FI-MO 2019

Solve $x(8\sqrt{1-x}+\sqrt{1+x}) \le 11\sqrt{1+x}-16\sqrt{1-x}$ when $0<x\le 1$

Prove that the number $\lfloor (2+\sqrt5)^{2019} \rfloor$ is not prime.

Let $ABCD$ be a cyclic quadrilateral whose side $AB$ is at the same time the diameter of the circle. The lines $AC$ and $BD$ intersect at point $E$ and the extensions of lines $AD$ and $BC$ intersect at point $F$ . Segment $EF$ intersects the circle at $G$ and the extension of segment $EF$ intersects $AB$ at $H$ . Show that if $G$ is the midpoint of $FH$ , then $E$ is the midpoint of $GH$ .

Define a sequence $a_n = n^n + (n - 1)^{n+1}$ when $n$ is a positive integer. Define all those positive integer $m$ , for which this sequence of numbers is eventually periodic modulo $m$ , e.g. there are such positive integers $K$ and $s$ such that $a_k \equiv a_{k+s}$ ( $mod m$ ), where $k$ is an integer with $k \ge K$ .

A teacher is known to have $2^k$ apples for some $k \in \mathbb{N}$ . He ets one of the apples and distributes the rest of the apples to his students $A$ and $B$ . The students do not see how many apples the other gets, and they do not know the number $k$ . However, they have pre-selected a discreet way to reveal one another something about the number of apples: each of the students scratches their head either by their right, left or both hands, depending on the number of apples they have received. To the teacher's surprise, the students will always know which one of the students got more apples, or that the teacher ate the only apple by herself. How is this possible?