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I'm preparing myself to a combinatorics test. A part of it will concentrate on the pigeonhole principle. Thus, I need some hard to very hard problems in the subject to solve. I would be thankful if you can send me links\books\or just a lone problem.

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This will be very difficult to answer since "hard to very hard" is not well-defined if we don't know more about your background. To start, have a look at these: jan.ucc.nau.edu/~mk84/226/worksheets/pigeonholesol.pdf and math.utah.edu/mathcircle/notes/pigeon.pdf. Are these too easy? Are they suitable? – M Turgeon Sep 11 '12 at 19:51
These aren't so taugh.. – Tomer Galanti Sep 13 '12 at 7:34

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There are many such problems (with solutions!) in the Math.SE archives. You can try other keywords in the search box at the top right of the page to look for more problems.

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The famous Proofs from the book contains a chapter on Pigeon-hole and double counting. You can find there several cute applications of the pigeon-hole principle.

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This turned up in a routine google search of the phrase "pidgeonhole principle exercise" and appears to be training problems for the New Zealand olympiad team. It contains numerous problems and has some solutions in the back.

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