even odd even odd
 asif e elahi
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 Location: Sylhet,Bangladesh
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.

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

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 asif e elahi
 Posts: 183
 Joined: Mon Aug 05, 2013 12:36 pm
 Location: Sylhet,Bangladesh
Re: even odd even odd
I don't see how to use this hint. Write your whole solution please.Nayeemul Islam Swad wrote:Hint:

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 asif e elahi
 Posts: 183
 Joined: Mon Aug 05, 2013 12:36 pm
 Location: Sylhet,Bangladesh
Re: even odd even odd
okaNayeemul Islam Swad wrote:Solution:
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.

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Re: even odd even odd
Nice solu joss application
^ Little typo which might confuse others...
"Black squares are counted twice and all the white squares are counted once."
^ Little typo which might confuse others...
"Black squares are counted twice and all the white squares are counted once."
Why so SERIOUS?!??!