CONO-SUR 2017

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?

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 $n \ge 1$ [/*] [/list]

[list = a]

• 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 $a_1,a_2,a_3,\ldots$