pow:problem1s21 [Department of Mathematics and Statistics, Binghamton University]

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<box 80% round orange| Problem 1 (due Monday, March 1)> 

We say that a vector in $\mathbb R^3$ is ${\rm positive}$ (${\rm negative}$)
if all its coordinates are positive (resp. negative). Let $v_1,v_2,v_3,v_4$ be vectors in $\mathbb R^3$ such that the sum of any two of these vectors is either positive or negative.
Prove that at least one of the vectors $v_1,v_2,v_3,v_4, v_1+v_2+v_3+v_4$ is either positive
or negative. 

             


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The problem was solved by Chris Eppolito, Yuqiao Huang, Ashton Keith, Maxwell T Meyers, and Wei
Yang.
One solver provided an (almost complete) solution in which it was observed that for vectors in $\mathbb R^3$ actually 
one of the vectors $v_1,v_2,v_3,v_4$ is either positive or negative. The other solvers provided complete solutions which work for vectors in $\mathbb R^n$ for any $n$. Detailed solutions are discussed in the following link {{:pow:2021sproblem1.pdf|Solution}}.