Toán học, Đề thi toán vô địch thế giới IMO, 1998

A1. In the convex quadrilateral ABCD, the diagonals AC and BD are perpendicular and the opposite sides AB and DC are not parallel. The point P, where the perpendicular bisectors of AB and DC meet, is inside ABCD. Prove that ABCD is cyclic if and only if the triangles ABP and CDP have equal areas.

A2. In a competition there are a contestants and b judges, where b e" 3 is an odd integer. Each judge rates each contestant as either "pass" or "fail". Suppose k is a number such that for any two judges their ratings coincide for at most k contestants. Prove k/a e" (b-1)/2b.

A3. For any positive integer n, let d(n) denote the number of positive divisors of n (including 1 and n). Determine all positive integers k such that d(n2) = k d(n) for some n.

B1. Determine all pairs (a, b) of positive integers such that ab2 + b + 7 divides a2b + a + b.

B2. Let I be the incenter of the triangle ABC. Let the incircle of ABC touch the sides BC, CA, AB at K, L, M respectively. The line through B parallel to MK meets the lines LM and LK at R and S respectively. Prove that the angle RIS is acute.

B3. Consider all functions f from the set of all positive integers into itself satisfying f(t2f(s)) = s f(t)2 for all s and t. Determine the least possible value of f(1998).