This one is probably simple, but I don't see it yet. 


Is a locally compact, pseudocompact Hausdorff space necessarily paracompact?



Does locally compact plus pseudocompact imply paracompact?
Any pseudocompact paracompact Hausdorff space is compact. So $\omega\_1$ with the order topology is a counterexample to your question since it is locally compact, pseudocompact (any real valued continuous function is eventually constant) and not compact.


