1978 USAMO #5
Source:
August 16, 2011
AMCUSA(J)MOUSAMOpigeonhole principlefloor functionRamsey Theory
Problem Statement
Nine mathematicians meet at an international conference and discover that among any three of them, at least two speak a common language. If each of the mathematicians speak at most three languages, prove that there are at least three of the mathematicians who can speak the same language.