TH-TMO 2008

Two positive integers $m, n$ satisfy the two equations $m^2 + n^2 = 3789$ and $gcd (m, n) + lcm (m, n) = 633$ . Compute $m + n$ .

Prove that $2551 \cdot 543^n -2008\cdot 7^n$ is never a perfect square, where $n$ varies over the set of positive integers

Find the number of pairs of sets $(A, B)$ satisfying $A \subseteq B \subseteq \{1, 2, ...,10\}$

On the sides of triangle $\vartriangle ABC$ , $17$ points are added, so that there are $20$ points in total (including the vertices of $\vartriangle ABC$ .) What is the maximum possible number of (nondegenerate) triangles that can be formed by these points.