28th International Mathematical Olympiad 1987 Problems & Solutions

A1.  Let pn(k) be the number of permutations of the set {1, 2, 3, ... , n} which have exactly k fixed points. Prove that the sum from k = 0 to n of (k pn(k) ) is n!. [A permutation f of a set S is a one-to-one mapping of S onto itself. An element i of S is called a fixed point if f(i) = i.] A2.  In an acute-angled triangle ABC the interior bisector of angle A meets BC at L