Found problems: 1800
2006 International Zhautykov Olympiad, 1
In a pile you have 100 stones. A partition of the pile in $ k$ piles is [i]good[/i] if:
1) the small piles have different numbers of stones;
2) for any partition of one of the small piles in 2 smaller piles, among the $ k \plus{} 1$ piles you get 2 with the same number of stones (any pile has at least 1 stone).
Find the maximum and minimal values of $ k$ for which this is possible.
2012 Indonesia TST, 2
Let $T$ be the set of all 2-digit numbers whose digits are in $\{1,2,3,4,5,6\}$ and the tens digit is strictly smaller than the units digit. Suppose $S$ is a subset of $T$ such that it contains all six digits and no three numbers in $S$ use all six digits. If the cardinality of $S$ is $n$, find all possible values of $n$.
2001 Baltic Way, 3
The numbers $1, 2, \ldots 49$ are placed in a $7\times 7$ array, and the sum of the numbers in each row and in each column is computed. Some of these $14$ sums are odd while others are even. Let $A$ denote the sum of all the odd sums and $B$ the sum of all even sums. Is it possible that the numbers were placed in the array in such a way that $A = B$?
2006 ITAMO, 1
Two people play the following game: there are $40$ cards numbered from $1$ to $10$ with $4$ different signs. At the beginning they are given $20$ cards each. Each turn one player either puts a card on the table or removes some cards from the table, whose sum is $15$. At the end of the game, one player has a $5$ and a $3$ in his hand, on the table there's a $9$, the other player has a card in his hand. What is it's value?
2005 China Northern MO, 4
Let $A$ be the set of $n$-digit integers whose digits are all from $\{ 1, 2, 3, 4, 5 \}$. $B$ is subset of $A$ such that it contains digit $5$, and there is no digit $3$ in front of digit $5$ (i.e. for $n = 2$, $35$ is not allowed, but $53$ is allowed). How many elements does set $B$ have?
2010 JBMO Shortlist, 2
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 $n$.
2016 SGMO, Q3
In Simoland there are $2017n$ cities arranged in a $2017\times n$ lattice grid. There are $2016$ MRT (train) tracks and each track can only go north and east, or south and east. Suppose that all the tracks together pass through all the cities. Determine the largest possible value of $n$.
2011 CentroAmerican, 1
Consider a cube with a fly standing at each of its vertices. When a whistle blows, each fly moves to a vertex in the same face as the previous one but diagonally opposite to it. After the whistle blows, in how many ways can the flies change position so that there is no vertex with 2 or more flies?
2002 Tournament Of Towns, 4
There are $n$ lamps in a row. Some of which are on. Every minute all the lamps already on go off. Those which were off and were adjacent to exactly one lamp which was on will go on. For which $n$ one can find an initial configuration of lamps which were on, such that at least one lamp will be on at any time?
2009 Junior Balkan Team Selection Test, 3
On each field of the board $ n\times n$ there is one figure, where $n\ge 2$. In one move we move every figure on one of its diagonally adjacent fields. After one move on one field there can be more than one figure. Find the least number of fields on which there can be all figures after some number of moves.
2011 Iran MO (3rd Round), 6
Every bacterium has a horizontal body with natural length and some nonnegative number of vertical feet, each with nonnegative (!) natural length, that lie below its body. In how many ways can these bacteria fill an $m\times n$ table such that no two of them overlap?
[i]proposed by Mahyar Sefidgaran[/i]
2008 Kazakhstan National Olympiad, 3
Find the maximum number of planes in the space, such there are $ 6$ points, that satisfy to the following conditions:
[b]1.[/b]Each plane contains at least $ 4$ of them
[b]2.[/b]No four points are collinear.
2008 Romania Team Selection Test, 3
Let $ n \geq 3$ be a positive integer and let $ m \geq 2^{n\minus{}1}\plus{}1$. Prove that for each family of nonzero distinct subsets $ (A_j)_{j \in \overline{1, m}}$ of $ \{1, 2, ..., n\}$ there exist $ i$, $ j$, $ k$ such that $ A_i \cup A_j \equal{} A_k$.
2005 ITAMO, 3
In each cell of a $4 \times 4$ table a digit $1$ or $2$ is written. Suppose that the sum of the digits in each of the four $3 \times 3$ sub-tables is divisible by $4$, but the sum of the digits in the entire table is not divisible by $4$. Find the greatest and the smallest possible value of the sum of the $16$ digits.
2008 Bosnia And Herzegovina - Regional Olympiad, 3
A rectangular table $ 9$ rows $ \times$ $ 2008$ columns is fulfilled with numbers $ 1$, $ 2$, ...,$ 2008$ in a such way that each number appears exactly $ 9$ times in table and difference between any two numbers from same column is not greater than $ 3$. What is maximum value of minimum sum in column (with minimal sum)?
