Question 93
Given an integer $N \ge 4$, determine the largest value the sum
$$
\sum_{i=1}^{\lfloor k/2 \rfloor + 1} (\lfloor n_i/2 \rfloor + 1)
$$
may achieve, where $k, n_1, \dots, n_k$ run through the integers subject to $k \ge 3, n_1 \ge \dots \ge n_k \ge 1$, and $n_1 + \dots + n_k = N$.
$$
\sum_{i=1}^{\lfloor k/2 \rfloor + 1} (\lfloor n_i/2 \rfloor + 1)
$$
may achieve, where $k, n_1, \dots, n_k$ run through the integers subject to $k \ge 3, n_1 \ge \dots \ge n_k \ge 1$, and $n_1 + \dots + n_k = N$.