Problem

Source: China Girls Mathematical Olympiad 2009, Problem 3

Tags: analytic geometry, inequalities unsolved, inequalities



Let $ n$ be a given positive integer. In the coordinate set, consider the set of points $ \{P_{1},P_{2},...,P_{4n+1}\}=\{(x,y)|x,y\in \mathbb{Z}, xy=0, |x|\le n, |y|\le n\}.$ Determine the minimum of $ (P_{1}P_{2})^{2} + (P_{2}P_{3})^{2} +...+ (P_{4n}P_{4n+1})^{2} + (P_{4n+1}P_{1})^{2}.$