The picture below shows the way Juan wants to divide a square field in three regions, so that all three of them share a well at vertex $B$. If the side length of the field is $60$ meters, and each one of the three regions has the same area, how far must the points $M$ and $N$ be from $D$? Note: the area of each region includes the area the well occupies. [asy][asy] pair A=(0,0),B=(60,0),C=(60,-60),D=(0,-60),M=(0,-40),N=(20,-60); pathpen=black; D(MP("A",A,W)--MP("B",B,NE)--MP("C",C,SE)--MP("D",D,SW)--cycle); D(B--MP("M",M,W)); D(B--MP("N",N,S)); D(CR(B,3));[/asy][/asy]

A series of figures is shown in the picture below, each one of them created by following a secret rule. If the leftmost figure is considered the first figure, how many squares will the 21st figure have? Note: only the little squares are to be counted (i.e., the $2 \times 2$ squares, $3 \times 3$ squares, $\dots$ should not be counted) Extra (not part of the original problem): How many squares will the 21st figure have, if we consider all $1 \times 1$ squares, all $2 \times 2$ squares, all $3 \times 3$ squares, and so on?.

In a triangle $ABC$, let $M$ be the midpoint of $AC$. If $BC = \frac{2}{3} MC$ and $\angle{BMC}=2 \angle{ABM}$, determine $\frac{AM}{AB}$.

Find all 4-digit numbers $\overline{abcd}$ that are multiples of $11$, such that the 2-digit number $\overline{ac}$ is a multiple of $7$ and $a + b + c + d = d^2$.

In a triangle $ABC$, let $D$, $E$ and $F$ be the feet of the altitudes from $A$, $B$ and $C$ respectively. Let $D'$, $E'$ and $F'$ be the second intersection of lines $AD$, $BE$ and $CF$ with the circumcircle of $ABC$. Show that the triangles $DEF$ and $D'E'F'$ are similar.