26th Spanish Mathematical Olympiad Problems 1990

A1.  Show that √x + √y + √(xy) = √x + √(y + xy + 2y√x). Hence show that √3 + √(10 + 2√3) = √(5 + √22) + √(8 - √22 + 2√(15 - 3√22)).

A2.  Every point of the plane is painted with one of three colors. Can we always find two points a distance 1 apart which are the same color?

A3.  Show that [(4 + √11)ⁿ] is odd for any positive integer n.

B1.  Show that ((a+1)/2 + ((a+3)/6)√((4a+3)/3) )^1/3 + ((a+1)/2 - ((a+3)/6)√((4a+3)/3) )^1/3 is independent of a for a ≥ 3/4 and find it.

B2.  ABC is a triangle with area S. Points A', B', C' are taken on the sides BC, CA, AB, so that AC'/AB = BA'/BC = CB'/CA = k, where 0 < k < 1. Find the area of A'B'C' in terms of S and k. Find the value of k which minimises the area. The line through A' parallel to AB and the line through C' parallel to AC meet at P. Find the locus of P as k varies.

B3.  There are n points in the plane so that no two pairs are the same distance apart. Each point is connected to the nearest point by a line. Show that no point is connected to more than 5 points.