9

Part of 2007 Hanoi Open Mathematics Competitions

Problems(2)

HOMC (Hanoi Open Mathematical Competition)

Source:

A triangle is said to be the Heron triangle if it has integer sides and integer area. In a Heron triangle, the sides a; b; c satisfy the equation b=a(a-c). Prove that the triangle is isosceles.

Prove that $ a_k\in[2006;2007]$

Source: HOMC

a_1,a_2,...,a_{2007}

be real number such that

a_1+a_2+...+a_{2007}\geq 2007^{2}

a_1^{2}+a_2^{2}+...+a_{2007}^{2}\leq 2007^{3}-1

a_k\in[2006;2008]

k\in\left \{ 1,2,...,2007 \right \}