Nikolay Bogdanov-Belsky

In 1895, Nikolay Bogdanov-Belsky painted the famous: "Mental Arithmetic. In the Public School of S. Rachinsky." The problem on the blackboard is (10²+11²+12²+13²+14²)/365, can you find a way to solve it with mental arithmetic?

$\eqalign{ & \frac{{10^2 + 11^2 + 12^2 + 13^2 + 14^2 }} {{365}} = \cr & \frac{{10^2 + \left( {10 + 1} \right)^2 + (10 + 2)^2 + (10 + 3)^2 + (10 + 4)^2 }} {{365}} = \cr & \frac{{5 \cdot 10^2 + 2 \cdot 10 + 4 \cdot 10 + 6 \cdot 10 + 8 \cdot 10 + 1 + 4 + 9 + 16}} {{365}} = \cr & \frac{{500 + 200 + 30}} {{365}} = \cr & \frac{{730}} {{365}} = 2 \cr}$