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<b<c<d 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.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s