Let $\triangle{ABC}$ be a triangle with orthocentre $H$. The tangent lines from $A$ to the circle with diameter $BC$ touch this circle at $P$ and $Q$. Prove that $H,P$ and $Q$ are collinear.

Find the smallest positive integer $ K$ such that every $ K$-element subset of $ \{1,2,...,50 \}$ contains two distinct elements $ a,b$ such that $ a+b$ divides $ ab$.

Suppose that the function $f:\mathbb{R}\to\mathbb{R}$ satisfies \[f(x^3 + y^3)=(x+y)(f(x)^2-f(x)f(y)+f(y)^2)\] for all $x,y\in\mathbb{R}$. Prove that $f(1996x)=1996f(x)$ for all $x\in\mathbb{R}$.

$8$ singers take part in a festival. The organiser wants to plan $m$ concerts. For every concert there are $4$ singers who go on stage, with the restriction that the times of which every two singers go on stage in a concert are all equal. Find a schedule that minimises $m$.

Let $n$ be a natural number. Suppose that $x_0=0$ and that $x_i>0$ for all $i\in\{1,2,\ldots ,n\}$. If $\sum_{i=1}^nx_i=1$ , prove that \[1\leq\sum_{i=1}^{n} \frac{x_i}{\sqrt{1+x_0+x_1+\ldots +x_{i-1}}\sqrt{x_i+\ldots+x_n}} < \frac{\pi}{2} \]

In the triangle $ABC$, $\angle{C}=90^{\circ},\angle {A}=30^{\circ}$ and $BC=1$. Find the minimum value of the longest side of all inscribed triangles (i.e. triangles with vertices on each of three sides) of the triangle $ABC$.