MK-MO 2001

Prove that if $m$ and $s$ are integers with $ms=2000^{2001}$ , then the equation $mx^2-sy^2=3$ has no integer solutions.

Does there exist a function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that $$f(f(n-1)=f(n+1)-f(n) \text{for all}\ n\ge 2\text{?}$$

Let $ABC$ be a scalene triangle and $k$ be its circumcircle. Let $t_A,t_B,t_C$ be the tangents to $k$ at $A, B, C,$ respectively. Prove that points $AB\cap t_C$ , $CA\cap t_B$ , and $BC\cap t_A$ exist, and that they are collinear.

Let $\Omega$ be a family of subsets of $M$ such that: $(\text{i})$ If $|A\cap B|\ge 2$ for $A,B\in\Omega$ , then $A=B$ ; $(\text{ii})$ There exist different subsets $A,B,C\in\Omega$ with $|A\cap B\cap C|=1$ ; $(\text{iii})$ For every $A\in\Omega$ and $a\in M \ A$ , there is a unique $B\in\Omega$ such that $a\in B$ and $A\cap B=\emptyset$ .

Prove that there are numbers $p$ and $s$ such that: $(1)$ Each $a\in M$ is contained in exactly $p$ sets in $\Omega$ ; $(2)$ $|A|=s$ for all $A\in\Omega$ ; $(3)$ $s+1\ge p$ .