Problems

Let $M$ be the point of intersection of the medians of the triangle $ABC$, and $O$ an arbitrary point on a plane. Prove that $$O{M}^2=1/3(O{A}^2+O{B}^2+O{C}^2)-1/9(A{B}^2+B{C}^2+A{C}^2).$$

Three non-coplanar vectors are given. Is it possible to find a fourth vector perpendicular to the three vectors given?

Find the volume of an inclined triangular prism whose base is an equilateral triangle with sides equal to a if the side edge of the prism is equal to the side of the base and is inclined to the plane of the base at an angle of ${60}^{\circ }$.