RO-NMO 2006 Problem 94

$2n$ students $(n \geq 5)$ participated at table tennis contest, which took $4$ days. In every day, every student played a match. (It is possible that the same pair meets twice or more times, in different days) Prove that it is possible that the contest ends like this:

• there is only one winner;

• there are $3$ students on the second place;

• no student lost all $4$ matches.

How many students won only a single match and how many won exactly $2$ matches? (In the above conditions)