IMO 1987 LL GDR28

In a chess tournament there are n \geq5 players, and they have

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IMO 1987 LL GDR28

Origin: GDR

Problem

In a chess tournament there are n \geq5 players, and they have already played  n2 

  • 2 games (each pair have played each other at most once). (a) Prove that there are five players a, b, c, d, e for which the pairs ab, ac, bc, ad, ae, de have already played. (b) Is the statement also valid for the  n2 
  • 1 games played? Make the proof by induction over n.