BG-MO 2002

Let $a_1, a_2...$ be an infinite sequence of real numbers such that $a_{n+1}=\sqrt{{a_n}^2+a_n-1}$ . Prove that $a_1 \notin (-2,1)$ Proposed by Oleg Mushkarov and Nikolai Nikolov

Consider the orthogonal projections of the vertices $A$ , $B$ and $C$ of triangle $ABC$ on external bisectors of $\angle ACB$ , $\angle BAC$ and $\angle ABC$ , respectively. Prove that if $d$ is the diameter of the circumcircle of the triangle, which is formed by the feet of projections, while $r$ and $p$ are the inradius and the semiperimeter of triangle $ABC$ , prove that $r^2+p^2=d^2$ Proposed by Alexander Ivanov

Given are $n^2$ points in the plane, such that no three of them are collinear, where $n \geq 4$ is the positive integer of the form $3k+1$ . What is the minimal number of connecting segments among the points, such that for each $n$ -plet of points we can find four points, which are all connected to each other?

Proposed by Alexander Ivanov and Emil Kolev

Let $I$ be the incenter of a non-equilateral triangle $ABC$ and $T_1$ , $T_2$ , and $T_3$ be the tangency points of the incircle with the sides $BC$ , $CA$ and $AB$ , respectively. Prove that the orthocenter of triangle $T_1T_2T_3$ lies on the line $OI$ , where $O$ is the circumcenter of triangle $ABC$ .

Proposed by Georgi Ganchev

Find all pairs $(b,c)$ of positive integers, such that the sequence defined by $a_1=b$ , $a_2=c$ and $a_{n+2}= \left| 3a_{n+1}-2a_n \right|$ for $n \geq 1$ has only finite number of composite terms.

Proposed by Oleg Mushkarov and Nikolai Nikolov

Find the smallest number $k$ , such that $\frac{l_a+l_b}{a+b}<k$ for all triangles with sides $a$ and $b$ and bisectors $l_a$ and $l_b$ to them, respectively.

Proposed by Sava Grodzev, Svetlozar Doichev, Oleg Mushkarov and Nikolai Nikolov