By Pablo Soberón

Every yr there's a minimum of one combinatorics challenge in all of the significant overseas mathematical olympiads. those difficulties can in basic terms be solved with a really excessive point of wit and creativity. This publication explains the entire problem-solving recommendations essential to take on those difficulties, with transparent examples from fresh contests. it is usually a wide challenge part for every subject, together with tricks and whole ideas in order that the reader can perform the fabric lined within the book. the fabric might be priceless not just to contributors within the olympiads and their coaches but additionally in collage classes on combinatorics.

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**Example text**

Vk−1 ) (or backwards) every time it was used. Thus we obtain a walk in the subgraph. With this we are no longer using the edge {v0 , vk−1 }, so the subgraph is connected. If we repeat this argument until no cycles remain, we obtain the tree we were looking for. The trees that satisfy this condition are called spanning trees of G. A connected graph can have more than one spanning tree. 3 Prove that every tree with n vertices has exactly n − 1 edges. Solution We prove this by induction on n. If n = 1, the assertion is clear.

Find the minimum number of squares that must be colored under these two rules. 6 (Germany 2009) On a table there are 100 coins. A and B are going to remove coins from the table by turns. In each turn they can remove 2, 5 or 6 coins. The first one that cannot make a move loses. Determine who has a winning strategy if A plays first. 7 (Middle European Mathematical Olympiad 2010) All positive divisors of a positive integer N are written on a blackboard. Two players A and B play the following game, taking alternate moves.

Since we have ab + 1 pairs, by the pigeonhole principle at least two must be equal, which is a contradiction. 2 Given any two positive integers a and b, find a sequence of ab different real numbers with no increasing subsequences of length a + 1 or more and no decreasing subsequences of length b + 1 or more. 4 23 An Application in Number Theory Besides its importance in combinatorics, the pigeonhole principle has various strong applications in number theory. One of the most known ones is in showing that the decimal representation of any rational number2 is periodic after some point.