A closed recticular polygon with 100 sides (may be concave) is given such that it's vertices have integer coordinates, it's sides are parallel to the axis and all it's sides have odd length. Prove that it's area is odd.
it's true for every polygon with $ 4(2k+1)$ sides. induction: the first case in the pic(area is ood obviously). then we can make longer one side or shorter-and it doesn't change the parity of the area cause we add even length to the side, that means even number of area. also if we change the direction of a side we add or subtract and even area because odd+odd=even. hope it's true and clear. we didn't use pick's theorem!
