Some sum of two elements exceeds a third one in a finite multiset with bounded values

Question

For a multiset $S = \{ a_1,a_2,a_3,\ldots,a_k \}$ where $k = 13$ and $1 \leqslant a < 32$, prove that there exists a subset $s = \{ a_i, a_j, a_k \}$ such that each sum of two elements exceeds the third, i.e. $a_i+a_j>a_k$ and $a_j+a_k>a_i$ and $a_i+a_k>a_j$.

I was wondering how to solve this using Pigeon-Hole Principle. Any insight would be appreciated.