TH-TMO 2023 Problem 7

Let $n$ be positive integer and $S$ = { $0,1,…,n$ }, Define set of point in the plane. $$A = \{(x,y) \in S \times S \mid -1 \leq x-y \leq 1 \}$$ , We want to place a electricity post on a point in $A$ such that each electricity post can shine in radius 1.01 unit. Define minimum number of electricity post such that every point in $A$ is in shine area