There are lots of hints in the problem to help you.

To help you understand the problem, you can start by working out some sets of lengths which can, or cannot be the lengths of the sides of a triangle.  This should lead you to the Triangle Inequality which states a relationship between the longest side of the triangle and the two shorter ones which must be true if the lengths are the lengths of the sides of a triangle.

Try sketching a tetrahedron where none of the vertices have three edges whose lengths could be the lengths of the sides of a triangle.  Can it be done?  You could assume that such a tetrahedron exists, and then show that this is actually impossible (called a proof by contradiction).