9th Balkan 1992


A1. Let a(n) = 3⁴ⁿ. For which n is (m^a(n)+6 - m^a(n)+4 - m⁵ + m³) always divisible by 1992?
A2. Prove that (2n² + 3n + 1)ⁿ ≥ 6ⁿn! n! for all positive integers.
A3. ABC is a triangle area 1. Take D on BC, E on CA, F on AB, so that AFDE is cyclic. Prove that: area DEF ≤ EF²/(4 AD²).
A4. For each n > 2 find the smallest f(n) such that any subset of {1, 2, 3, ... , n} with f(n) elements must have three which are relatively prime (in pairs).

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.