RO-TST 1991

In a triangle $A_1A_2A_3$ , the excribed circles corresponding to sides $A_2A_3$ , $A_3A_1$ , $A_1A_2$ touch these sides at $T_1$ , $T_2$ , $T_3$ , respectively. If $H_1$ , $H_2$ , $H_3$ are the orthocenters of triangles $A_1T_2T_3$ , $A_2T_3T_1$ , $A_3T_1T_2$ , respectively, prove that lines $H_1T_1$ , $H_2T_2$ , $H_3T_3$ are concurrent.

Let $n \ge 3$ be an integer. A finite number of disjoint arcs with the total sum of length $1 -\frac{1}{n}$ are given on a circle of perimeter $1$ . Prove that there is a regular $n$ -gon whose all vertices lie on the considered arcs

Let $x_1,x_2,...,x_{2n}$ be positive real numbers with the sum $1$ . Prove that $$x_1^2x_2^2...x_n^2+x_2^2x_3^2...x_{n+1}^2+...+x_{2n}^2x_1^2...x_{n-1}^2 <\frac{1}{n^{2n}}$$

Let $n, a, b$ be integers with $n \geq 2$ and $a \notin \{0, 1\}$ and let $u(x) = ax + b$ be the function defined on integers. Show that there are infinitely many functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that $f_n(x) = \underbrace{f(f(\cdots f}_{n}(x) \cdots )) = u(x)$ for all $x$ . If $a = 1$ , show that there is a $b$ for which there is no $f$ with $f_n(x) \equiv u(x)$ .

The diagonals of a pentagon $ABCDE$ determine another pentagon $MNPQR$ . If $MNPQR$ and $ABCDE$ are similar, must $ABCDE$ be regular?

Let $a_1<a_2<\cdots<a_n$ be positive integers. Some colouring of $\mathbb{Z}$ is periodic with period $t$ such that for each $x\in \mathbb{Z}$ exactly one of $x+a_1,x+a_2,\dots,x+a_n$ is coloured. Prove that $n\mid t$ .

Andrei Radulescu-Banu