Problem
COM-B2-M04-P008 A Maximal Unextendable Set
In the set \(\{1,2,\ldots,2n\}\), a subset \(A\) is chosen so that no new number can be added while still having no two numbers with sum \(2n+1\). Prove that \(A\) contains exactly one number from each pair \(\{1,2n\},\{2,2n-1\},\ldots,\{n,n+1\}\).
First prove that at most one number is taken from each pair, then use the impossibility of adding a number.
From each pair \(\{i,2n+1-i\}\), we cannot take both numbers because their sum is \(2n+1\). Thus at most one number is taken from each pair.
Suppose some pair contributes no number. Then one number from this pair can be added to \(A\): its only partner for the sum \(2n+1\) is also absent. The condition would remain true, contradicting maximality by inclusion.
Therefore exactly one number is taken from each pair.
Here it is important to distinguish maximal by inclusion from maximum size.