There are 17 mathematicians and 3 official languages. Every pair of mathematicians communicate in one of the official languages. Prove that there are 3 mathematicians communicating in the same language, pairwise.
I guess pigeonhole principle must be applied but I cannot see how to move forward with it. So, let us consider the three languages (say, A, B and C) as pigeonholes. By pigeonhole principle, at least one of A, B and C must contain at least ceil(17/3) = 6 pigeons. So, 6 mathematicians must be there and all of them know the same language. I can only see this.