FI-MO 2002

A function $f$ satisfies $f(\cos x) = \cos (17x)$ for every real $x$ . Show that $f(\sin x) =\sin (17x)$ for every $x \in \mathbb{R}.$

Show that if $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a + b + c},$ then also $$\frac{1}{a^n} +\frac{1}{b^n} +\frac{1}{c^n} =\frac{1}{a^n + b^n + c^n},$$ provided $n$ is an odd positive integer.

$n$ pairs are formed from $n$ girls and $n$ boys at random. What is the probability of having at least one pair of girls? For which $n$ the probability is over $0,9?$

Convex figure $\mathcal{K}$ has the following property: if one looks at $\mathcal{K}$ from any point of the certain circle $\mathcal{Y}$ , then $\mathcal{K}$ is seen in the right angle. Show that the figure is symmetric with respect to the centre of $\mathcal{Y.}$

There is a regular $17$ -gon $\mathcal{P}$ and its circumcircle $\mathcal{Y}$ on the plane. The vertices of $\mathcal{P}$ are coloured in such a way that $A,B \in \mathcal{P}$ are of different colour, if the shorter arc connecting $A$ and $B$ on $\mathcal{Y}$ has $2^k+1$ vertices, for some $k \in \mathbb{N},$ including $A$ and $B.$ What is the least number of colours which suffices?