International Mathematical Olympiad 2015

We say that a finite set $\mathcal{S}$ of points in the plane is balanced if, for any two different points $A$ and $B$ in $\mathcal{S}$, there is a point $C$ in $\mathcal{S}$ such that $AC = BC$. We say that $\mathcal{S}$ is centre-free if for any three different points $A$, $B$ and $C$ in $\mathcal{S}$, there is no point $P$ in $\mathcal{S}$ such that $PA = PB = PC$.

(a) Show that for all integers $n \geqslant 3$, there exists a balanced set consisting of $n$ points.

(b) Determine all integers $n \geqslant 3$ for which there exists a balanced centre-free set consisting of $n$ points.

Determine all triples $(a, b, c)$ of positive integers such that each of the numbers $$ab - c, \quad bc - a, \quad ca - b$$ is a power of 2.

(A power of 2 is an integer of the form $2^n$, where $n$ is a non-negative integer.)

Let $ABC$ be an acute triangle with $AB > AC$. Let $\Gamma$ be its circumcircle, $H$ its orthocentre, and $F$ the foot of the altitude from $A$. Let $M$ be the midpoint of $BC$. Let $Q$ be the point on $\Gamma$ such that $\angle HQA = 90°$, and let $K$ be the point on $\Gamma$ such that $\angle HKQ = 90°$. Assume that the points $A$, $B$, $C$, $K$ and $Q$ are all different, and lie on $\Gamma$ in this order.

Prove that the circumcircles of triangles $KQH$ and $FKM$ are tangent to each other.

Triangle $ABC$ has circumcircle $\Omega$ and circumcentre $O$. A circle $\Gamma$ with centre $A$ intersects the segment $BC$ at points $D$ and $E$, such that $B$, $D$, $E$ and $C$ are all different and lie on line $BC$ in this order. Let $F$ and $G$ be the points of intersection of $\Gamma$ and $\Omega$, such that $A$, $F$, $B$, $C$ and $G$ lie on $\Omega$ in this order. Let $K$ be the second point of intersection of the circumcircle of triangle $BDF$ and the segment $AB$. Let $L$ be the second point of intersection of the circumcircle of triangle $CGE$ and the segment $CA$.

Suppose that the lines $FK$ and $GL$ are different and intersect at the point $X$. Prove that $X$ lies on the line $AO$.

Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f \colon \mathbb{R} \to \mathbb{R}$ satisfying the equation $$f \big (x + f (x + y) \big) + f (xy) = x + f (x + y) + yf (x)$$ for all real numbers $x$ and $y$.

The sequence $a_1, a_2, \ldots$ of integers satisfies the following conditions:

(i) $1 \leqslant a_{j} \leqslant 2015$ for all $j \geqslant 1$;

(ii) $k + a_{k} \neq \ell + a_{\ell}$ for all $1 \leqslant k < \ell$.

Prove that there exist two positive integers $b$ and $N$ such that $$\left| \sum_{j = m + 1}^{n} (a_{j} - b) \right| \leqslant 1007^{2}$$ for all integers $m$ and $n$ satisfying $n > m \geqslant N$.