Given a convex polygon $ G$, show that there are three vertices of $ G$ which form a triangle so that it's perimeter is not less than 70% of the polygon's perimeter.
1972 Bundeswettbewerb Mathematik, 3
$2^{n-1}$ subsets are choosen from a set with $n$ elements, such that every three of these subsets have an element in common. Show that all subsets have an element in common.
1971 IMO Longlists, 37
Let $S$ be a circle, and $\alpha =\{A_1,\ldots ,A_n\}$ a family of open arcs in $S$. Let $N(\alpha )=n$ denote the number of elements in $\alpha$. We say that $\alpha$ is a covering of $S$ if $\bigcup_{k=1}^n A_k\supset S$.
Let $\alpha=\{A_1,\ldots ,A_n\}$ and $\beta =\{B_1,\ldots ,B_m\}$ be two coverings of $S$.
Show that we can choose from the family of all sets $A_i\cap B_j,\ i=1,2,\ldots ,n,\ j=1, 2,\ldots ,m,$ a covering $\gamma$ of $S$ such that $N(\gamma )\le N(\alpha)+N(\beta)$.
2012 China Team Selection Test, 3
In some squares of a $2012\times 2012$ grid there are some beetles, such that no square contain more than one beetle. At one moment, all the beetles fly off the grid and then land on the grid again, also satisfying the condition that there is at most one beetle standing in each square. The vector from the centre of the square from which a beetle $B$ flies to the centre of the square on which it lands is called the [i]translation vector[/i] of beetle $B$.
For all possible starting and ending configurations, find the maximum length of the sum of the [i]translation vectors[/i] of all beetles.
2002 Romania National Olympiad, 1
Eight card players are seated around a table. One remarks that at some moment, any player and his two neighbours have altogether an odd number of winning cards.
Show that any player has at that moment at least one winning card.
2007 Bulgaria Team Selection Test, 4
Let $G$ is a graph and $x$ is a vertex of $G$. Define the transformation $\varphi_{x}$ over $G$ as deleting all incident edges with respect of $x$ and drawing the edges $xy$ such that $y\in G$ and $y$ is not connected with $x$ with edge in the beginning of the transformation. A graph $H$ is called $G-$[i]attainable[/i] if there exists a sequece of such transformations which transforms $G$ in $H.$ Let $n\in\mathbb{N}$ and $4|n.$ Prove that for each graph $G$ with $4n$ vertices and $n$ edges there exists $G-$[i]attainable[/i] graph with at least $9n^{2}/4$ triangles.