Given that $\displaystyle m,n$ are positive numbers and given that $\displaystyle A_1, A_2, \ldots, A_m$ $\displaystyle \subseteq{\{1,2,\ldots,n\}}$ such that $\displaystyle A_i \neq A_j$ for $\displaystyle i \neq j$.

We know that $\displaystyle \exists$ a constant $\displaystyle L$ such that for every $\displaystyle i\neq j$:

$\displaystyle \sum_{x\in A_i \cap A_j}{x^3 = L}$

Prove $\displaystyle m \leq n$