AT-OEMO-REG 2010

Let $0 \le a$ , $b \le 1$ be real numbers. Prove the following inequality: $$\sqrt{a^3b^3}+ \sqrt{(1-a^2)(1-ab)(1-b^2)} \le 1.$$ (41th Austrian Mathematical Olympiad, regional competition, problem 1)

Solve the following in equation in $\mathbb{R}^3$ : $$4x^4-x^2(4y^4+4z^4-1)-2xyz+y^8+2y^4z^4+y^2z^2+z^8=0.$$

Let $\triangle ABC$ be a triangle and let $D$ be a point on side $\overline{BC}$ . Let $U$ and $V$ be the circumcenters of triangles $\triangle ABD$ and $\triangle ADC$ , respectively. Show, that $\triangle ABC$ and $\triangle AUV$ are similar.

(41th Austrian Mathematical Olympiad, regional competition, problem 3)

Let $(b_n)_{n \ge 0}=\sum_{k=0}^{n} (a_0+kd)$ for positive integers $a_0$ and $d$ . We consider all such sequences containing an element $b_i$ which equals $2010$ . Determine the greatest possible value of $i$ and for this value the integers $a_0$ and $d$ .

(41th Austrian Mathematical Olympiad, regional competition, problem 4)