I'm confused as to whose "inability" you are referring to. Yours or those who are coming to you with the problems.

Ok, so you quickly solved the problem. Just ask "Is there anything else I can help you with?", if there is then fix that too. Otherwise just move on.

I get the impression from your description that it is you "giving your 2 cents worth" that triggers the competitive behaviors. Would it be possible for you to just say "Glad I could help." and leave it at that? Perhaps some of the people you are helping with their problems are less competent than they claim to be and that is why they try to befuddle you by trying to appear smart.

Truly smart people don't act in a condescending manner. Only those who have something to hide put on "the act". That's my 2 cents worth :-)