3. Let $a_{1}, a_{2}, \ldots, a_{n}$ be positive integers, not necessarily distinct but with at least five distinct values. Suppose that for any $1 \leq i<j \leq n$, there exist $k, \ell$, both different from $i$ and $j$ such that $a_{i}+a_{j}=a_{k}+a_{\ell}$. What is the smallest possible value of $n$ ?
3. $a_{1} \leq a_{2} \leq \cdots \leq a_{n}$. Suppose $x<y$ are the two smallest values. Then $a_{1}=x$ and let $s$ be the smallest index such that $a_{s}=y$. Now there are two other terms whose sum is $x+y$. Thus we have $a_{2}=x$ and $a_{s+1}=y$. Since $a_{1}+a_{2}=2 x$, we must have $a_{3}=a_{4}=x$. Similarly, by considering the largest two values $w<z$, we have $a_{n}=a_{n-1}=a_{n-2}=a_{n-3}=z$ and another two terms equal to $w$. Since there is one other value, there are at least $4+2+4+2+1=13$ terms. The following 13 numbers satisfy the required property: $1,1,1,1,2,2,3,4,4,5,5,5,5$. Thus the smallest possible value of $n$ is 13 .