IMO 2008

Let $H$ be the orthocenter of an acute-angled triangle $ABC$ . The circle $\Gamma_{A}$ centered at the midpoint of $BC$ and passing through $H$ intersects the sideline $BC$ at points $A_{1}$ and $A_{2}$ . Similarly, define the points $B_{1}$ , $B_{2}$ , $C_{1}$ and $C_{2}$ .

Prove that the six points $A_{1}$ , $A_{2}$ , $B_{1}$ , $B_{2}$ , $C_{1}$ and $C_{2}$ are concyclic.

Author: Andrey Gavrilyuk, Russia

(a) Prove that $$\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} \geq 1$$ for all real numbers $x$ , $y$ , $z$ , each different from $1$ , and satisfying $xyz=1$ .(b) Prove that equality holds above for infinitely many triples of rational numbers $x$ , $y$ , $z$ , each different from $1$ , and satisfying $xyz=1$ .

Author: Walther Janous, Austria

Prove that there are infinitely many positive integers $n$ such that $n^{2} + 1$ has a prime divisor greater than $2n + \sqrt {2n}$ .

Author: Kestutis Cesnavicius, Lithuania

Determine all functions $f : (0, \infty) \to (0, \infty)$ such that $$\frac{(f(p))^2 + (f(q))^2}{f(r^2) + f(s^2)} = \frac{p^2 + q^2}{r^2 + s^2}$$ for all positive real numbers $p$, $q$, $r$, $s$ satisfying $pq = rs$.

Let $n$ and $k$ be positive integers with $k \geq n$ and $k - n$ an even number. Let $2n$ lamps labelled $1$ , $2$ , ..., $2n$ be given, each of which can be either on or off. Initially all the lamps are off. We consider sequences of steps: at each step one of the lamps is switched (from on to off or from off to on).

Let $N$ be the number of such sequences consisting of $k$ steps and resulting in the state where lamps $1$ through $n$ are all on, and lamps $n + 1$ through $2n$ are all off.

Let $M$ be number of such sequences consisting of $k$ steps, resulting in the state where lamps $1$ through $n$ are all on, and lamps $n + 1$ through $2n$ are all off, but where none of the lamps $n + 1$ through $2n$ is ever switched on.

Determine $\frac {N}{M}$ .

Author: Bruno Le Floch and Ilia Smilga, France

Let $ABCD$ be a convex quadrilateral with $BA\neq BC$ . Denote the incircles of triangles $ABC$ and $ADC$ by $\omega_{1}$ and $\omega_{2}$ respectively. Suppose that there exists a circle $\omega$ tangent to ray $BA$ beyond $A$ and to the ray $BC$ beyond $C$ , which is also tangent to the lines $AD$ and $CD$ . Prove that the common external tangents to $\omega_{1}$ and $\omega_{2}$ intersect on $\omega$ .

Author: Vladimir Shmarov, Russia