38th Spanish 2002


A1. Find all polynomials p(x) such that p(x²-y²) ≡ p(x+y) p(x-y).
A2. AD is an altitude of the triangle ABC and H is the orthocenter. Find a relation between ∠B and ∠C in terms of k = AD/HD. Given B, C and k, find the locus of A.
A3. The function f is defined on the positive integers and satisfies f(2) = 1, f(2n) = f(n), f(2n+1) = f(2n) + 1. Find the maximum value M of f(n) for 1 ≤ n ≤ 2002 and find how many n satisfy f(n) = M.
B1. r(n) is the number obtained by writing the digits of n in reverse order, and s(n) is the sum of the digits of n. Find all 3-digit numbers n such that 2r(n) + s(n) = n.
B2. Given 2002 line segments in the plane with total length 1. Show that there is a line L such that the projections of the segments onto L have total length < 2/3.
B3. r vertices of a regular (6n+1)-gon are colored red and the remaining vertices are colored blue. Show that the number of isosceles triangles with all vertices the same color depends only on n and r.

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