Let
![n \geq 1](/media/m/a/9/8/a982fcac3e2c9e0d94e965d6efb5a582.png)
be an integer. What is the maximum number of disjoint pairs of elements of the set
![\{ 1,2,\ldots , n \}](/media/m/4/b/b/4bbed8a5a4e6d4e1022a3948ea0ee617.png)
such that the sums of the different pairs are different integers not exceeding
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
?
%V0
Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?