Here are some interesting problems I encountered during the last week:

1.Several identical coins are placed on the table without overlapping. Prove that there is a coin which touches no more than three other coins.

2.Several coins are placed on the table without overlapping.Prove that there is a coin that touches no more than five other coins.

3.Several coins are placed on the table without overlapping.Prove that one can slide one of the coins to the edge of table without moving any other coin.

In next problem one can apply the method of small perturbations which is related to the method of extreme case (see previous post):

4.A straight line intersects a polygon exactly 2001 times.Prove that there is a straight line which is not parallel to any side of polygon and has more than 2001 common points with it.

5.There are 100 chords in a circle such that any two of them intersect.Is it always possible to draw one more chord such that it intersects all of them?

## Friday, January 25, 2008

### Coins

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