AT-OEMO-BEG 2012

Let $a, b, c$ and $d$ be four integers such that $7a + 8b = 14c + 28d$ . Prove that the product $a\cdot b$ is always divisible by $14$ .

A postman wants to divide $n$ packages with weights $1, 2, 3, 4, n$ into three groups of exactly the same weight. Can he do this if (a) $n = 2011$ ? (b) $n = 2012$ ?

Let $a$ and $b$ be two positive real numbers with $a \le 2b \le 4a$ . Prove that $4ab \le2 (a^2+ b^2) \le 5 ab$ .

A segment $AB$ is given. We erect the equilateral triangles $ABC$ and $ADB$ above and below $AB$ . We denote the midpoints of $AC$ and $BC$ by $E$ and $F$ respectively. Prove that the straight lines $DE$ and $DF$ divide the segment $AB$ into three parts of equal length .