7
Part of 2003 Tournament Of Towns
Problems(3)
Can it happen that all these numbers on vertices are even?
Source: Tournament of Towns Spring 2003 - Senior A-Level - Problem 7
6/15/2011
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?
combinatorics proposedcombinatorics
Winning strategy for a card game
Source: ToT 2003-JA-7, SA-5
6/19/2011
Two players in turn play a game. First Player has cards with numbers while Second Player has cards with numbers . 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 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?
combinatorics unsolvedcombinatorics
Reducing a table by a given operation.
Source: ToT 2003 SA-7
7/3/2011
A table is filled with signs and . 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).
It is allowed to flip all the signs in a row or in a column. Prove that an irreducible table contains an irreducible sub table.
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 ). Prove that an irreducible table contains an irreducible sub table.
inductionlinear algebramatrixcombinatorics unsolvedcombinatorics