PL-OM 2004

A point $D$ is taken on the side $AB$ of a triangle $ABC$ . Two circles passing through $D$ and touching $AC$ and $BC$ at $A$ and $B$ respectively intersect again at point $E$ . Let $F$ be the point symmetric to $C$ with respect to the perpendicular bisector of $AB$ . Prove that the points $D,E,F$ lie on a line.

Let $P$ be a polynomial with integer coefficients such that there are two distinct integers at which $P$ takes coprime values. Show that there exists an infinite set of integers, such that the values $P$ takes at them are pairwise coprime.

On a tournament with $n \ge 3$ participants, every two participants played exactly one match and there were no draws. A three-element set of participants is called a draw-triple if they can be enumerated so that the first defeated the second, the second defeated the third, and the third defeated the first. Determine the largest possible number of draw-triples on such a tournament.

Let real numbers $a,b,c$ . Prove that $\sqrt{2(a^2+b^2)}+\sqrt{2(b^2+c^2)}+\sqrt{2(c^2+a^2)}\ge \sqrt{3(a+b)^2+3(b+c)^2+3(c+a)^2}$ .

Find the greatest possible number of lines in space that all pass through a single point and the angle between any two of them is the same.

An integer $m > 1$ is given. The infinite sequence $(x_n)_{n\ge 0}$ is defined by $x_i=2^i$ for $i<m$ and $x_i=x_{i-1}+x_{i-2}+\cdots +x_{i-m}$ for $i\ge m$ . Find the greatest natural number $k$ such that there exist $k$ successive terms of this sequence which are divisible by $m$ .