If Only I Could See In 81 Dimensions, Then This Would Be Easy

Algebra Level 5

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

\[ \left[ 9 \sum_{i=1}^{81} a_i ^2 \right] + \left[ \sum_{1 \leq j < k \leq 81 } ( a_k - a_j + 1)^2 \right] ? \]

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