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ASU 434 All Soviet Union MO 1986 vector sum zero in n-gon A_1A_2...A_n

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August 7, 2019
vectorpolygongeometry

Problem Statement

Given a regular nn-gon A1A2...AnA_1A_2...A_n. Prove that if
a) nn is even number, than for the arbitrary point MM in the plane, it is possible to choose signs in an expression ±MA1→±MA2→±...±MAn→\pm \overrightarrow{MA_1} \pm \overrightarrow{MA_2} \pm ... \pm \overrightarrow{MA_n}to make it equal to the zero vector .
b) nn is odd, than the abovementioned expression equals to the zero vector for the finite set of MM points only.
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