MathDB

The old +1 and -1 problem on the board - [IMO LongList 1971]

Source:

January 1, 2011

modular arithmeticinvariantcombinatorics proposedcombinatorics

Problem Statement

We are given an

n \times n

board, where

n

is an odd number. In each cell of the board either

+1

-1

is written. Let

a_k

and

b_k

denote them products of numbers in the

k^{th}

row and in the

k^{th}

column respectively. Prove that the sum

a_1 +a_2 +\cdots+a_n +b_1 +b_2 +\cdots+b_n

cannot be equal to zero.