Question 65
Given a point $X$ and $n$ vectors $\mathbf{x}_{\mathrm{i}}$ with sum zero in the plane. For each permutation of the vectors we form a set of $n$ points, by starting at $X$ and adding the vectors in order. For example, with the original ordering we get $\mathbf{X}_1$ such that $\mathbf{X}\mathbf{X}_1 = \mathbf{x}_1$, $\mathbf{X}_2$ such that $\mathbf{X}_1\mathbf{X}_2 = \mathbf{x}_2$ and so on. Show that for some permutation we can find two points $Y$, $Z$ with angle $\mathbf{Y}\mathbf{X}\mathbf{Z} = 60^\circ$, so that all the points lie inside or on the triangle $XYZ$.