I'm looking for sites, or books, for bright 5th graders or above, with math problems and respective solutions.
Check out the three books by James Fixx (Solve It, Games for the Super-intelligent, and More Games for the Super-intelligent). I don't think they are super difficult, but of course please examine and make sure they are not too tough for your advanced 5th graders. (If so, look for some other puzzle books a little easier.)
Aimed at students or teachers?
For the latter, Cut-the-knot is quite good, with many resources at this level. For the former, Khan Academy is free, and Beast Academy from the Art of Problem Solving is good (only workbooks for now, but online classes are supposedly coming soon).
In terms of puzzle books, I can add Smullyan's books to the other recommednations. Those are all logic puzzles about truth-tellers and liars (which he calls knights and knaves) and are very entertaining to reason through together with children. The Lady or the Tiger is a good one to start with.
A slightly different approach is to watch short videos like those in Numberphile together with the child, to stop often and discuss the material. This requires more involvement from the adult, but is a much more open-ended learning experience.
The basic idea comes from research about how children interact with educational tv programs; and it (i.e., my recollection of the take-away conclusion) seems to suggests that it's not very effective in general unless accompanied with parental (or equivalent) interaction around the material. I think Numberphile videos are particularly suitable, as they are short and often contain very simple proof ideas which lend themselves to further examples, discussion, and generalisations.