What is Hume’s challenge, as laid out by Kant in the Preface to the Prolegomena

What is Hume’s challenge, as laid out by Kant in the Preface to the Prolegomena and the Introduction to the Critique of Pure Reason? Explain why Kant thinks that Hume’s “skeptical solution” (from section V of Hume’s Enquiry) is inadequate. What is the difference between a priori knowledge and a posteriori knowledge? Explain Kant’s distinction between analytic judgments and synthetic judgments. Why does Kant think that synthetic, a priori judgments are possible? Why does he think this defeats Hume’s skeptical problem?
Explain the distinction between appearances (phenomena) and things-in-themselves (noumena). In what way does Kant believe that space and time are a priori structures of the mind, rather than properties of things-in-themselves? How does his example of the hand in the mirror (Prolegomena, Part I) help to prove this point? How does Kant’s position differ from those that du Châtelet outlines in Chapter Five of Foundations of Physics?