PMO 2009

The sequence ${a_0, a_1, a_2, ...}$ of real numbers satisfies the recursive relation $$n(n+1)a_{n+1}+(n-2)a_{n-1} = n(n-1)a_n$$ for every positive integer $n$ , where $a_0 = a_1 = 1$ . Calculate the sum $$\frac{a_0}{a_1} + \frac{a_1}{a_2} + ... + \frac{a_{2008}}{a_{2009}}$$ .

(a) Find all pairs $(n,x)$ of positive integers that satisfy the equation $2^n + 1 = x^2$ .(b) Find all pairs $(n,x)$ of positive integers that satisfy the equation $2^n = x^2 + 1$ .

Each point of a circle is colored either red or blue.(a) Prove that there always exists an isosceles triangle inscribed in this circle such that all its vertices are colored the same.(b) Does there always exist an equilateral triangle inscribed in this circle such that all its vertices are colored the same?

Let $k$ be a positive real number such that $$\frac{1}{k+a} + \frac{1}{k+b} + \frac{1}{k+c} \leq 1$$ for any positive positive real numbers $a$ , $b$ and $c$ with $abc = 1$ . Find the minimum value of $k$ .

Segments $AC$ and $BD$ intersect at point $P$ such that $PA = PD$ and $PB = PC$ . Let $E$ be the foot of the perpendicular from $P$ to the line $CD$ . Prove that the line $PE$ and the perpendicular bisectors of the segments $PA$ and $PB$ are concurrent.