NO-ABEL 2018

For an odd number n, we write $n!! = n\cdot (n-2)...3 \cdot 1$ . How many different residues modulo $1000$ do you get from $n!!$ for $n= 1, 3, 5, …$ ?

The circumcentre of a triangle $ABC$ is called $O$ . The points $A',B'$ and $C'$ are the reflections of $O$ in $BC, CA$ , and $AB$ , respectively. Show that the three lines $AA' , BB'$ , and $CC'$ meet in a common point.

Find all polynomials $P$ such that $P(x) + \binom{2018}{2}P(x+2)+...+\binom{2018}{2106}P(x+2016)+P(x+2018)=$ $=\binom{2018}{1}P(x+1)+\binom{2018}{3}P(x+3)+...+\binom{2018}{2105}P(x+2015)+\binom{2018}{2107}P(x+2017)$ for all real numbers $x$ .