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 *center-free* if for any three different points $A$, $B$ and $C$ in $\mathcal{S}$, there are no points $P$ in $\mathcal{S}$ such that $PA=PB=PC$.
(a) Show that for all integers $n\ge 3$, there exists a balanced set consisting of $n$ points.
(b) Determine all integers $n\ge 3$ for which there exists a balanced center-free set consisting of $n$ points.
Find all positive integers $a$, $b$, $c$ such that each of $ab-c$, $bc-a$, $ca-b$ is a power of $2$ (possibly including $2^0=1$). A power of 2 is an integer of the form $2^r$ for a nonnegative integer $r$.
Let $ABC$ be an acute triangle with $AB \gt AC$. Let $\Gamma$ be its circumcircle, $H$ its orthocenter, and $F$ the foot of the altitude from $A$. Let $M$ be the midpoint of $\overline{BC}$. Let $Q$ be the point on $\Gamma$ such that $\angle HQA = 90^\circ$ and let $K$ be the point on $\Gamma$ such that $\angle HKQ = 90^\circ$. 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 $B,D,E,C$ 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 $A,F,B,C,G$ 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 $AO$.
Solve the functional equation $$
f(x+f(x+y)) + f(xy) = x + f(x+y) + yf(x)
$$ for $f \colon \mathbb{R} \to \mathbb{R}$. The equation must hold for all real numbers $x,y$.
The sequence $a_1,a_2,\dots$ of integers satisfies the conditions:
(i) $1\le a_j\le2015$ for all $j\ge1$,
(ii) $k+a_k\neq \ell+a_\ell$ for all $1\le k\lt \ell$.
Prove that there exist two positive integers $b$ and $N$ for which $$
\left\lvert\sum_{j=m+1}^n (a_j-b) \right\rvert \le 1007^2
$$ for all integers $m$ and $n$ such that $n \gt m\ge N$.