39th Spanish 2003


A1. For which primes p ≠ 2 or 5 is there a multiple of p whose digits are all 9? For example, 999999 = 13·76923.
A2. Does there exist a finite set M of at least two real numbers such that if a, b ∈ M, then 2a - b² ∈ M?
A3. H is the orthocenter of ABC and AB = CH. Find ∠C.
B1. α is a real root of x³ + 2x² + 10x - 20 = 0. Show that α² is irrational.
B2. A hexagon has all its angles equal and sides 1, 2, 3, 4, 5, 6 in that order. What is its area?
B3. 2n white balls and 2n black balls are arranged in a line. Show that however they are arranged it is possible to find 2n consecutive balls, just n of which are white.

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