BG-MO 1971

A natural number is called triangular if it may be presented in the form $\frac{n(n+1)}2$ . Find all values of $a$ $(1\le a\le9)$ for which there exist a triangular number all digit of which are equal to $a$ .

Prove that the equation $$\sqrt{2-x^2}+\sqrt[3]{3-x^3}=0$$ has no real solutions.

There are given $20$ points in the plane, no three of which lie on a single line. Prove that there exist at least $969$ quadrilaterals with vertices from the given points.

It is given a triangle $ABC$ . Let $R$ be the radius of the circumcircle of the triangle and $O_1,O_2,O_3$ be the centers of excircles of the triangle $ABC$ and $q$ is the perimeter of the triangle $O_1O_2O_3$ . Prove that $q\le6R\sqrt3$ . When does equality hold?

Let $A_1,A_2,\ldots,A_{2n}$ are the vertices of a regular $2n$ -gon and $P$ is a point from the incircle of the polygon. If $\alpha_i=\angle A_iPA_{i+n}$ , $i=1,2,\ldots,n$ . Prove the equality $$\sum_{i=1}^n\tan^2\alpha_i=2n\frac{\cos^2\frac\pi{2n}}{\sin^4\frac\pi{2n}}.$$

In a triangular pyramid $SABC$ one of the plane angles with vertex $S$ is a right angle and the orthogonal projection of $S$ on the base plane $ABC$ coincides with the orthocenter of the triangle $ABC$ . Let $SA=m$ , $SB=n$ , $SC=p$ , $r$ is the inradius of $ABC$ . $H$ is the height of the pyramid and $r_1,r_2,r_3$ are radii of the incircles of the intersections of the pyramid with the plane passing through $SA,SB,SC$ and the height of the pyramid. Prove that

(a) $m^2+n^2+p^2\ge18r^2$ ; (b) $\frac{r_1}H,\frac{r_2}H,\frac{r_3}H$ are in the range $(0.4,0.5)$ .