AT-OEMO-REG 2012

Prove that the inequality $$a + a^3 - a^4 - a^6 < 1$$ holds for all real numbers $a$ .

Determine all integer solutions $(x, y)$ of the equation $$(x - 1)x(x + 1) + (y - 1)y(y + 1) = 24 - 9xy\text{.}$$

In an arithmetic sequence, the difference of consecutive terms in constant. We consider sequences of integers in which the difference of consecutive terms equals the sum of the differences of all preceding consecutive terms. Which of these sequences with $a_0 = 2012$ and $1\leqslant d = a_1-a_0 \leqslant 43$ contain square numbers?

In a triangle $ABC$ , let $H_a$ , $H_b$ and $H_c$ denote the base points of the altitudes on the sides $BC$ , $CA$ and $AB$ , respectively. Determine for which triangles $ABC$ two of the lengths $H_aH_b$ , $H_bH_c$ and $H_aH_c$ are equal.