CZ-SK-MO 2020

Two positive integers $m$ and $n$ are written on the board. We replace one of two numbers in each step on the board by either their sum, or product, or ratio (if it is an integer). Depending on the numbers $m$ and $n$ , specify all the pairs that can appear on the board in pairs.

(Radovan Švarc)

The triangle $ABC$ is given. Inside its sides $AB$ and $AC$ , the points $X$ and $Y$ are respectively selected Let $Z$ be the intersection of the lines $BY$ and $CX$ . Prove the inequality $$[BZX] + [CZY]> 2 [XY Z]$$ , where $[DEF]$ denotes the content of the triangle $DEF$ .

(David Hruska, Josef Tkadlec)

Consider the system of equations $\begin{cases} x^2 - 3y + p = z, y^2 - 3z + p = x, z^2 - 3x + p = y \end{cases}$ with real parameter $p$ . a) For $p \ge 4$ , solve the considered system in the field of real numbers. b) Prove that for $p \in (1, 4)$ every real solution of the system satisfies $x = y = z$ .

(Jaroslav Svrcek)

Positive integers $a, b$ satisfy equality $b^2 = a^2 + ab + b$ . Prove that $b$ is a square of a positive integer.

(Patrik Bak)

Given an isosceles triangle $ABC$ with base $BC$ . Inside the side $BC$ is given a point $D$ . Let $E, F$ be respectively points on the sides $AB, AC$ that $|\angle BED | = |\angle DF C| > 90^o$ . Prove that the circles circumscribed around the triangles $ABF$ and $AEC$ intersect on the line $AD$ at a point different from point $A$ .

(Patrik Bak, Michal Rolínek)

For each positive integer $k$ , denote by $P (k)$ the number of all positive integers $4k$ -digit numbers which can be composed of the digits $2, 0$ and which are divisible by the number $2 020$ . Prove the inequality $$P (k) \ge \binom{2k - 1}{k}^2$$ and determine all $k$ for which equality occurs.

(Note: A positive integer cannot begin with a digit of $0$ .)

(Jaromir Simsa)