25th USAMO 1996


A1. Let k = 1^o. Show that 2 sin 2k + 4 sin 4k + 6 sin 6k + ... + 180 sin 180k = 90 cot k.
A2. Let S be a set of n positive integers. Let P be the set of all integers which are the sum of one or more distinct elements of S. Show that we can find n subsets of P whose union is P such that if a, b belong to the same subset, then a ≤ 2b.
A3. Given a triangle, show that we can reflect it in some line so that the area of the intersection of the triangle and its reflection has area greater than 2/3 the area of the triangle.
B1. A type 1 sequence is a sequence with each term 0 or 1 which does not have 0, 1, 0 as consecutive terms. A type 2 sequence is a sequence with each term 0 or 1 which does not have 0, 0, 1, 1 or 1, 1, 0, 0 as consecutive terms. Show that there are twice as many type 2 sequences of length n+1 as type 1 sequences of length n.
B2. D lies inside the triangle ABC. ∠BAC = 50^o. ∠DAB = 10^o, ∠DCA = 30^o, ∠DBA = 20^o. Show that ∠DBC = 60^o.
B3. Does there exist a subset S of the integers such that, given any integer n, the equation n = 2s + s' has exactly one solution in S? For example, if T = {-3, 0, 1, 4), then there are unique solutions -3 = 2·0 - 3, -1 = 2·1 - 3, 0 = 2·0 + 0, 1 = 2·0 + 1, 2 = 0 + 2·1, 3 = 2·1 + 1, 4 = 2·0 + 4, 5 = 2·-3 + 1, but not for 6 = 2·1 + 4 = 2·-3 + 0, so T cannot be a subset of S.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.