ILL 1987 Problem 28

In a chess tournament there are $n \geq 5$ players, and they have already played $\left[ \frac{n^2}{4} \right] +2$ 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 $\left[ \frac{n^2}{4} \right] +1$ games played?

Make the proof by induction over $n.$