Date: Feb 1, 2013 10:50 AM
Author: mina_world
Subject: Discrete math with IMO.
Hello~ teacher...

2n ambassadors are invited to a banquet.

Every ambassador has at most (n-1) enemies.

Prove that the ambassadors can be seated around a round table,

so that nobody sits next to an enemy.

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Solution)

First, we seat the ambassadors in any way.

Let H be the number of neighboring hostile couples.

We must find an algorithm which reduces this number whenever H > 0.

Let (A, B) be a hostile couple with ? sitting to the right of A(Fig. 1.3).

We must separate them so as to cause as little disturbance as possible.

This will be achieved if we reverse some arc ? A' getting Fig. 1.4.

H will be reduced(***) if (A, A') and (?, ?') in Fig. 1.4 are friendly

couples.

It remains to be shown that

such a couple always exists with B' sitting to the right of A'. (***)

We start in A and go around the table counterclockwise.

We will encounter at least n friends of A.

To their right, there are at least n seats.

They cannot all be occupied by enemies of ? since ? has at most (n-1)

enemies.

Thus, there is a friend A' of A with right neighbor B', a friend of B.

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Hm... can you understand this solution ?

I need your explanation. specially (***) part.

For reference(text copy)

http://board-2.blueweb.co.kr/user/math565/data/math/olimo.jpg