"Sport" vs "Sports" Origin

This is really old, but if you go to Cambridge Dictionary 'sports' is the adjective (only before noun) and 'sport' is the noun. So, 'sports day' but 'do you like sport?'. That's for BrE, I've got no idea about AmE.

sports

adjective [before noun] UK
​relating to ​sport:

Sports ​equipment.

It's the ​school sports ​day on ​Monday.

sport

noun A1 [U] UK

all ​types of ​physical ​activity that ​people do to ​keep ​healthy or for ​enjoyment:

She used to do/​play a lot of sport when she was ​younger.

"sport" or "sports" is a tangled problem. "sport" has several meanings. As referring to physical activity there is some difference in usage between BrE and AmE. Actually only usage notes of dictionaries can give some help.

But I can't imagine that things are as simple as OALD says (sport BrE, sports AmE). Usage in BrE and in AmE probably vary according to region and individual usage and probably usage is changing.

sport no. 2 can be used for a person as in "He's a good sport".

sport no. 3 can mean fun or joke.

http://www.oxfordlearnersdictionaries.com/definition/english/sport_1?q=sport

There is also the problem of compound words of the type sportsman, sports car and sport shirt.