Hard Number Theory · Superchallenge problem

Cubing permutes the nonzero residues mod $11$. Use this to evaluate $\left\lfloor \dfrac{1^3}{11} \right\rfloor + \left\lfloor \dfrac{2^3}{11} \right\rfloor + \cdots + \left\lfloor \dfrac{10^3}{11} \right\rfloor$.