ISL 1997 Problem 3

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 $2n.$