BALKAN 2001

Let $a,b,n$ be positive integers such that $2^n - 1 =ab$ . Let $k \in \mathbb N$ such that $ab+a-b-1 \equiv 0 \pmod {2^k}$ and $ab+a-b-1 \neq 0 \pmod {2^{k+1}}$ . Prove that $k$ is even.

A convex pentagon $ABCDE$ has rational sides and equal angles. Show that it is regular.

Let $a$ , $b$ , $c$ be positive real numbers with $abc \leq a+b+c$ . Show that $$a^2 + b^2 + c^2 \geq \sqrt 3 abc.$$ Cristinel Mortici, Romania

A cube side 3 is divided into 27 unit cubes. The unit cubes are arbitrarily labeled 1 to 27 (each cube is given a different number). A move consists of swapping the cube labeled 27 with one of its 6 neighbours. Is it possible to find a finite sequence of moves at the end of which cube 27 is in its original position, but cube $n$ has moved to the position originally occupied by $27-n$ (for each $n = 1, 2, \ldots , 26$ )?