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Here is the proof. Let us associate to our function $F$ the curve $C$ in unite $S^{n-1}$, given by the vector $\frac{F}{|F|}$. Then it follows from $F(0)=F(1)$ that the convex hull of the curve $C$ containes the point $0$. It follows that there is a simplex with vertices on $C$ that containes $0$ -- because the convex hull is by definition the union of all such simplexs. End of proof