## Tough Combinatorics

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

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

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

Prove $m \leq n$