IMO Shortlist 1997 problem 3

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Dodao/la: arhiva
2. travnja 2012.
For each finite set U of nonzero vectors in the plane we define l(U) to be the length of the vector that is the sum of all vectors in U. Given a finite set V of nonzero vectors in the plane, a subset B of V is said to be maximal if l(B) is greater than or equal to l(A) for each nonempty subset A of V.

(a) Construct sets of 4 and 5 vectors that have 8 and 10 maximal subsets respectively.

(b) Show that, for any set V consisting of n \geq 1 vectors the number of maximal subsets is less than or equal to 2n.
Izvor: Međunarodna matematička olimpijada, shortlist 1997