IR-MO 2002 Problem 13

$f,g$ are two permutations of set $X=\{1,\dots,n\}$ . We say $f,g$ have common points iff there is a $k\in X$ that $f(k)=g(k)$ . a) If $m>\frac{n}{2}$ , prove that there are $m$ permutations $f_{1},f_{2},\dots,f_{m}$ from $X$ that for each permutation $f\in X$ , there is an index $i$ that $f,f_{i}$ have common points. b) Prove that if $m\leq\frac{n}{2}$ , we can not find permutations $f_{1},f_{2},\dots,f_{m}$ satisfying the above condition.