International Mathematical Olympiad 2018 Problem 6

A convex quadrilateral $ABCD$ satisfies $AB \cdot CD = BC \cdot DA$. Point $X$ lies inside $ABCD$ so that $$\angle XAB = \angle XCD \quad \text{and} \quad \angle XBC = \angle XDA.$$ Prove that $\angle BXA + \angle DXC = 180°$.