Let $H$ be the orthocenter of an acute-angled triangle $ABC$. The circle $\Gamma_{A}$ centered at the midpoint of $\overline{BC}$ and passing through $H$ intersects the sideline $BC$ at points $A_1$ and $A_2$. Similarly, the circle centred at the midpoint of $CA$ and passing through $H$ meets line $CA$ at $B_1,B_2$, and the circle centred at the midpoint of $AB$ and passing through $H$ meets line $AB$ at $C_1,C_2$. Prove that six points $A_1$, $A_2$, $B_1$, $B_2$, $C_1$, $C_2$ are concyclic.
Let $x$, $y$, $z$ be real numbers with $xyz = 1$, all different from $1$. Prove that $$
\frac{x^2}{(x-1)^2} + \frac{y^2}{(y-1)^2} + \frac{z^2}{(z-1)^2} \ge 1
$$ and show that equality holds for infinitely many choices of rational numbers $x$, $y$, $z$.
Find all functions $f$ from the positive reals to the positive reals such that $$
\frac{f(w)^2 + f(x)^2}{f(y^2)+f(z^2)} = \frac{w^2+x^2}{y^2+z^2}
$$ for all positive real numbers $w$, $x$, $y$, $z$ satisfying $wx=yz$.
Let $n$ and $k$ be positive integers with $k \geq n$ and $k - n$ an even number. There are $2n$ lamps labelled $1$, $2$, …, $2n$ each of which can be either on or off. Initially all the lamps are off. We consider sequences of steps: at each step one of the lamps is switched (from on to off or from off to on). Let $N$ be the number of such sequences consisting of $k$ steps and resulting in the state where lamps $1$ through $n$ are all on, and lamps $n + 1$ through $2n$ are all off. Let $M$ be number of such sequences consisting of $k$ steps, resulting in the state where lamps $1$ through $n$ are all on, and lamps $n + 1$ through $2n$ are all off, but where none of the lamps $n + 1$ through $2n$ is ever switched on. Determine $\frac{N}{M}$.
Let $ABCD$ be a convex quadrilateral with $BA \neq BC$. Denote the incircles of triangles $ABC$ and $ADC$ by $\omega_1$ and $\omega_2$ respectively. Suppose that there exists a circle $\omega$ tangent to ray $BA$ beyond $A$ and to the ray $BC$ beyond $C$, which is also tangent to the lines $AD$ and $CD$. Prove that the common external tangents to $\omega_1$ and $\omega_2$ intersect on $\omega$.