24th Brasil 2002


A1. Show that there is a set of 2002 distinct positive integers such that the sum of one or more elements of the set is never a square, cube, or higher power.
A2. ABCD is a cyclic quadrilateral and M a point on the side CD such that ADM and ABCM have the same area and the same perimeter. Show that two sides of ABCD have the same length.
A3. The squares of an m x n board are labeled from 1 to mn so that the squares labeled i and i+1 always have a side in common. Show that for some k the squares k and k+3 have a side in common.
B1. For any non-empty subset A of {1, 2, ... , n} define f(A) as the largest element of A minus the smallest element of A. Find ∑ f(A) where the sum is taken over all non-empty subsets of {1, 2, ... , n}.
B2. A finite collection of squares has total area 4. Show that they can be arranged to cover a square of side 1.
B3. Show that we cannot form more than 4096 binary sequences of length 24 so that any two differ in at least 8 positions.

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