# The 54th International Mathematical Olympiad: Problems and Solutions

## Day 1 (July 23th, 2013)

Problem 1

Assume that $$k$$ and $$n$$ are two positive integers. Prove that there exist positive integers $$m_1$$, $$\dots$$, $$m_k$$ such that $1+\frac{2^k-1}{n}=\left(1+\frac1{m_1}\right)\cdots \left(1+\frac1{m_k}\right).$

Problem 2

Given $$2013$$ red and $$2014$$ blue points in the plane, assume that no three of the given points lie on a line. A partition of the plane is called perfect if no region contains points of different colors. Determine the smallest number $$k$$ for which a perfect partition can always be achieved by drawing $$k$$ straight lines.

Problem 3

Let $$A_1$$, $$B_1$$, and $$C_1$$ be the points at which the excircles touch the sides $$BC$$, $$CA$$, and $$AB$$ of the triangle $$ABC$$. Prove that if the circumcenter of $$\triangle A_1B_1C_1$$ belongs to the circumcircle of $$\triangle ABC$$, then one of the angles of $$\triangle ABC$$ is $$90^{\circ}$$.

## Day 2 (July 24th, 2013)

Problem 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 perpendiculars from $$B$$ and $$C$$, respectively. Let $$\omega_1$$ be the circumcircle of $$\triangle BWN$$, and let $$X$$ be the point such that $$WX$$ is a diameter of $$\omega_1$$. Let $$\omega_2$$ be the circumcircle of $$\triangle CWM$$, and let $$Y$$ be the point such that $$WY$$ is a diameter of $$\omega_2$$. Prove that the points $$X$$, $$Y$$, and $$H$$ are collinear.

Problem 5

Let $$\mathbb Q_+$$ be the set of all positive rational numbers. Let $$f:\mathbb Q_+\to\mathbb R$$ be a function that satisfies the following three conditions:

• (i) $$f(x)f(y)\geq f(xy)$$ for all $$x,y\in\mathbb Q_+$$,

• (ii) $$f(x+y)\geq f(x)+f(y)$$ for all $$x,y\in\mathbb Q_+$$,

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

Prove that $$f(x)=x$$ for all $$x\in\mathbb Q_+$$.

Problem 6

Given an integer $$n\geq 3$$, assume that $$n+1$$ equally spaced points are marked on a circle. Consider all labelings of these points with the numbers $$0$$, $$1$$, $$\dots$$, $$n$$ such that each label is used exactly once. Two labelings are considered the same if one can be obtained from the other by a rotation of the circle. A labeling is beautiful if, for any four labels $$a< b< c< d$$ with $$a+d=b+c$$, the chord joining points labeled $$a$$ and $$d$$ does not intersect the chord joining points labeled $$b$$ and $$c$$.

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

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