2001 Balkan MO

Part of Balkan MO

Subcontests

(4)

1

A cube side 3 is divided into 27 unit cubes

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

n = 1, 2, \ldots , 26

1

An easy and classical inequality in 3 variables abc<=a+b+c

a

b

c

be positive real numbers with

abc \leq a+b+c

a^2 + b^2 + c^2 \geq \sqrt 3 abc.

Cristinel Mortici, Romania

1

A convex pentagon has rational sides and equal angles

A convex pentagon

ABCDE

has rational sides and equal angles. Show that it is regular.

1

An easy diophantine-like equation: find the exponent of 2

a,b,n

be positive integers such that

2^n - 1 =ab

k \in \mathbb N

ab+a-b-1 \equiv 0 \pmod {2^k}

ab+a-b-1 \neq 0 \pmod {2^{k+1}}

k