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n points, vectors, 2 player game - All-Russian MO 2002 Regional (R4) 11.6

Source:

September 17, 2024
vectorcombinatoricscombinatorial geometry

Problem Statement

There are n>1n > 1 points on the plane. Two take turns connecting more an unconnected pair of points by a vector of one of two possible directions. If after the next move of a player the sum of all drawn vectors is zero, then the second one wins; if it's another move is impossible, and there was no zero sum, then the first one wins. Who wins when played correctly?