Found problems: 1488
2010 Tuymaada Olympiad, 1
Misha and Sahsa play a game on a $100\times 100$ chessboard. First, Sasha places $50$ kings on the board, and Misha places a rook, and then they move in turns, as following (Sasha begins):
At his move, Sasha moves each of the kings one square in any direction, and Misha can move the rook on the horizontal or vertical any number of squares. The kings cannot be captured or stepped over. Sasha's purpose is to capture the rook, and Misha's is to avoid capture.
Is there a winning strategy available for Sasha?
2010 Contests, 1
A table $2 \times 2010$ is divided to unit cells. Ivan and Peter are playing the following game. Ivan starts, and puts horizontal $2 \times 1$ domino that covers exactly two unit table cells. Then Peter puts vertical $1 \times 2$ domino that covers exactly two unit table cells. Then Ivan puts horizontal domino. Then Peter puts vertical domino, etc. The person who cannot put his domino will lose the game. Find who have winning strategy.
2001 Turkey MO (2nd round), 3
We wish to color the cells of a $n \times n$ chessboard with $k$ different colors such that for every $i\in \{1,2,...,n\}$, the $2n-1$ cells on $i$. row and $i$. column have all different colors.
a) Prove that for $n=2001$ and $k=4001$, such coloring is not possible.
b) Show that for $n=2^{m}-1$ and $k=2^{m+1}-1$, such coloring is possible.
1987 China National Olympiad, 3
Some players participate in a competition. Suppose that each player plays one game against every other player and there is no draw game in the competition. Player $A$ is regarded as an excellent player if the following condition is satisfied: for any other player $B$, either $A$ beats $B$ or there exists another player $C$ such that $C$ beats $B$ and $A$ beats $C$. It is known that there is only one excellent player in the end, prove that this player beats all other players.
2003 Tournament Of Towns, 4
A chocolate bar in the shape of an equilateral triangle with side of the length $n$, consists of triangular chips with sides of the length $1$, parallel to sides of the bar. Two players take turns eating up the chocolate. Each player breaks off a triangular piece (along one of the lines), eats it up and passes leftovers to the other player (as long as bar contains more than one chip, the player is not allowed to eat it completely).
A player who has no move or leaves exactly one chip to the opponent, loses. For each $n$, find who has a winning strategy.
2009 Czech-Polish-Slovak Match, 6
Let $n\ge 16$ be an integer, and consider the set of $n^2$ points in the plane: \[ G=\big\{(x,y)\mid x,y\in\{1,2,\ldots,n\}\big\}.\] Let $A$ be a subset of $G$ with at least $4n\sqrt{n}$ elements. Prove that there are at least $n^2$ convex quadrilaterals whose vertices are in $A$ and all of whose diagonals pass through a fixed point.
2003 Tournament Of Towns, 2
Two players in turns color the sides of an $n$-gon. The first player colors any side that has $0$ or $2$ common vertices with already colored sides. The second player colors any side that has exactly $1$ common vertex with already colored sides. The player who cannot move, loses. For which $n$ the second player has a winning strategy?
1987 China National Olympiad, 2
We are given an equilateral triangle ABC with the length of its side equal to $1$. There are $n-1$ points on each side of the triangle $ABC$ that equally divide the side into $n$ segments. We draw all possible lines that pass through any two of all those $3(n-1)$ points such that they are parallel to one of three sides of triangle $ABC$. All such lines divide triangle $ABC$ into some lesser triangles whose vertices are called [i]nodes[/i]. We assign a real number for each [i]node[/i] such that the following conditions are satisfied:
(I) real numbers $a,b,c$ are assigned to $A,B,C$ respectively;
(II) for any rhombus that is consisted of two lesser triangles that share a common side, the sum of the numbers of vertices on its one diagonal is equal to that of vertices on the other diagonal.
1) Find the minimum distance between the [i]node[/i] with the maximal number to the [i]node[/i] with the minimal number;
2) Denote by $S$ the sum of the numbers of all [i]nodes[/i], find $S$.
1999 China Team Selection Test, 3
Let $S = \lbrace 1, 2, \ldots, 15 \rbrace$. Let $A_1, A_2, \ldots, A_n$ be $n$ subsets of $S$ which satisfy the following conditions:
[b]I.[/b] $|A_i| = 7, i = 1, 2, \ldots, n$;
[b]II.[/b] $|A_i \cap A_j| \leq 3, 1 \leq i < j \leq n$
[b]III.[/b] For any 3-element subset $M$ of $S$, there exists $A_k$ such
that $M \subset A_k$.
Find the smallest possible value of $n$.
2003 Korea - Final Round, 3
There are $n$ distinct points on a circumference. Choose one of the points. Connect this point and the $m$th point from the chosen point counterclockwise with a segment. Connect this $m$th point and the $m$th point from this $m$th point counterclockwise with a segment. Repeat such steps until no new segment is constructed. From the intersections of the segments, let the number of the intersections - which are in the circle - be $I$. Answer the following questions ($m$ and $n$ are positive integers that are relatively prime and they satisfy $6 \leq 2m < n$).
1) When the $n$ points take different positions, express the maximum value of $I$ in terms of $m$ and $n$.
