Question

Problem 1. Given any set A = {a1, a2, a3, a4} of four distinct positive integers, we denote the sum
a1 +a2 +a3 +a4 by sA. Let nA denote the number of pairs (i, j) with 1  i < j  4 for which ai +aj
divides sA. Find all sets A of four distinct positive integers which achieve the largest possible value
of nA.

Answer 1

For starters, I don't think there is any case where n(A)=4, at least where all the elements are distinct.

Answer 2

Without loss of generality, we can say that
\[a_1<a_2<a_3<a_4\]
If we assume that any pair of these elements will divide the sum, then
\[a_3+a_4|a_1+a_2+a_3+a_4\]\[a_1+a_2+a_3+a_4<a_3+a_4+a_3+a_4<2(a_3+a_4)\]so
\[a_3+a_4|x, x<2(a_3+a_4)\]
which is not possible.

Answer 3

thanks,but