Jack of all trades & Remarkable athlete

[Mechaviking]

Do they stack with each other?

Does using half a proficiency bonus on x count as using a proficiency bonus on x?

[DireSickFish]

Do they stack with each other?

Does using half a proficiency bonus on x count as using a proficiency bonus on x?

They don't stack because having half prof bonus is having a proficiency bonus to a skill. So you'd just use whichever one is higher at the time, as Remarkable athlete rounds up and Jack of all trades rounds down.

[Mechaviking]

I was thinking Initiative mostly :D

Doing a mental exercise of the quickest man alive :D

Also Remarkable athlete should have just been add half your prof. bonus to all ability or skill checks using str, dex or con(that´s basically half an expertise and would at least make this ability slightly more interesting).