Prove that inequality $AM \cdot AN + BM \cdot BN + CM \cdot CN \geq DM \cdot DN$

Let $ABCD$ be the regular tetrahedron, and $M, N$ points in space. Prove that: $$AM \cdot AN + BM \cdot BN + CM \cdot CN \geq DM \cdot DN$$

Maybe use Ptolemy's inequality solve it?

Yes this is just Ptolemy's inequality.

Suppose that the isometry (i.e. rotation/reflection) swapping $(A,B)$ and $(C,D)$ sends $M$ to $L$. Now it suffices to show that $$BL\cdot AN+AL\cdot BN+DL\cdot CN\geq CL\cdot DN.$$ Ptolemy's inequality gives the following: $$\begin{align*}BL\cdot AN+AL\cdot BN&\geq LN\cdot AB\\LN\cdot CD+DL\cdot CN&\geq CL\cdot DN.\end{align*}$$ Since $AB=CD$, adding the two gives the desired inequality.