OPM 2016

To unlock his cell phone, Joao slides his finger horizontally or vertically across a numerical box, similar to the one represented in the figure, describing a $7$ -digit code, without ever passing through the same digit twice. For example, to indicate the code $1452369$ , Joao follows the path indicated in the figure. João forgot his code, but he remembers that it is divisible by $9$ . How many codes are there under these conditions?

In how many different ways can you write $2016$ as the sum of a sequence of consecutive natural numbers?

Let $[ABC]$ be an equilateral triangle on the side $1$ . Determine the length of the smallest segment $[DE]$ , where $D$ and $E$ are on the sides of the triangle, which divides $[ABC]$ into two figures with equal area.

Determine all natural numbers $x, y$ and $z$ such that the number $2^x +4^y +8^z +16^2$ is a power of $2$ .