International Mathematical Olympiad 2013 — Official contest

Let $k$ and $n$ be positive integers. Prove that there exist positive integers $m_1$, …, $m_k$ such that $$
1 + \frac{2^k-1}{n} = \left( 1 + \frac{1}{m_1} \right) \left( 1 + \frac{1}{m_2} \right)
\dots \left( 1 + \frac{1}{m_k} \right).
$$ The integers $m_1,\ldots,m_k$ need not be different.

A configuration of $4027$ points in the plane is called *Colombian* if it consists of $2013$ red points and $2014$ blue points, and no three of the points of the configuration are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines is *good* for a Colombian configuration if the following two conditions are satisfied:

(i) No line passes through any point of the configuration.

(ii) No region contains points of both colors.

Find the least value of $k$ such that for any Colombian configuration of $4027$ points, there is a good arrangement of $k$ lines.

Let the excircle of triangle $ABC$ opposite the vertex $A$ be tangent to the side $BC$ at the point $A_1$. Define the points $B_1$ on $CA$ and $C_1$ on $AB$ analogously, using the excircles opposite $B$ and $C$, respectively. Suppose that the circumcenter of triangle $A_1B_1C_1$ lies on the circumcircle of triangle $ABC$. Prove that triangle $ABC$ is right-angled. The excircle opposite $A$ is tangent to segment $BC$, to ray $AB$ beyond $B$, and to ray $AC$ beyond $C$. The other excircles are defined similarly.

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

Suppose a function $f \colon \mathbb{Q}_{\gt 0} \to \mathbb{R}$ satisfies:

1. [(i)] If $x,y \in \mathbb{Q}_{\gt 0}$, then $f(x)f(y) \ge f(xy)$.

2. [(ii)] If $x,y \in \mathbb{Q}_{\gt 0}$, then $f(x+y) \ge f(x) + f(y)$.

3. [(iii)] There exists a rational number $a \gt 1$ with $f(a) = a$.

Prove that $f(x) = x$ for all positive rational numbers $x$.

Let $n \ge 3$ 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 $a \lt b \lt c \lt d$ 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 $x + y \le n$ and $\gcd(x, y) = 1$. Prove that $M = N + 1$.