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The distinct positive integers $a_1,a_2,...,a_n,b_1,b_2,...,b_n$ with $n\ge2$ have the property that the $\binom{n}2$ sums $a_i+a_j$ are the same as the $\binom{n}2$ sums $b_i+b_j$ (in some order). Prove that $n$ is a power of $2$.

I've seen a solution using generating function and some calculus, but is there a more intuitive proof why $n$ has to be a power of $2$?

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