62nd Ukrainian National Mathematical Olympiad, Third Round, First Tour

Let $p_1,p_2,\dots ,p_{2n-1}$ be distinct prime integers larger than $2$. Then for a set $(1,p_1-1,p_2-1,\dots ,p_{2n-1}-1)$ it's not possible to form $n$ such pairs, as no number can be paired with number $1$.

From the other side, we can always form at least $n-1$ pairs from the numbers of the same parity. In each such pair the sum will be even and not less than $4$, therefore composite.