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Moderator Note: At the time that this question was posted, it was from an ongoing contest. The relevant deadline has now passed.

In the $xy$ plane, suppose $M_{i,j}=\{(x,y)\mid i\le x\le j\}$. For $i= 1,2,3,\ldots, 2012$, color $M_{i,i+1}$ red if $i$ is even and blue if $i$ is odd. For a convex polygon $P$ in the plane, let $d(P)$ denote the fraction of its area that is red. Consider polygons that lie completely in the region $M_{0,2013}$ and that have at least one vertex on each of the lines $x=0$ and $x=2013$. What is the minimum value of $d(P)$ over all non-degenerate convex polygons of this type?

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is there a picture –  binn Sep 26 '12 at 0:38
    
No... am not sure how to draw one. –  Aria Fitzpatrick Sep 26 '12 at 3:06
    
I would suggest two things. 1) see if you can answer the question with 2012 replaced with something easier like 2 or 4, and whether that gives you any ideas for the original question. 2) consider doing something about that 33 percent accept rate. –  Gerry Myerson Sep 27 '12 at 6:52
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