17th Balkan 2000


A1. Find all real-valued functions on the reals which satisfy f( xf(x) + f(y) ) = f(x)² + y for all x, y.
A2. ABC is an acute-angled triangle which is not isosceles. M is the midpoint of BC. X is any point on the segment AM. Y is the foot of the perpendicular from X to BC. Z is any point on the segment XY. U and V are the feet of the perpendiculars from Z to AB and AC. Show that the bisectors of angles UZV and UXV are parallel.
A3. How many 1 by 10√2 rectangles can be cut from a 50 x 90 rectangle using cuts parallel to its edges.
A4. Show that for any n we can find a set X of n distinct integers greater than 1, such that the average of the elements of any subset of X is a square, cube or higher power.

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