Given 81 variables that satisfy
\[ 0 \leq a_1 \leq a_2 \leq \ldots \leq a_{81} \leq 1, \]
what is the maximum value of
\[ \left[ 9 \sum_{i=1}^{81} a_i ^2 \right] + \left[ \sum_{1 \leq j < k \leq 81 } ( a_k - a_j + 1)^2 \right] ? \]
by
Calvin Lin
