Lowest product of pair multiplication This is kind of an algebra question, and I am interested in an algebric proof to it.
Suppose we have $k$ natural numbers that are all greater than $0$.
We would like to arrange them in multiplication-pairs of two, such that the sum of each pair's product is the lowest possible.
For example: Given $A = \{5,9,1,3,6,12\}$, a minimal product of pairs multiplication is taking the pairs $(1,12), (3,9), (5,6)$, such that $ 1 \cdot 12 + 3 \cdot 9 + 5 \cdot 6$ is the lowest possible.
Is it safe to say, that for each pair selection out of the set of the natural numbers, we pair the minimal with the maximal, then remove them from the set and go on?

?: 
A={1,2,3}
Suppose you have made one pair (1,2) and removed it from the set. How do you make another pair?
Your reasoning requires additional conditions.
Saying set to be containing even number of elements would be better.
