BG-MO 1978

We are given the sequence $a_1,a_2,a_3,\ldots$ , for which: $$a_n=\frac{a^2_{n-1}+c}{a_{n-2}}\enspace\text{for all }n>2.$$ Prove that the numbers $a_1$ , $a_2$ and $\frac{a_1^2+a_2^2+c}{a_1a_2}$ are whole numbers.

$k_1$ denotes one of the arcs formed by intersection of the circumference $k$ and the chord $AB$ . $C$ is the middle point of $k_1$ . On the half line (ray) $PC$ is drawn the segment $PM$ . Find the locus formed from the point $M$ when $P$ is moving on $k_1$ .

G. Ganchev

On the name day of a man there are $5$ people. The men observed that of any $3$ people there are $2$ that knows each other. Prove that the man may order his guests around circular table in such way that every man have on its both side people that he knows.

N. Nenov, N. Hazhiivanov

Find the greatest possible real value of $S$ and smallest possible value of $T$ such that for every triangle with sides $a,b,c$ $(a\le b\le c)$ to be true the inequalities: $$S\le\frac{(a+b+c)^2}{bc}\le T.$$

Prove that for every convex polygon can be found such three sequential vertices for which a circle that they lie on covers the polygon.

Jordan Tabov

The base of the pyramid with vertex $S$ is a pentagon $ABCDE$ for which $BC>DE$ and $AB>CD$ . If $AS$ is the longest edge of the pyramid prove that $BS>CS$ .

Jordan Tabov