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Topic: Discrete math with IMO.
Replies: 5   Last Post: Feb 2, 2013 4:51 AM

 Messages: [ Previous | Next ]
 mina_world Posts: 2,144 Registered: 12/13/04
Discrete math with IMO.
Posted: Feb 1, 2013 10:50 AM

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

Date Subject Author
2/1/13 mina_world
2/2/13 William Elliot
2/2/13 quasi
2/2/13 mina_world
2/2/13 fom
2/2/13 fom