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Suppose that we are given 40 points equally spaced around the perimeter of a square, so that four of them are located at the vertices and the remaining points divide each side into ten congruent segments. If P, Q, and R are chosen to be any three of these points which are not collinear, then how many different possible positions are there for the centroid of triangle PQR?

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    I think that if you take coordinates so your square is $[0,10]\times[0,10]$ and use that the centroid of $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$ is $\left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right),$then everything reduces to a combinatorics problem. My conjecture is that all points $(x,y)\in(0,10)\times(0,10)$ such that $3x,3y\in\mathbb{Z}$ are centroids of some triangle.2012-11-02

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