A square of $3 \times 3$ is subdivided into 9 small squares of $1 \times 1$. It is desired to distribute the nine digits $1, 2, . . . , 9$ in each small square of $1 \times 1$, a number in each small square. Find the number of different distributions that can be formed in such a way that the difference of the digits in cells that share a side in common is less than or equal to three. Two distributions are distinct even if they differ by rotation and/or reflection.

Javiera and Claudio play on a board consisting of a row with $2019$ cells. Claudio starts by placing a token anywhere on the board. Next Javiera says a natural number $k$, $1 \le k \le n$ and Claudio must move the token to the right or to the left at your choice $k$ squares and so on. Javiera wins if she manages to remove the piece that Claudio moves from the board. Determine the smallest $n$ such that Javiera always wins after a finite number of moves.

Find all solutions $x,y,z$ in the positive integers of the equation $$3^x -5^y = z^2$$

In the convex quadrilateral $ABCD$ , $\angle ADC = \angle BCD > 90^o$ . Let $E$ be the intersection of the line $AC$ with the line parallel to $AD$ that passes through $B$. Let $F$ be the intersection of line $BD$ with the line parallel to $BC$ passing through $A$. Prove that $EF$ is parallel to $CD$.