Prove, that in the subset of $\{1,\ldots,150\}$ there are two disjoint pairs with the same sums.

Question

Prove, that in the subset with cardinality $25$ of $\{1,\ldots,150\}$ there are two disjoint pairs with the same sums.

Well, there are at most $150+149=299$ possibilities of sums. But if we have a pair there are ${{23}\choose{2}}=253$ ways of choosing another disjoint to it, so theoretically it would be possible for all other pairs to have other sums that the one we chose.

Answer 1

There are $\binom{25}{2}=300$ pairs. So two at least have the same sum. They must be disjoint. For if they share an element, and have the same sum, they are identical.