Question 110
Let $n$ be a given positive integer. In the coordinate plane, consider the set of the points
$$
\{P_1, P_2, \dots, P_{4n+1}\} = \{(x, y) \mid x \text{ and } y \text{ are integers with } xy = 0, |x| \le n, |y| \le n\}.
$$
Determine the minimum of $(P_1P_2)^2 + (P_2P_3)^2 + \dots + (P_{4n}P_{4n+1})^2 + (P_{4n+1}P_1)^2$. (Posed by Wang Xinmao)
$$
\{P_1, P_2, \dots, P_{4n+1}\} = \{(x, y) \mid x \text{ and } y \text{ are integers with } xy = 0, |x| \le n, |y| \le n\}.
$$
Determine the minimum of $(P_1P_2)^2 + (P_2P_3)^2 + \dots + (P_{4n}P_{4n+1})^2 + (P_{4n+1}P_1)^2$. (Posed by Wang Xinmao)