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Minimum area of a convex grid polygon

 Published on Saturday, 9th January 2021, 07:00 pm; Solved by 97

Problem 742

A symmetrical convex grid polygon is a polygon such that:

  • All its vertices have integer coordinates.
  • All its internal angles are strictly smaller than $180°$.
  • It has both horizontal and vertical symmetry.

For example, the left polygon is a convex grid polygon which has neither horizontal nor vertical symmetry, while the right one is a valid symmetrical convex grid polygon with six vertices:

Define $A(N)$, the minimum area of a symmetrical convex grid polygon with $N$ vertices.

You are given $A(4) = 1$, $A(8) = 7$, $A(40) = 1039$ and $A(100) = 17473$.

Find $A(1000)$.