ASMO Math 2022 — Grade 8 — School Round Question 6

Let $a_1,a_2,\ldots,a_n$ be positive integers, not necessarily distinct, with at least five distinct values. Suppose that for every pair of indices $1\leq i\lt j\leq n$, there exist indices $1\leq k\lt l\leq n$, both different from $i$ and $j$, such that

$$a_i+a_j=a_k+a_l.$$

Thus, the four indices $i,j,k,l$ are pairwise distinct. Find the smallest possible value of $n$.