BALKAN 1992

For all positive integers $m,n$ define $f(m,n) = m^{3^{4n}+6} - m^{3^{4n}+4} - m^5 + m^3$ . Find all numbers $n$ with the property that $f(m, n)$ is divisible by 1992 for every $m$ .

Bulgaria

Prove that for all positive integers $n$ the following inequality takes place $$(2n^2+3n+1)^n \geq 6^n (n!)^2 .$$ Cyprus

Let $D$ , $E$ , $F$ be points on the sides $BC$ , $CA$ , $AB$ respectively of a triangle $ABC$ (distinct from the vertices). If the quadrilateral $AFDE$ is cyclic, prove that $$\frac{ 4 \mathcal A[DEF] }{\mathcal A[ABC] } \leq \left( \frac{EF}{AD} \right)^2 .$$ Greece

For each integer $n\geq 3$ , find the least natural number $f(n)$ having the property $\star$ For every $A \subset \{1, 2, \ldots, n\}$ with $f(n)$ elements, there exist elements $x, y, z \in A$ that are pairwise coprime.