31st Spanish 1995


A1. X is a set of 100 distinct positive integers such that if a, b, c ∈ X (not necessarily distinct), then there is a triangle with sides a, b, c which is not obtuse. Let S(X) be the sum of the perimeters of all the possible triangles. Find the minimum possible value of S(X).
A2. A finite number of paper disks are arranged so that no disk lies inside another, but there is some overlapping. Show that if we cut out the parts which do not overlap we cannot rearrange them to form disks.
A3. ABC is a triangle with centroid G. A line through G meets the side AB at P and the side AC at Q. Show that (PB/PA)(QC/QA) ≤ 1/4.
B1. p is a prime number. Find all integral solutions to p(m+n) = mn.
B2. Given that the equations x³ + mx - n = 0, nx³ - 2m²x² - 5mnx - 2m³ - n² = 0 (where n ≠ 0) have a common root, show that the first must have two equal roots and find the roots of the two equations in terms of n.
B3. C is a variable point on the segment AB. Equilateral triangles AB'C and BA'C are constructed on the same side of AB, and the equilateral triangle ABC' is constructed on the opposite side of AB. Show that AA', BB', CC' meet at some point P. Find the locus of P as C varies. Show that the centers of the three equilateral triangles form an equilateral triangle and lie on a fixed circle (as C varies).

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