Question 51
For an integer $n \ge 3$ and real numbers $a_1, \dots, a_n$ and $b_1, \dots, b_n$, show the following inequality.
$$
\sum_{i=1}^{n} a_i (b_i - b_{i+3}) \le \frac{3n}{8} \sum_{i=1}^{n} \left[ (a_i - a_{i+1})^2 + (b_i - b_{i+1})^2 \right] \\
(a_{n+1} = a_1 \text{ and } b_{n+i} = b_i \text{ for } i = 1, 2, 3)
$$
$$
\sum_{i=1}^{n} a_i (b_i - b_{i+3}) \le \frac{3n}{8} \sum_{i=1}^{n} \left[ (a_i - a_{i+1})^2 + (b_i - b_{i+1})^2 \right] \\
(a_{n+1} = a_1 \text{ and } b_{n+i} = b_i \text{ for } i = 1, 2, 3)
$$