45th Eötvös 1941

45th Eötvös Competition 1941


1. Prove that (1+x)(1+x²)(1+x⁴) ... (1+x^{2^n-1}) = 1 + x + x² + x³ + ... + x^2ⁿ-1.
2. The a parallelogram has its vertices at lattice points and there is at least one other lattice point inside the parallelogram or on its sides. Show that its area is greater than 1.
3. ABCDEF is a hexagon with vertices on a circle radius R (in that order). The three sides AB, CD, EF have length R. Show that the midpoints of BC, DE, FA form an equilateral triangle.

The original problems are in Hungarian. They are available on the KöMaL archive on the web. They are also available in English (with solutions) in: Andy Liu, Hungarian Problem Book III, 1929-1943, MAA 2001. ISBN 0883856441.