(BEL 3) Consider a regular $2n$-gon and the $n$ diagonals of it that pass through its center. Let $P$ be a point of the inscribed circle and let $a_1,a_2,\ldots,a_n$ be the angles in which the diagonals mentioned are visible from the point $P$. Prove that
$$\sum_{i=1}^n\tan^2a_i=2n\frac{\cos^2\frac{\pi}{2n}}{\sin^4\frac{\pi}{2n}}.$$
(ROM 1; IMO 2) Let $a$ and $b$ be the bases of two number systems and let
$$A_n=\overline{x_1x_2\ldots x_n}^{(a)},\quad A_{n+1}=\overline{x_0x_1x_2\ldots x_n}^{(a)},$$
$$B_n=\overline{x_1x_2\ldots x_n}^{(b)},\quad B_{n+1}=\overline{x_0x_1x_2\ldots x_n}^{(b)},$$
be numbers in the number systems with respective bases $a$ and $b$, so that $x_0,x_1,x_2,\ldots,x_n$ denote digits in the number system with base $a$ as well as in the number system with base $b$. Suppose that neither $x_0$ nor $x_1$ is zero. Prove that $a\gt b$ if and only if
$$\frac{A_n}{A_{n+1}}\lt \frac{B_n}{B_{n+1}}.$$
(BUL 6; IMO 5) In the tetrahedron $SABC$ the angle $BSC$ is a right angle, and the projection of the vertex $S$ to the plane $ABC$ is the intersection of the altitudes of the triangle $ABC$. Let $z$ be the radius of the inscribed circle of the triangle $ABC$. Prove that
$$SA^2+SB^2+SC^2\ge18z^2.$$
(CZS 1; IMO 4) For what natural numbers $n$ can the product of some of the numbers $n,n+1,n+2,n+3,n+4,n+5$ be equal to the product of the remaining ones?
(CZS 3) Let $M$ be an interior point of the tetrahedron $ABCD$. Prove that
$$\overrightarrow{MA}\operatorname{vol}(MBCD)+\overrightarrow{MB}\operatorname{vol}(MACD)+\overrightarrow{MC}\operatorname{vol}(MABD)+\overrightarrow{MD}\operatorname{vol}(MABC)=0$$
($\operatorname{vol}(PQRS)$ denotes the volume of the tetrahedron $PQRS$).
(FRA 1) In the triangle $ABC$ let $B'$ and $C'$ be the midpoints of the sides $AC$ and $AB$ respectively and $H$ the foot of the altitude passing through the vertex $A$. Prove that the circumcircles of the triangles $AB'C'$, $BC'H$, and $B'CH$ have a common point $I$ and that the line $HI$ passes through the midpoint of the segment $B'C'$.
(POL 2; IMO 1) Given a point $M$ on the side $AB$ of the triangle $ABC$, let $r_1$ and $r_2$ be the radii of the inscribed circles of the triangles $ACM$ and $BCM$ respectively and let $\rho_1$ and $\rho_2$ be the radii of the excircles of the triangles $ACM$ and $BCM$ at the sides $AM$ and $BM$ respectively. Let $r$ and $\rho$ denote the radii of the inscribed circle and the excircle at the side $AB$ of the triangle $ABC$ respectively. Prove that
$$\frac{r_1r_2}{\rho_1\rho_2}=\frac r\rho.$$
(GDR 3) Let $u_1,u_2,\ldots,u_n,v_1,v_2,\ldots,v_n$ be real numbers. Prove that
$$1+\sum_{i=1}^n(u_i+v_i)^2\le\frac43\left(1+\sum_{i=1}^nu_i^2\right)\left(1+\sum_{i=1}^nv_i^2\right).$$
In what case does equality hold?
(SWE 4; IMO 3) Let $1=a_0\le a_1\le a_2\le\cdots\le a_n\le\cdots$ be a sequence of real numbers. Consider the sequence $b_1,b_2,\ldots$ defined by:
$$b_n=\sum_{k=1}^n\left(1-\frac{a_{k-1}}{a_k}\right)\frac1{\sqrt{a_k}}.$$
Prove that:
(a) For all natural numbers $n$, $0\le b_n\lt 2$.
(b) Given an arbitrary $0\le b\lt 2$, there is a sequence $a_0,a_1,\ldots,a_n,\ldots$ of the above type such that $b_n\gt b$ is true for infinitely many natural numbers $n$.
(SWE 6) Let $P,Q,R$ be polynomials and let $S(x)=P(x^3)+xQ(x^3)+x^2R(x^3)$ be a polynomial of degree $n$ whose roots $x_1,\ldots,x_n$ are distinct. Construct with the aid of the polynomials $P,Q,R$ a polynomial $T$ of degree $n$ that has the roots $x_1^3,x_2^3,\ldots,x_n^3$.
(USS 4; IMO 6) We are given 100 points in the plane, no three of which are on the same line. Consider all triangles that have all vertices chosen from the 100 given points. Prove that at most 70% of these triangles are acute angled.