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Polygon Area Ponder (Posted on 2016-09-01) Difficulty: 3 of 5
A convex polygon having precisely 2N (N ≥2) sides has each of its vertices at lattice points.

Can the area of the polygon be < N3/100?
If so, give an example.
If not, prove it.

No Solution Yet Submitted by K Sengupta    
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Solution Solution | Comment 1 of 2
Before even looking for a solution it stands to reason that since the limit grows as a cubic, all we need to do is find a way of increasing the area as a lower degree function of N and a solution will occur.

This is easy to do with a zigzag.
(0,0) (1,0) 
        (1,1) (2,1)
(0,2) (1,2)
        (1,3) (2,3)

When N=10, the area of this 20-gon is 9 < 10^3/100 = 10

  Posted by Jer on 2016-09-01 13:25:49
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