[makar]

A square is dissected in to 9 rectangles by lines parallel to its sides such that all these rectangles have integer sides. Prove that there are always two congruent rectangles.

[-Elixir-]

Sorry, this solution is kind of incomplete. I assumed there were only four lines.

Very nice.

If the sides are cut into and , you will form rectangles of for in .

Then,



Assume no congruent rectangles exist. We then know that no two of or can be equal. However, this implies and are permutations of .

But, you can always pick such that and , giving two congruent rectangles of different orientation. Hence there will always be two congruent rectangles.

[makar]

http://www.artofproblemsolving.com/viewtopic.php?s ... 2&t=122541

[-Elixir-]

makar wrote:

http://www.artofproblemsolving.com/viewtopic.php?search_id=820928092&t=122541

I can't believe the solution was that simple.