IMO 1989 LL ISR59

Let v1, v2, . . . , v1989 be a set of coplanar vectors with |vr| \leq1

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IMO 1989 LL ISR59

Origin: ISR

Problem

Let v1, v2, . . . , v1989 be a set of coplanar vectors with |vr| \leq1 for 1 \leqr \leq1989. Show that it is possible to find ϵr (1 \leqr \leq1989), each equal to \pm1, such that 1989  r=1 ϵrvr \leq \sqrt 3.