DE-BUNDESWETTBEWERB 2018

Find the largest positive integer with the property that each digit apart from the first and the last one is smaller than the arithmetic mean of her neighbours.

Find all real numbers $x$ satisfying the equation $$\left\lfloor \frac{20}{x+18}\right\rfloor+\left\lfloor \frac{x+18}{20}\right\rfloor=1.$$

Let $H$ be the orthocenter of the acute triangle $ABC$ . Let $H_a$ be the foot of the perpendicular from $A$ to $BC$ and let the line through $H$ parallel to $BC$ intersect the circle with diameter $AH_a$ in the points $P_a$ and $Q_a$ . Similarly, we define the points $P_b, Q_b$ and $P_c,Q_c$ .

Show that the six points $P_a,Q_a,P_b,Q_b,P_c,Q_c$ lie on a common circle.