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 while $\rho_1$ and $\rho_2$ are the radii of the excircles of the triangles $ACM$ and $BCM$ at the sides $AM$ and $BM$ respectively. Let $r$ and $\rho$ denote the respective radii of the inscribed circle and the excircle at the side $AB$ of the triangle $ABC$. Prove that
$$\frac{r_1r_2}{\rho_1\rho_2}=\frac r\rho.$$
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
$$\frac{A_n}{A_{n+1}}\lt \frac{B_n}{B_{n+1}}.$$
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 an infinity of natural numbers $n$.
In the tetrahedron $ABCD$, the edges $BD$ and $CD$ are mutually perpendicular, and the projection of the vertex $D$ to the plane $ABC$ is the intersection of the altitudes of the triangle $ABC$. Prove that
$$(AB+BC+CA)^2\le6(DA^2+DB^2+DC^2).$$
For which tetrahedra does equality hold?
Given 100 points in the plane, no three of which are on the same line, consider all triangles that have all their vertices chosen from the 100 given points. Prove that at most 70% of these triangles are acute-angled.