1998 Baltic Way, 19
Consider a ping-pong match between two teams, each consisting of $1000$ players. Each player played against each player of the other team exactly once (there are no draws in ping-pong). Prove that there exist ten players, all from the same team, such that every member of the other team has lost his game against at least one of those ten players.
2007 Vietnam Team Selection Test, 5
Let $A\subset \{1,2,\ldots,4014\}$, $|A|=2007$, such that $a$ does not divide $b$ for all distinct elements $a,b\in A$. For a set $X$ as above let us denote with $m_{X}$ the smallest element in $X$. Find $\min m_{A}$ (for all $A$ with the above properties).
2005 Kyiv Mathematical Festival, 3
Two players by turn paint the vertices of triangles on the given picture each with his colour. At the end, each of small triangles is painted by the colour of the majority of its vertices. The winner is one who gets at least 6 triangles of his colour. If both players get at most 5, then it is a draw. Does any of them have winning strategy? If yes, then who wins?
\[ \begin{picture}(40,50) \put(2,2){\put(0,0){\line(6,0){42}} \put(7,14){\line(6,0){28}} \put(14,28){\line(6,0){14}} \put(0,0){\line(1,2){21}} \put(14,0){\line(1,2){14}} \put(28,0){\line(1,2){7}} \put(14,28){\line(1,2){7}} \put(14,0){\line( \minus{} 1,2){7}} \put(28,0){\line( \minus{} 1,2){14}} \put(42,0){\line( \minus{} 1,2){21}} \put(0,0){\circle*{3}} \put(14,0){\circle*{3}} \put(28,0){\circle*{3}} \put(42,0){\circle*{3}} \put(7,14){\circle*{3}} \put(21,14){\circle*{3}} \put(35,14){\circle*{3}} \put(14,28){\circle*{3}} \put(28,28){\circle*{3}} \put(21,42){\circle*{3}}} \end{picture}\]
2010 Baltic Way, 7
There are some cities in a country; one of them is the capital. For any two cities $A$ and $B$ there is a direct flight from $A$ to $B$ and a direct flight from $B$ to $A$, both having the same price. Suppose that all round trips with exactly one landing in every city have the same total cost. Prove that all round trips that miss the capital and with exactly one landing in every remaining city cost the same.
2014 Contests, 2
Let $ k\geq 1 $ and let $ I_{1},\dots, I_{k} $ be non-degenerate subintervals of the interval $ [0, 1] $. Prove that
\[ \sum \frac{1}{\left | I_{i}\cup I_{j} \right |} \geq k^{2} \]
where the summation is over all pairs $ (i, j) $ of indices such that $I_i\cap I_j\neq \emptyset$.
2012 Iran Team Selection Test, 1
Consider $m+1$ horizontal and $n+1$ vertical lines ($m,n\ge 4$) in the plane forming an $m\times n$ table. Cosider a closed path on the segments of this table such that it does not intersect itself and also it passes through all $(m-1)(n-1)$ interior vertices (each vertex is an intersection point of two lines) and it doesn't pass through any of outer vertices. Suppose $A$ is the number of vertices such that the path passes through them straight forward, $B$ number of the table squares that only their two opposite sides are used in the path, and $C$ number of the table squares that none of their sides is used in the path. Prove that
\[A=B-C+m+n-1.\]
[i]Proposed by Ali Khezeli[/i]
2008 China Team Selection Test, 2
Prove that for arbitary integer $ n > 16$, there exists the set $ S$ that contains $ n$ positive integers and has the following property:if the subset $ A$ of $ S$ satisfies for arbitary $ a,a'\in A, a\neq a', a \plus{} a'\notin S$ holds, then $ |A|\leq4\sqrt n.$
2010 Baltic Way, 10
Let $n$ be an integer with $n\ge 3$. Consider all dissections of a convex $n$-gon into triangles by $n-3$ non-intersecting diagonals, and all colourings of the triangles with black and white so that triangles with a common side are always of a different colour. Find the least possible number of black triangles.
2006 Austrian-Polish Competition, 9
We have an 8x8 chessboard with 64 squares. Then we have 3x1 dominoes which cover exactly 3 squares. Such dominoes can only be moved parallel to the borders of the chessboard and also only if the passing squares are free. If no dominoes can be moved, then the position is called stable.
a. Find the smalles number of covered squares neccessary for a stable position.
b. Prove: There exist a stable position with only one square uncovered.
c. Find all Squares which are uncoverd in at least one position of b).
2003 All-Russian Olympiad, 3
On a line are given $2k -1$ white segments and $2k -1$ black ones. Assume that each white segment intersects at least $k$ black segments, and each black segment intersects at least $k$ white ones. Prove that there are a black segment intersecting all the white ones, and a white segment intersecting all the black ones.