On a mathematical competition n problems were given. The final results showed that:
(i) on each problem, exactly three contestants scored 7 points;
(ii) for each pair of problems, exactly one contestant scored 7 points on both problems.
Prove that if n≥8, then there is a contestant who got 7 points on each problem. Is this statement necessarily true if n \equal{} 7? combinatorics unsolvedcombinatorics