A square is divided into $25$ unit squares by drawing lines parallel to the sides of the square. Some diagonals of unit squares are drawn from such that two diagonals do not share points. What is the maximum number diagonals that can be drawn with this property?

Let $ABCD$ be a square. Let $M$ and $K$ be points on segments $BC$ and $CD$ respectively, such that $MC = KD$. Let $P$ be the intersection of the segments $MD$ and $BK$. Prove that $AP$ is perpendicular to $MK$.

Find the largest value that the expression can take $a^3b + b^3a$ where $a, b$ are non-negative real numbers, with $a + b = 3$.

Rectangle $ABCD$ has sides $AB = 3$, $BC = 2$. Point $P$ lies on side $AB$ is such that the bisector of the angle $CDP$ passes through the midpoint $M$ of $BC$. Find $BP$.

The wizard Gandalf has a necklace that is shaped like a row of magic pearls. The necklace has $2019$ pearls, $2018$ are black and the last one is white. Everytime that the magician Gandalf touches the necklace, the following occurs: the pearl in position $i$ is move to position $i-1$, for $1 <i <2020$, furthermore the pearl in position $1$ moves to position $2019$. But something else happens, if the pearl in position $1$ now is white, then the last pearl turns white without the need for Gandalf to touch the necklace again. How many times does Gandalf have to touch the necklace to be sure that all pearls are white?

Starting from a pyramid $T_0$ whose edges are all of length $2019$, we construct the Figure $T_1$ when considering the triangles formed by the midpoints of the edges of each face of $T_0$, building in each of these new pyramid triangles with faces identical to base. Then the bases of these new pyramids are removed. Figure $T_2$ is constructed by applying the same process from $T_1$ on each triangular face resulting from $T_1$, and so on for $T_3, T_4, ...$ Let $D_0= \max \{d(x,y)\}$, where $x$ and $y$ are vertices of $T_0$ and $d(x,y)$ is the distance between $x$ and $y$. Then we define $D_{n + 1} = \max \{d (x, y) |d (x, y) \notin \{D_0, D_1,...,D_n\}$, where $x, y$ are vertices of $T_{n+1}$. Find the value of $D_n$ for all $n$.