RS-SMO 2017

Prove that for positive real numbers $a,b,c$ such that $a+b+c=1$ , $$a\sqrt{2b+1}+b\sqrt{2c+1}+c\sqrt{2a+1}\le \sqrt{2-(a^2+b^2+c^2)}.$$

Let $ABCD$ be a convex and cyclic quadrilateral. Let $AD\cap BC=\{E\}$ , and let $M,N$ be points on $AD,BC$ such that $AM:MD=BN:NC$ . Circle around $\triangle EMN$ intersects circle around $ABCD$ at $X,Y$ prove that $AB,CD$ and $XY$ are either parallel or concurrent.

There are $2n-1$ lamps in a row. In the beginning only the middle one is on (the $n$ -th one), and the other ones are off. Allowed move is to take two non-adjacent lamps which are turned off such that all lamps between them are turned on and switch all of their states from on to off and vice versa. What is the maximal number of moves until the process terminates?

Let $k$ be the circumcircle of $\triangle ABC$ and let $k_a$ be A-excircle .Let the two common tangents of $k,k_a$ cut $BC$ in $P,Q$ .Prove that $\measuredangle PAB=\measuredangle CAQ$ .