OlympiadBench / 2336 / Kerry has a list of n integers a₁,a₂,…,a_(n) satisfying …

Kerry has a list of n integers a₁,a₂,…,a_(n) satisfying a₁≤a₂≤…≤a_(n). Kerry calculates the pairwise sums of all m=(1)/(2)n(n−1) possible pairs of integers in her list and orders these pairwise sums as s₁≤s₂≤…≤s_(m). For example, if Kerry's list consists of the three integers 1,2,4, the three pairwise sums are 3,5,6.


Suppose that n=4 and that the 6 pairwise sums are s₁=8,s₂=104,s₃=106, s₄=110,s₅=112, and s₆=208. Determine two possible lists (a₁,a₂,a₃,a₄) that Kerry could have.

Kerry has a list of $n$ integers $a_{1}, a_{2}, \ldots, a_{n}$ satisfying $a_{1} \leq a_{2} \leq \ldots \leq a_{n}$. Kerry calculates the pairwise sums of all $m=\frac{1}{2} n(n-1)$ possible pairs of integers in her list and orders these pairwise sums as $s_{1} \leq s_{2} \leq \ldots \leq s_{m}$. For example, if Kerry's list consists of the three integers $1,2,4$, the three pairwise sums are $3,5,6$.


Suppose that $n=4$ and that the 6 pairwise sums are $s_{1}=8, s_{2}=104, s_{3}=106$, $s_{4}=110, s_{5}=112$, and $s_{6}=208$. Determine two possible lists $(a_{1}, a_{2}, a_{3}, a_{4})$ that Kerry could have.