59th Kürschák 1959


1. a, b, c are three distinct integers and n is a positive integer. Show that aⁿ/((a - b)(a - c)) + bⁿ/((b - a)(b - c)) + cⁿ/((c - a)(c - b)) is an integer.
2. The angles subtended by a tower at distances 100, 200 and 300 from its foot sum to 90^o. What is its height?
3. Three brothers and their wives visited a friend in hospital. Each person made just one visit, so that there were six visits in all. Some of the visits overlapped, so that each of the three brothers met the two other brothers' wives during a visit. Show that one brother must have met his own wife during a visit.

The original problems are in Hungarian. Many thanks to Péter Dombi for the translation.