AT-OEMO-FED1 2002

Determine all integers $a$ and $b$ such that $$(19a + b)^{18} + (a + b)^{18} + (a + 19b)^{18}$$ is a perfect square.

Find the greatest real number $C$ such that, for all real numbers $x$ and $y \neq x$ with $xy = 2$ it holds that $$\frac{((x + y)^2 - 6)((x - y)^2 + 8)}{(x-y)^2}\geq C.$$ When does equality occur?

Let $f(x)=\frac{9^x}{9^x+3}$ . Compute $\sum_{k} f \biggl( \frac{k}{2002} \biggr)$ , where $k$ goes over all integers $k$ between $0$ and $2002$ which are coprime to $2002$ .

Let $A,C, P$ be three distinct points in the plane. Construct all parallelograms $ABCD$ such that point $P$ lies on the bisector of angle $DAB$ and $\angle APD = 90^\circ$ .