FI-MO 2001

In the right triangle $ABC,$ $CF$ is the altitude based on the hypotenuse $AB.$ The circle centered at $B$ and passing through $F$ and the circle with centre $A$ and the same radius intersect at a point of $CB.$ Determine the ratio $FB : BC.$

Equations of non-intersecting curves are $y = ax^2 + bx + c$ and $y = dx^2 + ex + f$ where $ad < 0.$ Prove that there is a line of the plane which does not meet either of the curves.

Numbers $a, b$ and $c$ are positive integers and $\frac{1}{a}+\frac{1}{b}+\frac{ 1}{c}< 1.$ Show that $$\frac{1}{a}+\frac{1}{b}+\frac{ 1}{c}\leq \frac{41}{42}.$$

A sequence of seven digits is randomly chosen in a weekly lottery. Every digit can be any of the digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9.$ What is the probability of having at most five different digits in the sequence?

Determine $n \in \Bbb{N}$ such that $n^2 + 2$ divides $2 + 2001n.$