34th Spanish 1998


A1. A square side 1 is rotated through an angle θ about its center. Find the area common to the original and rotated squares.
A2. Find all 4-digit numbers which equal the cube of the sum of their digits.
A3. ABC is a triangle. D, E are points on the line BC such that AD, AE are parallel to the tangents to the circumcircle at C, B. Show that BE/CD = (AB/AC)².
B1. A triangle has angles A, B, C such that tan A, tan B, tan C are all positive integers. Find A, B, C.
B2. Let N be the set of positive integers. Find all functions f : N → N which are strictly increasing and which satisfy f(n + f(n)) = 2 f(n) for all n.
B3. For which values of n is it possible to tile an n x n square with tiles of the type:

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