RS-SMO 2022

Let $k$ be incircle of acute triangle $ABC$ , $AC\neq BC$ , and $l$ be excircle that touches $AB$ . Line $p$ through the $C$ is orthogonal to $AB$ , $p\cap k = \{X, Y\}$ , $p\cap l = \{Z, T\}$ and the point arrangement is $X-Y-Z-T$ . Circle $m$ through $X$ and $Z$ intersects $AB$ at $D$ and $E$ . Prove that points $D,Y,E,T$ are concyclic.

Let $a$ , $b$ and $c$ be positive real numbers and $a^3+b^3+c^3=3$ . Prove $$\frac{1}{3-2a}+\frac{1}{3-2b}+\frac{1}{3-2c}\geq 3$$

Let $f(n)$ be number of numbers $x \in \{1,2,\cdots ,n\}$ , $n\in\mathbb{N}$ , such that $gcd(x, n)$ is either $1$ or prime. Prove $$\sum_{d|n} f(d) + \varphi(n) \geq 2n$$ For which $n$ does equality hold?

On the board are written $n$ natural numbers, $n\in \mathbb{N}$ . In one move it is possible to choose two equal written numbers and increase one by $1$ and decrease the other by $1$ . Prove that in this the game cannot be played more than $\frac{n^3}{6}$ moves.

Let $p$ and $q$ be different primes, and $\alpha\in (0, 3)$ a real number. Prove that in sequence $$\left[ \alpha \right] , \left[ 2\alpha \right] , \left[ 3\alpha \right] \dots$$ exists number less than $2pq$ , divisible by $p$ or $q$ .