Wasn't sure where to put this...profound apologies if this is in the wrong forum...
Suppose there are $n$ distinct odd, positive integers $\{a_i\}_{i=1}^n$ where the absolute value of the difference between each pair of numbers is distinct i.e. each $|a_k - a_i|, \ 1\leq k < i \leq n$ is distinct.
Prove that $$\sum_{i=1}^n a_i \geq \tfrac{1}{3} n(n^2 + 2)$$
I tried doing this by induction and have gotten to the stage of my induction hypothesis but am not sure how to proceed. I have tried to prove that $a_{k+1} \geq 1 + k + k^2$ (as this would help yield the result for my induction) but, again, no success....
Can anyone give me a hint as to how I would go about the induction (or even if induction is the correct way and I should try a different approach)?
Thank you