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
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We will evaluate solutions written in ancient Greek to the best of our ability.
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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
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