18th Balkan 2001


A1. If 2ⁿ - 1 = ab and 2^k is the highest power of 2 dividing 2ⁿ - 2 + a - b then k is even.
A2. A convex pentagon has rational sides and equal angles. Show that it is regular.
A3. a, b, c are positive reals whose product does not exceed their sum. Show that a² + b² + c² ≥ (√3) abc.
A4. 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 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 n = 1, 2, ... , 26)?

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