CO-MO 2020

A positive integer is called sabrosoif when it is added to the number obtained when its digits are interchanged from one side of its written form to the other, the result is a perfect square. For example, $143$ is sabroso, since $143 + 341 =484 = 22^2$ . Find all two-digit sabroso numbers.

Given a regular $n$ -sided polygon with $n \ge 3$ . Maria draws some of its diagonals in such a way that each diagonal intersects at most one of the other diagonals drawn in the interior of the polygon. Determine the maximum number of diagonals that Maria can draw in such a way.

Note: Two diagonals can share a vertex of the polygon. Vertices are not part of the interior of the polygon.

A number is said to be triangular if it can be expressed in the form $1 + 2 +...+n$ for some positive integer $n$ . We call a positive integer $a$ retriangular if there exists a fixed positive integer $b$ such that $aT +b$ is a triangular number whenever $T$ is a triangular number. Determine all retriangular numbers.