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

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William Elliot

Posts: 2,212
Registered: 1/8/12
Re: Discrete math with IMO.
Posted: Feb 2, 2013 3:23 AM
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On Sat, 2 Feb 2013, mina_world wrote:

> 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.

Let n = 2 and a,b,c all have the enemy d while d has no enemies.

For example, Alice, Betty, Carol and Diane all are after John.
D dates J and now A,B,C have the enemy D while D, out with J,
has no enemy; she has J.

Who will sit next to d?

> 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.
> -----------------------------------------------------------------------------------------------
> 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

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