# IMO 2013 – Day 2

4. Let $ABC$ be an acute triangle with orthocenter $H$, and let $W$ be a point on the side $BC$, between $B$ and $C$. The points $M$ and $N$ are the feet of the altitudes drawn from $B$ and $C$, respectively. $\omega_1$ is the circumcircle of triangle $BWN$, and $X$ is a point such that $WX$ is a diameter of $\omega_1$. Similarly, $\omega_2$ is the circumcircle of triangle $CWM$, and $Y$ is a point such that $WY$ is a diameter of $\omega_2$. Show that the points $X, Y$, and $H$ are collinear.

5. Let $\mathbb Q_{>0}$ be the set of all rational numbers greater than zero. Let $f:\mathbb Q_{>0} \to \mathbb R$ be a function satisfying the following conditions

(i) $f(x)f(y) \geq f(xy)$ for all $x, y \in \mathbb Q_{>0}$,

(ii) $f(x+y) \geq f(x) + f(y)$ for all $x, y \in \mathbb Q_{>0}$,

(iii) There exists a rational number $a> 1$ such that $f (a) = a$.

Show that $f(x) = x$ for all $x\in\mathbb Q_{>0}$.

6. Let $n\geq 3$ be an integer, and consider a circle with $n+1$ equally spaced points marked on it. Consider all labellings of these points with the numbers $0,1,\dots, n$ such that each label is used exactly once; two such labellings are considered to be the same if one can be obtained from the other by a rotation of the circle. A labelling is called beautiful if, for any four labels $a with $a+d=b+c$, the chord joining the points labelled $a$ and $d$ does not intersect the chord joining the points labelled $b$ and $c$.

Let $M$ be the number of beautiful labellings and let $N$ be the number of ordered pairs $(x,y)$ of positive integers such that $x+y\leq n$ and $\gcd(x,y)=1$. Prove that $M=N+1.$