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Various geometry problems


1. Suppose we have a polygon P in plane, not necessarily convex such that its area is strictly less than 1. Prove that we can translate the polygon in such a way that there are no points with integer coordinates inside or on its edges.

Solution: Imagine we break the lattice into unit squares and overlap all of them on the square [0,1]\times [0,1]. Then, because the area of our polygon is strictly smaller than 1, there exists one point A \in [0,1]\times [0,1] such that A is not covered. Consider all the points \mathcal{P}=\{ (a,b)+\overrightarrow{OA}: a,b \in \Bbb{Z}\} in the plane; by the property of A, these points are all outside P. Consider the translation of vector \overrightarrow{OA}, where O is the origin of the lattice. Then all the lattice points are translated in the set \mathcal{P}, which does not intersect P. This means that a translation of P with vector -\overrightarrow{OA} leaves all lattice points outside P.

2. We have two sets of points in plane A which has 2n elements and B which has 2m elements, with m,n positive integers such that no three points from their union are collinear. Prove that there exists a line which splits the plane in two regions each of which contain n elements of A and m elements of B.

3. Inside the unit square consider n^2 points. Prove that there exists a polygonal line joining these points which has length smaller than 3n.

4. A plane convex figure has the property that any segment which divides it in two figures of equal areas has length at most 1. Prove that the figure has area at most 2.

These problems are solvable, but I don’t have any solutions momentarily. If you solve them, you can post a hint in comments. Thanks

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  1. Jeff
    October 17, 2010 at 4:04 pm

    3. Inside the unit square consider n^2 points. Prove that there exists a polygonal line joining these points which has length smaller than 3n.

    hint – prove that n^2+1 points has a length greater than 3n

    • October 17, 2010 at 10:55 pm

      Can you prove this? Still I don’t think that this is enough. If n^2+1 points always give a polygonal line with length greater than 1 then this does not tell us anything about the length of a polygonal line of length n^2.

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