

A1. Let p_{n}(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 p_{n}(k) ) is n!.
[A permutation f of a set S is a onetoone mapping of S onto itself. An element i of S is called a fixed point if f(i) = i.]


A2. In an acuteangled triangle ABC the interior bisector of angle A meets BC at L and meets the circumcircle of ABC again at N. From L perpendiculars are drawn to AB and AC, with feet K and M respectively. Prove that the quadrilateral AKNM and the triangle ABC have equal areas.


A3. Let x_{1}, x_{2}, ... , x_{n} be real numbers satisfying x_{1}^{2} + x_{2}^{2} + ... + x_{n}^{2} = 1. Prove that for every integer k ≥ 2 there are integers a_{1}, a_{2}, ... , a_{n}, not all zero, such that a_{i} ≤ k  1 for all i, and a_{1}x_{1} + a_{2}x_{2} + ... + a_{n}x_{n} ≤ (k  1)√n/(k^{n}  1).


B1. Prove that there is no function f from the set of nonnegative integers into itself such that f(f(n)) = n + 1987 for all n.


B2. Let n be an integer greater than or equal to 3. Prove that there is a set of n points in the plane such that the distance between any two points is irrational and each set of 3 points determines a nondegenerate triangle with rational area.


B3. Let n be an integer greater than or equal to 2. Prove that if k^{2} + k + n is prime for all integers k such that 0 ≤ k ≤ √(n/3), then k^{2} + k + n is prime for all integers k such that 0 ≤ k ≤ n2.

