If you say that someone has a keen mind, you mean that they are very clever and aware of what is happening around them.

If you have a keen eye or ear, you are able to notice things that are difficult to detect.

A keen interest or emotion is one that is very intense.

If you are keen on doing something, you very much want to do it.

If you are keen on something, you like it a lot and are very enthusiastic about it.

You use keen to indicate that someone has a lot of enthusiasm for a particular activity and spends a lot of time doing it.

A keen fight or competition is one in which the competitors are all trying very hard to win, and it is not easy to predict who will win.