A positive integer $n$ is called guayaquilean if the sum of the digits of $n$ is equal to the sum of the digits of $n^2$. Find all the possible values that the sum of the digits of a guayaquilean number can take.

Let $A(XYZ)$ be the area of the triangle $XYZ$. A non-regular convex polygon $P_1 P_2 \ldots P_n$ is called guayaco if exists a point $O$ in its interior such that \[A(P_1OP_2) = A(P_2OP_3) = \cdots = A(P_nOP_1).\]Show that, for every integer $n \ge 3$, a guayaco polygon of $n$ sides exists.

Let $n$ be a positive integer. In how many ways can a $4 \times 4n$ grid be tiled with the following tetromino? [asy][asy] size(4cm); draw((1,0)--(3,0)--(3,1)--(0,1)--(0,0)--(1,0)--(1,2)--(2,2)--(2,0)); [/asy][/asy]

Let $ABC$ an acute triangle with circumcenter $O$. Points $X$ and $Y$ are chosen such that: $\angle XAB = \angle YCB = 90^\circ$ $\angle ABC = \angle BXA = \angle BYC$ $X$ and $C$ are in different half-planes with respect to $AB$ $Y$ and $A$ are in different half-planes with respect to $BC$ Prove that $O$ is the midpoint of $XY$.

Let $a$, $b$ and $c$ positive integers. Three sequences are defined as follows: $a_1=a$, $b_1=b$, $c_1=c$ $a_{n+1}=\lfloor{\sqrt{a_nb_n}}\rfloor$, $\:b_{n+1}=\lfloor{\sqrt{b_nc_n}}\rfloor$, $\:c_{n+1}=\lfloor{\sqrt{c_na_n}}\rfloor$ for $n \ge 1$ Prove that for any $a$, $b$, $c$, there exists a positive integer $N$ such that $a_N=b_N=c_N$. Find the smallest $N$ such that $a_N=b_N=c_N$ for some choice of $a$, $b$, $c$ such that $a \ge 2$ y $b+c=2a-1$.

The infinite sequence $a_1,a_2,a_3,\ldots$ of positive integers is defined as follows: $a_1=1$, and for each $n \ge 2$, $a_n$ is the smallest positive integer, distinct from $a_1,a_2, \ldots , a_{n-1}$ such that: $$\sqrt{a_n+\sqrt{a_{n-1}+\ldots+\sqrt{a_2+\sqrt{a_1}}}}$$is an integer. Prove that all positive integers appear on the sequence $a_1,a_2,a_3,\ldots$