Given any set of four distinct positive integers, we denote the sum by . Let denote the number of pairs with for which divides . Find all sets of four distinct positive integers which achieve the largest possible value of .
Obviously .We may assume without loss of generality that .
Well
So ;
But either we have or
In the first case we have
We note and we have that and so we find depending on
In this we had 4 pairs and all the solutions;
In the second case we would have 3 pairs so it doesn't matter.