27th Spanish Mathematical Olympiad Problems 1991

A1.  Let S be the set of all points in the plane with integer coordinates. Let T be the set of all segments AB, where A, B ∈ S and AB has integer length. Prove that we cannot find two elements of T making an angle 45^o. Is the same true in three dimensions?

A2.  a, b are distinct elements of {0,1,-1}. A is the matrix:

a+b    a+b2    a+b3   ...    a+bm
a2+b   a2+b2   a2+b3   ...   a2+bm
a3+b   a3+b2   a3+b3   ...   a3+bm
 ...
an+b   an+b2   an+b3   ...   an+bm

Find the smallest possible number of columns of A such that any other column is a linear combination of these columns with integer coefficients.

A3.  What condition must be satisfied by the coefficients u, v, w if the roots of the polynomial x³ - ux² + vx - w can be the sides of a triangle?

B1.  The incircle of ABC touches BC, CA, AB at A', B', C' respectively. The line A'C' meets the angle bisector of A at D. Find ∠ADC.

B2.  Let s(n) be the sum of the binary digits of n. Find s(1) + s(2) + s(3) + ... + s(2^k) for each positive integer k.

B3.  Find the integral part of 1/√1 + 1/√2 + 1/√3 + ... + 1/√1000.