Question 1
Let $ABCD$ be a convex quadrilateral. A circle passing through the points $A$ and $D$ and a circle passing through the points $B$ and $C$ are externally tangent at a point $P$ inside the quadrilateral. Suppose that
$$
\angle PAB + \angle PDC \leq 90^{\circ} \quad \text{ and } \quad \angle PBA + \angle PCD \leq 90^{\circ}.
$$
Prove that $AB + CD \geq BC + AD$.
$$
\angle PAB + \angle PDC \leq 90^{\circ} \quad \text{ and } \quad \angle PBA + \angle PCD \leq 90^{\circ}.
$$
Prove that $AB + CD \geq BC + AD$.