99th Kürschák 1999


1. Let e(k) be the number of positive even divisors of k, and let o(k) be the number of positive odd divisors of k. Show that the difference between e(1) + e(2) + ... + e(n) and o(1) + o(2) + ... + o(n) does not exceed n.
2. ABC is an arbitrary triangle. Construct an interior point P such that if A' is the foot of the perpendicular from P to BC, and similarly for B' and C', then the centroid of A'B'C' is P.
3. Prove that every set of integers with more than 2^k members has a subset B with k+2 members such that any two non-empty subsets of B with the same number of members have different sums.

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