Question 67
Let $n$ be a positive integer greater than or equal to $2$. Determine the maximum possible value the quantity
$$
\left( \sum_{i=1}^{n} i a_i \right) \left( \sum_{i=1}^{n} \frac{a_i}{i} \right)^2
$$
can take where $a_1, a_2, \dots, a_n$ are non-negative real numbers satisfying $a_1 + a_2 + \dots + a_n = 1$.
$$
\left( \sum_{i=1}^{n} i a_i \right) \left( \sum_{i=1}^{n} \frac{a_i}{i} \right)^2
$$
can take where $a_1, a_2, \dots, a_n$ are non-negative real numbers satisfying $a_1 + a_2 + \dots + a_n = 1$.