XK-MO 2020

Compare the following two numbers: $2^{2^{2^{2^{2}}}}$ and $3^{3^{3^{3}}}$ .

A natural number $n$ is written on the board. Ben plays a game as follows: in every step, he deletes the number written on the board, and writes either the number which is three greater or two less than the number he has deleted. Is it possible that for every value of $n$ , at some time, he will get to the number $2020$ ?

Let $\triangle ABC$ be a triangle. Let $O$ be the circumcenter of triangle $\triangle ABC$ and $P$ a variable point in line segment $BC$ . The circle with center $P$ and radius $PA$ intersects the circumcircle of triangle $\triangle ABC$ again at another point $R$ and $RP$ intersects the circumcircle of triangle $\triangle ABC$ again at another point $Q$ . Show that points $A$ , $O$ , $P$ and $Q$ are concyclic.

Let $p$ and $q$ be prime numbers. Show that $p^2+q^2+2020$ is composite.

Let $x\in\mathbb{R}$ . What is the maximum value of the following expression: $\sqrt{x-2018} + \sqrt{2020-x}$ ?

Ana baked 15 pasties. She placed them on a round plate in a circular way: 7 with cabbage, 7 with meat and 1 with cherries in that exact order and put the plate into a microwave. She doesn't know how the plate has been rotated in the microwave. She wants to eat a pasty with cherries. Is it possible for Ana, by trying no more than three pasties, to find exactly where the pasty with cherries is?

Find all prime numbers $p$ such that $3^p + 5^p -1$ is a prime number.

Let $B'$ and $C'$ be points in the circumcircle of triangle $\triangle ABC$ such that $AB=AB'$ and $AC=AC'$ . Let $E$ and $F$ be the foot of altitudes from $B$ and $C$ to $AC$ and $AB$ , respectively. Show that $B'E$ and $C'F$ intersect on the circumcircle of triangle $\triangle ABC$ .

Some positive integers, sum of which is $23$ , are written in sequential form. Neither one of the terms nor the sum of some consecutive terms in the sequence is equal to $3$ . **a)**Is it possible that the sequence contains exactly $11$ terms?**b)**Is it possible that the sequence contains exactly $12$ terms?

Find all positive integers $x$ , $y$ such that $2^x+5^y+2$ is a perfect square.

Let $a$ and $b$ be real numbers such that $a+b=\log_2( \log_2 3)$ . What is the minimum value of $2^a + 3^b$ ?

Let $\triangle ABC$ be a triangle and $\omega$ its circumcircle. The exterior angle bisector of $\angle BAC$ intersects $\omega$ at point $D$ . Let $X$ be the foot of the altitude from $C$ to $AD$ and let $F$ be the intersection of the internal angle bisector of $\angle BAC$ and $BC$ . Show that $BX$ bisects segment $AF$ .

Two players, Agon and Besa, choose a number from the set $\{1,2,3,4,5,6,7,8\}$ , in turns, until no number is left. Then, each player sums all the numbers that he has chosen. We say that a player wins if the sum of his chosen numbers is a prime and the sum of the numbers that his opponent has chosen is composite. In the contrary, the game ends in a draw. Agon starts first. Does there exist a winning strategy for any of the players?

Let $a_1,a_2,...,a_n$ be integers such that $a_1^{20}+a_2^{20}+...+a_n^{20}$ is divisible by $2020$ . Show that $a_1^{2020}+a_2^{2020}+...+a_n^{2020}$ is divisible by $2020$ .

Let $a_0$ be a fixed positive integer. We define an infinite sequence of positive integers $\{a_n\}_{n\ge 1}$ in an inductive way as follows: if we are given the terms $a_0,a_1,...a_{n-1}$ , then $a_n$ is the smallest positive integer such that $\sqrt[n]{a_0\cdot a_1\cdot ...\cdot a_n}$ is a positive integer. Show that the sequence $\{a_n\}_{n\ge 1}$ is eventually constant.Note: The sequence $\{a_n\}_{n\ge 1}$ is eventually constant if there exists a positive integer $k$ such that $a_n=c$ , for every $n\ge k$ .