You might be able to use game. Relevant definitions, from Merriam-Webster:
1 a (1) : activity engaged in for diversion or amusement : play
3 a (1) : a physical or mental competition conducted according to rules with the participants in direct opposition to each other
Although not all games would fit your requirement of mental exertion and skill, I think any activities that do fit your definition could be called games—and most games, even of the very physical type, also require at least some strategy/mental exertion, at least to remember the rules1. This term also seems a bit more specific to me than competition (which isn't always for entertainment).
1 There is a fair amount of overlap between game and sport, with the key difference largely hinging on
How organized the competition is: games require organization and rules, but not all definitions of sport have this requirement; and
Physical activity: most definitions of sport, including the one you use, require this—although there is a fair amount of dispute about the level of prowess/exertion required—while definitions of game don't generally require physical exertion, although they do allow for it.
There's a good discussion of the distinction in this answer to the question "Difference between 'game' and 'sport'".