1973 IMO Problems/Problem 1
Point lies on line are unit vectors such that points all lie in a plane containing and on one side of Prove that if is odd, Here denotes the length of vector
We prove it by induction on the number of vectors. The base step (when we have one vector) is clear, and for the induction step we use the hypothesis for the vectors obtained by disregarding the outermost two vectors. We thus get a vector with norm betwen two with norm . The sum of the two vectors of norm makes an angle of with the vector of norm , so their sum has norm , and we're done.
The above solution was posted and copyrighted by grobber. The original thread for this problem can be found here: 
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