2nd Iberoamerican 1987


A1. Find f(x) such that f(x)²f( (1-x)/(1+x) ) = 64x for x not 0, ±1.
A2. In the triangle ABC, the midpoints of AC and AB are M and N respectively. BM and CN meet at P. Show that if it is possible to inscribe a circle in the quadrilateral AMPN (touching every side), then ABC is isosceles.
A3. Show that if (2 + √3)^k = 1 + m + n√3, for positive integers m, n, k with k odd, then m is a perfect square.
B1. Define the sequence p₁, p₂, p₃, ... as follows. p₁ = 2, and p_n is the largest prime divisor of p₁p₂ ... p_n-1 + 1. Prove that 5 does not occur in the sequence.
B2. Show that the roots r, s, t of the equation x(x - 2)(3x - 7) = 2 are real and positive. Find tan^-1r + tan^-1s + tan^-1t.
B3. ABCD is a convex quadrilateral. P, Q are points on the sides AD, BC respectively such that AP/PD = BQ/QC = AB/CD. Show that the angle between the lines PQ and AB equals the angle between the lines PQ and CD.

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