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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

V

is said to be maximal if

l(B)

is greater than or equal to

l(A)

for each nonempty subset

A

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.

Problem Statement