Found problems: 1800
2004 Bulgaria Team Selection Test, 3
In any cell of an $n \times n$ table a number is written such that all the rows are distinct. Prove that we can remove a column such that the rows in the new table are still distinct.
2012 Bulgaria National Olympiad, 1
Let $n$ be an even natural number and let $A$ be the set of all non-zero sequences of length $n$, consisting of numbers $0$ and $1$ (length $n$ binary sequences, except the zero sequence $(0,0,\ldots,0)$). Prove that $A$ can be partitioned into groups of three elements, so that for every triad $\{(a_1,a_2,\ldots,a_n), (b_1,b_2,\ldots,b_n), (c_1,c_2,\ldots,c_n)\}$, and for every $i = 1, 2,\ldots,n$, exactly zero or two of the numbers $a_i, b_i, c_i$ are equal to $1$.
2009 Indonesia TST, 1
Ati has $ 7$ pots of flower, ordered in $ P_1,P_2,P_3,P_4,P_5,P_6,P_7$. She wants to rearrange the position of those pots to $ B_1,B_2,B_2,B_3,B_4,B_5,B_6,B_7$ such that for every positive integer $ n<7$, $ B_1,B_2,\dots,B_n$ is not the permutation of $ P_1,P_2,\dots,P_7$. In how many ways can Ati do this?
2002 Baltic Way, 12
A set $S$ of four distinct points is given in the plane. It is known that for any point $X\in S$ the remaining points can be denoted by $Y,Z$ and $W$ so that
$|XY|=|XZ|+|XW|$
Prove that all four points lie on a line.
2007 Kyiv Mathematical Festival, 3
a) One has a set of stones with weights $1, 2, \ldots, 20$ grams. Find all $k$ for which it is possible to place $k$ and the rest $20-k$ stones from the set respectively on the two pans of a balance so that equilibrium is achieved.
b) One has a set of stones with weights $1, 2, \ldots, 51$ grams. Find all $k$ for which it is possible to place $k$ and the rest $51-k$ stones from the set respectively on the two pans of a balance so that equilibrium is achieved.
c) One has a set of stones with weights $1, 2, \ldots, n$ grams ($n\in\mathbb{N}$). Find all $n$ and $k$ for which it is possible to place $k$ and the rest $n-k$ stones from the set respectively on the two pans of a balance so that equilibrium is achieved.
[size=75] a) and b) were proposed at the festival, c) is a generalization[/size]
2010 Kazakhstan National Olympiad, 2
Exactly $4n$ numbers in set $A= \{ 1,2,3,...,6n \} $ of natural numbers painted in red, all other in blue.
Proved that exist $3n$ consecutive natural numbers from $A$, exactly $2n$ of which numbers is red.
2005 Turkey Team Selection Test, 3
Initially the numbers 1 through 2005 are marked. A finite set of marked consecutive integers is called a block if it is not contained in any larger set of marked consecutive integers. In each step we select a set of marked integers which does not contain the first or last element of any block, unmark the selected integers, and mark the same number of consecutive integers starting with the integer two greater than the largest marked integer. What is the minimum number of steps necessary to obtain 2005 single integer blocks?
2004 Iran Team Selection Test, 5
This problem is generalization of [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=5918]this one[/url].
Suppose $G$ is a graph and $S\subset V(G)$. Suppose we have arbitrarily assign real numbers to each element of $S$. Prove that we can assign numbers to each vertex in $G\backslash S$ that for each $v\in G\backslash S$ number assigned to $v$ is average of its neighbors.
2014 Korea National Olympiad, 4
There is a city with $n$ metro stations, each located at a vertex of a regular n-polygon. Metro Line 1 is a line which only connects two non-neighboring stations $A$ and $B$. Metro Line 2 is a cyclic line which passes through all the stations in a shape of regular n-polygon. For each line metro can run in any direction, and $A$ and $B$ are the stations which one can transfer into other line. The line between two neighboring stations is called 'metro interval'. For each station there is one stationmaster, and there are at least one female stationmaster and one male stationmaster. If $n$ is odd, prove that for any integer $k$ $(0<k<n)$ there is a path that starts from a station with a male stationmaster and ends at a station with a female stationmaster, passing through $k$ metro intervals.
1991 IberoAmerican, 1
Each vertex of a cube is assigned an 1 or a -1, and each face is assigned the product of the numbers assigned to its vertices. Determine the possible values the sum of these 14 numbers can attain.
2004 Bulgaria Team Selection Test, 2
The edges of a graph with $2n$ vertices ($n \ge 4$) are colored in blue and red such that there is no blue triangle and there is no red complete subgraph with $n$ vertices. Find the least possible number of blue edges.
2013 Middle European Mathematical Olympiad, 3
There are $n \ge 2$ houses on the northern side of a street. Going from the west to the east, the houses are numbered from 1 to $n$. The number of each house is shown on a plate. One day the inhabitants of the street make fun of the postman by shuffling their number plates in the following way: for each pair of neighbouring houses, the currnet number plates are swapped exactly once during the day.
How many different sequences of number plates are possible at the end of the day?
