even odd even odd

For discussing Olympiad Level Combinatorics problems
asif e elahi
Posts: 183
Joined: Mon Aug 05, 2013 12:36 pm

even odd even odd

A $2015\times 2015$ grid is coloured like a chessboard so that the four corner squres are coloured black. We put pebbles in some of the cells so that every row and column contains an odd number of cells with pebbles. Prove that there are an even number of white cells with pebbles.

Golam Musabbir Joy
Posts: 11
Joined: Tue Jun 16, 2015 5:11 am

Re: even odd even odd

Can any row or column be empty?

asif e elahi
Posts: 183
Joined: Mon Aug 05, 2013 12:36 pm

Re: even odd even odd

Golam Musabbir Joy wrote:Can any row or column be empty?
No, as $0$ is an even number.

Posts: 22
Joined: Sat Dec 14, 2013 3:28 pm

Re: even odd even odd

Hint:
Why so SERIOUS?!??!

asif e elahi
Posts: 183
Joined: Mon Aug 05, 2013 12:36 pm

Re: even odd even odd

Nayeemul Islam Swad wrote:Hint:
I don't see how to use this hint. Write your whole solution please.

Posts: 22
Joined: Sat Dec 14, 2013 3:28 pm

Re: even odd even odd

Solution:
Why so SERIOUS?!??!

asif e elahi
Posts: 183
Joined: Mon Aug 05, 2013 12:36 pm

Re: even odd even odd

Nayeemul Islam Swad wrote:Solution:
oka
There is an easier solution. Just take the $1,3,5,\dots 2015th$ rows and $1,3,5\dots 2015th$ columns and prove that the number of squares with pebbles i these rows and columns is even. White squares are counted once and all the black squares are counted twice. The conclusion follows.

I used this idea to solve $IMO$ $2016$ $P2$ in the contest.