OMCPLP 2018

Fill in the corners of the square, so that the sum of the numbers in each one of the $5$ lines of the square is the same and the sum of the four corners is $123$ .

Attachments:

In a triangle $ABC$ , right in $A$ and isosceles, let $D$ be a point on the side $AC$ ( $A \ne D \ne C$ ) and $E$ be the point on the extension of $BA$ such that the triangle $ADE$ is isosceles. Let $P$ be the midpoint of segment $BD$ , $R$ be the midpoint of the segment $CE$ and $Q$ the intersection point of $ED$ and $BC$ . Prove that the quadrilateral $ARQP$ is a square.

For each positive integer $n$ , let $S(n)$ be the sum of the digits of $n$ . Determines the smallest positive integer $a$ such that there are infinite positive integers $n$ for which you have $S (n) -S (n + a) = 2018$ .

Determine the pairs of positive integer numbers $m$ and $n$ that satisfy the equation $m^2=n^2 +m+n+2018$ .

Determine the increasing geometric progressions, with three integer terms, such that the sum of these terms is $57$

In a $3 \times 25$ board, $1 \times 3$ pieces are placed (vertically or horizontally) so that they occupy entirely $3$ boxes on the board and do not have a common point. What is the maximum number of pieces that can be placed, and for that number, how many configurations are there?