40th Eötvös 1936

40th Eötvös Competition 1936


1. Show that 1/(1·2) + 1/(3·4) + 1/(5·6) + ... + 1/( (2n-1)·2n) = 1/(n+1) + 1/(n+2) + ... + 1/(2n).
2. ABC is a triangle. Show that, if the point P inside the triangle is such that the triangles PAB, PBC, PCA have equal area, then P must be the centroid.
3. Given any positive integer N, show that there is just one solution to m + ½(m + n - 1)(m + n - 2) = N in positive integers.

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.