2003 Tournament Of Towns

Part of Tournament Of Towns

Subcontests

(7)

3

Can it happen that all these numbers on vertices are even?

A square is triangulated in such way that no three vertices are collinear. For every vertex (including vertices of the square) the number of sides issuing from it is counted. Can it happen that all these numbers are even?

Winning strategy for a card game

Two players in turn play a game. First Player has cards with numbers

2, 4, \ldots, 2000

while Second Player has cards with numbers

1, 3, \ldots, 2001

. In each his turn, a player chooses one of his cards and puts it on a table; the opponent sees it and puts his card next to the first one. Player, who put the card with a larger number, scores 1 point. Then both cards are discarded. First Player starts. After

1000

turns the game is over; First Player has used all his cards and Second Player used all but one. What are the maximal scores, that players could guarantee for themselves, no matter how the opponent would play?

Reducing a table by a given operation.

m \times n

table is filled with signs

"+"

"-"

. A table is called irreducible if one cannot reduce it to the table filled with

"+"

, applying the following operations (as many times as one wishes).

a)

It is allowed to flip all the signs in a row or in a column. Prove that an irreducible table contains an irreducible

2\times 2

b)

It is allowed to flip all the signs in a row or in a column or on a diagonal (corner cells are diagonals of length

1

). Prove that an irreducible table contains an irreducible

4\times 4

4

7

8

7

8

5