| 1. Prove that (n + 1)3 ≠ n3 + (n - 1)3 for any positive integer n. |
|
| 2. α is acute. Show that α < (sin α + tan α)/2. |
|
| 3. ABC is a triangle. The feet of the altitudes from A, B, C are P, Q, R respectively, and P, Q, R are distinct points. The altitudes meet at O. Show that if ABC is acute, then O is the center of the circle inscribed in the triangle PQR, and that A, B, C are the centers of the other three circles that touch all three sides of PQR (extended if necessary). What happens if ABC is not acute? |
|
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): (Translated by Elvira Rapaport) József Kürschák, G Hajós, G Neukomm & J Surányi, Hungarian Problem Book 2, 1906-1928, MAA 1963. Out of print, but available in some university libraries.
Eötvös home
John Scholes
jscholes@kalva.demon.co.uk
20 Oct 1999