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UT Math Club Problem of the Month November 2012 Problem I (Dot products for Euclid ) Define the dot product of the arrows in the left figure to be the total area of the dark squares minus the area of the light square, all divided by two. Prove to an ancient Greek geometer that the area of the dark rectangle in the right figure is equal to the dot product of the two arrows. You may assume that the angle between the arrows is acute, and that the geometer speaks English.1 1 We will evaluate solutions written in ancient Greek to the best of our ability. 1 Problem II (Line party) I have two disjoint families A and B of lines in three-dimensional space. • Two distinct lines in A ∪ B intersect if and only if one is in A and the other is in B. • The union of all the lines in A is the same as the union of all the lines in B. Prove that the union of all the lines in A ∪ B is the zero-set of a polynomial. Submission Deadline: Friday, December 18th, 11:59pm 2