2013 All-Russian Olympiad, 4
$N$ lines lie on a plane, no two of which are parallel and no three of which are concurrent. Prove that there exists a non-self-intersecting broken line $A_0A_1A_2A_3...A_N$ with $N$ parts, such that on each of the $N$ lines lies exactly one of the $N$ segments of the line.
2006 India National Olympiad, 4
Some 46 squares are randomly chosen from a $9 \times 9$ chess board and colored in [color=red]red[/color]. Show that there exists a $2\times 2$ block of 4 squares of which at least three are colored in [color=red]red[/color].
2011 All-Russian Olympiad Regional Round, 9.3
A closed non-self-intersecting polygonal chain is drawn through the centers of some squares on the $8\times 8$ chess board. Every link of the chain connects the centers of adjacent squares either horizontally, vertically or diagonally, where the two squares are adjacent if they share an edge or a corner. For the interior polygon bounded by the chain, prove that the total area of black pieces equals the total area of white pieces. (Author: D. Khramtsov)
1985 Balkan MO, 4
There are $1985$ participants to an international meeting. In any group of three participants there are at least two who speak the same language. It is known that each participant speaks at most five languages. Prove that there exist at least $200$ participans who speak the same language.
1951 Miklós Schweitzer, 6
In lawn-tennis the player who scores at least four points, while his opponent scores at least two points less, wins a game. The player who wins at least six games, while his opponent wins at least two games less, wins a set. What minimum percentage of all points does the winner have to score in a set?
2006 Iran MO (3rd Round), 4
Let $D$ be a family of $s$-element subsets of $\{1.\ldots,n\}$ such that every $k$ members of $D$ have non-empty intersection. Denote by $D(n,s,k)$ the maximum cardinality of such a family.
a) Find $D(n,s,4)$.
b) Find $D(n,s,3)$.
2009 India IMO Training Camp, 12
Let $ G$ be a simple graph with vertex set $ V\equal{}\{0,1,2,3,\cdots ,n\plus{}1\}$ .$ j$and$ j\plus{}1$ are connected by an edge for $ 0\le j\le n$. Let $ A$ be a subset of $ V$ and $ G(A)$ be the induced subgraph associated with $ A$. Let $ O(G(A))$ be number of components of $ G(A)$ having an odd number of vertices.
Let
$ T(p,r)\equal{}\{A\subset V \mid 0.n\plus{}1 \notin A,|A|\equal{}p,O(G(A))\equal{}2r\}$ for $ r\le p \le 2r$.
Prove That $ |T(p,r)|\equal{}{n\minus{}r \choose{p\minus{}r}}{n\minus{}p\plus{}1 \choose{2r\minus{}p}}$.
2012 All-Russian Olympiad, 4
In a city's bus route system, any two routes share exactly one stop, and every route includes at least four stops. Prove that the stops can be classified into two groups such that each route includes stops from each group.
2010 Middle European Mathematical Olympiad, 8
Let $n$ be a positive integer. A square $ABCD$ is partitioned into $n^2$ unit squares. Each of them is divided into two triangles by the diagonal parallel to $BD$. Some of the vertices of the unit squares are colored red in such a way that each of these $2n^2$ triangles contains at least one red vertex. Find the least number of red vertices.
[i](4th Middle European Mathematical Olympiad, Team Competition, Problem 4)[/i]
2008 Poland - Second Round, 1
We have an $n \times n$ board, and in every square there is an integer. The sum of all integers on the board is $0$. We define an action on a square where the integer in the square is decreased by the number of neighbouring squares, and the number inside each of the neighbouring squares is increased by 1. Determine if there exists $n\geq 2$ such that we can turn all the integers into zeros in a finite number of actions.
2014 China Team Selection Test, 5
Find the smallest positive constant $c$ satisfying: For any simple graph $G=G(V,E)$, if $|E|\geq c|V|$, then $G$ contains $2$ cycles with no common vertex, and one of them contains a chord.
Note: The cycle of graph $G(V,E)$ is a set of distinct vertices ${v_1,v_2...,v_n}\subseteq V$, $v_iv_{i+1}\in E$ for all $1\leq i\leq n$ $(n\geq 3, v_{n+1}=v_1)$; a cycle containing a chord is the cycle ${v_1,v_2...,v_n}$, such that there exist $i,j, 1< i-j< n-1$, satisfying $v_iv_j\in E$.
2024 Austrian MO National Competition, 3
Initially, the numbers $1, 2, \dots, 2024$ are written on a blackboard. Trixi and Nana play a game, taking alternate turns. Trixi plays first.
The player whose turn it is chooses two numbers $a$ and $b$, erases both, and writes their (possibly negative) difference $a-b$ on the blackboard. This is repeated until only one number remains on the blackboard after $2023$ moves. Trixi wins if this number is divisible by $3$, otherwise Nana wins.
Which of the two has a winning strategy?
[i](Birgit Vera Schmidt)[/i]
2007 Romania Team Selection Test, 3
Three travel companies provide transportation between $n$ cities, such that each connection between a pair of cities is covered by one company only. Prove that, for $n \geq 11$, there must exist a round-trip through some four cities, using the services of a same company, while for $n < 11$ this is not anymore necessarily true.
[i]Dan Schwarz[/i]