36th Eötvös 1932

36th Eötvös Competition 1932


1. Show that if m is a multiple of aⁿ, then (a + 1)^m -1 is a multiple of aⁿ⁺¹.
2. ABC is a triangle with AB and AC unequal. AM, AX, AD are the median, angle bisector and altitude. Show that X always lies between D and M, and that if the triangle is acute-angled, then angle MAX < angle DAX.
3. An acute-angled triangle has angles A < B < C. Show that sin 2A > sin 2B > sin 2C.

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.