2) Prove that $I \geq n$. Prove that there is a case, which is $I=n$, when $m=3$ and $n$ is arbitrary even number that satisfies the condition.
1982 IMO Longlists, 21
Al[u][b]l[/b][/u] edges and all diagonals of regular hexagon $A_1A_2A_3A_4A_5A_6$ are colored blue or red such that each triangle $A_jA_kA_m, 1 \leq j < k < m\leq 6$ has at least one red edge. Let $R_k$ be the number of red segments $A_kA_j, (j \neq k)$. Prove the inequality
\[\sum_{k=1}^6 (2R_k-7)^2 \leq 54.\]
1994 USAMO, 2
The sides of a 99-gon are initially colored so that consecutive sides are red, blue, red, blue, $\,\ldots, \,$ red, blue, yellow. We make a sequence of modifications in the coloring, changing the color of one side at a time to one of the three given colors (red, blue, yellow), under the constraint that no two adjacent sides may be the same color. By making a sequence of such modifications, is it possible to arrive at the coloring in which consecutive sides
are red, blue, red, blue, red, blue, $\, \ldots, \,$ red, yellow, blue?
2001 India National Olympiad, 4
Show that given any nine integers, we can find four, $a, b, c, d$ such that $a + b - c - d$is divisible by $20$. Show that this is not always true for eight integers.
1993 All-Russian Olympiad, 3
What is the maximum number of checkers it is possible to put on a $ n \times n$ chessboard such that in every row and in every column there is an even number of checkers?
2003 CHKMO, 2
In conference there $n>2$ mathematicians. Every two mathematicians communicate in one of the $n$ offical languages of the conference. For any three different offical languages the exists three mathematicians who communicate with each other in these three languages. Find all $n$ such that this is possible.
2010 Costa Rica - Final Round, 6
Let $F$ be the family of all sets of positive integers with $2010$ elements that satisfy the following condition:
The difference between any two of its elements is never the same as the difference of any other two of its elements. Let $f$ be a function defined from $F$ to the positive integers such that $f(K)$ is the biggest element of $K \in F$. Determine the least value of $f(K)$.
2008 Ukraine Team Selection Test, 2
There is a row that consists of digits from $ 0$ to $ 9$ and Ukrainian letters (there are $ 33$ of them) with following properties: there aren’t two distinct digits or letters $ a_i$, $ a_j$ such that $ a_i > a_j$ and $ i < j$ (if $ a_i$, $ a_j$ are letters $ a_i > a_j$ means that $ a_i$ has greater then $ a_j$ position in alphabet) and there aren’t two equal consecutive symbols or two equal symbols having exactly one symbol between them. Find the greatest possible number of symbols in such row.
2009 All-Russian Olympiad, 1
In a country, there are some cities linked together by roads. The roads just meet each other inside the cities. In each city, there is a board which showing the shortest length of the road originating in that city and going through all other cities (the way can go through some cities more than one times and is not necessary to turn back to the originated city). Prove that 2 random numbers in the boards can't be greater or lesser than 1.5 times than each other.
2011 Philippine MO, 1
Find all nonempty finite sets $X$ of real numbers such that for all $x\in X$, $x+|x| \in X$.
2006 Lithuania Team Selection Test, 4
Prove that in every polygon there is a diagonal that cuts off a triangle and lies within the polygon.
2008 Croatia Team Selection Test, 4
Let $ S$ be the set of all odd positive integers less than $ 30m$ which are not multiples of $ 5$, where $ m$ is a given positive integer. Find the smallest positive integer $ k$ such that each $ k$-element subset of $ S$ contains two distinct numbers, one of which divides the other.
2007 Tournament Of Towns, 6
The audience arranges $n$ coins in a row. The sequence of heads and tails is chosen arbitrarily. The audience also chooses a number between $1$ and $n$ inclusive. Then the assistant turns one of the coins over, and the magician is brought in to examine the resulting sequence. By an agreement with the assistant beforehand, the magician tries to determine the number chosen by the audience.
[list][b](a)[/b] Prove that if this is possible for some $n$, then it is also possible for $2n$.
[b](b)[/b] Determine all $n$ for which this is possible.[/list]
2010 China Team Selection Test, 1
Let $G=G(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. Suppose $|V|=n$. A map $f:\,V\rightarrow\mathbb{Z}$ is called good, if $f$ satisfies the followings:
(1) $\sum_{v\in V} f(v)=|E|$;
(2) color arbitarily some vertices into red, one can always find a red vertex $v$ such that $f(v)$ is no more than the number of uncolored vertices adjacent to $v$.
Let $m(G)$ be the number of good maps. Prove that if every vertex in $G$ is adjacent to at least one another vertex, then $n\leq m(G)\leq n!$.
2014 Tuymaada Olympiad, 8
There are $m$ villages on the left bank of the Lena, $n$ villages on the right bank and one village on an island. It is known that $(m+1,n+1)>1$. Every two villages separated by water are connected by ferry with positive integral number.
The inhabitants of each village say that all the ferries operating in their village have different numbers and these numbers form a segment of the series of the integers. Prove that at least some of them are wrong.
[i](K. Kokhas)[/i]
1998 South africa National Olympiad, 4
In a group of people, every two people have exactly one friend in common. Prove that there is a person who is a friend of everyone else.