The 18th Balkan Mathematical Olympiad
2001年第18届巴尔干地区数学奥林匹克 |
- If 2n - 1 = ab and 2k is the highest power of 2 dividing 2n - 2 + a - b then k is even.
- A convex pentagon has rational sides and equal angles. Show that it is regular.
- a, b, c are positive reals whose product does not exceed their sum.
Show that a2 + b2 + c2 ≥ (√3)abc.
- 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) ?
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