A long time ago, there existed Martians with $3$ different colours: red, green and blue. As Mars was devastated by an intergalactic war, only $2$ Martians of each colours survived. In order to reconstruct the Martian population, they decided to use a machine that transforms two Martians of distinct colours into four Martians of colour different to the two initial ones. For example, if a red Martian and a blue Martian use the machine, they'll be transformed into four green Martians. a) Is it possible that, after using that machine finitely many times, we have $2022$ red Martians, $2022$ green Martians and $2022$ blue Martians? b) Is it possible that, after using that machine finitely many times, we have $2021$ red Martians, $2022$ green Martians and $2023$ blue Martians?

Let $D$ be a point on the inside of triangle $ABC$ such that $AD=CD$, $\angle DAB=70^{\circ}$, $\angle DBA=30^{\circ}$ and $\angle DBC=20^{\circ}$. Find the measure of angle $\angle DCB$.

An integer $n$ is called $k$-special, with $k$ a positive integer, if it's the sum of the squares of $k$ consecutive integers. For example, $13$ is $2$-special, since $13=2^2+3^2$, and $2$ is $3$-special, since $2=(-1)^2+0^2+1^2$. a) Prove that there's no perfect square that is $4$-special. b) Find a perfect square that is $I^2$-special, for some odd positive integer $I$ with $I\ge 3$.

A positive integer with 3 digits $\overline{ABC}$ is $Lusophon$ if $\overline{ABC}+\overline{CBA}$ is a perfect square. Find all $Lusophon$ numbers.

Let $ABCDEF$ be a regular hexagon with side 1. Point $X, Y$ are on sides $CD$ and $DE$ respectively, such that the perimeter of $DXY$ is $2$. Determine $\angle XAY$.

A calculator has two operations $A$ and $B$ and initially shows the number $1$. Operation $A$ turns $x$ into $x+1$ and operation B turns $x$ into $\dfrac{x}{x+1}$. a) Show all the ways we can get the number $\dfrac{20}{23}$. b) For every rational $r \neq 1$, determine if it is possible to get $r$ using only operations $A$ and $B$.