IMO 1972 LL GBR18

Origin: GBR

Problem

We have p players participating in a tournament, each player playing against every other player exactly once. A point is scored for each victory, and there are no draws. A sequence of nonnegative integers s1 \leqs2 \leqs3 \leq\cdot \cdot \cdot \leqsp is given. Show that it is possible for this sequence to be a set of final scores of the players in the tournament if and only if (i) p  i=1 si = 1 2p(p −1) and (ii) for all k < p, k  i=1 si \geq1 2k(k −1).