The thing is he's 3, he's just not gonna get that he can't win all the time, 5yo is lovely to let him win now and then, you just have to show him, heap praise on 5yo when he wins and ignore 3yo tantrum and then let 3yo win and heap praise on 3 yo, he will soon get the message. And at this age letting him win, or thinking he's has won is no big deal, it keeps the peace and your sanity, it won't stay with him for life!
Have 2 DS, 4 and 9, they used to be like this, but now DS 4.5 totally gets that he can't win all the time and DS9 backs off a lot and lets him think he's won eg: footy and connect 4 that sort of thing, to keep the peace! When ds2 was the same as yours he would be livid with rage and crying if his older brother scored a goal, the whole neighbourhood heard about it!
I too have banned races up and down stairs, first out the house, first in the car - all that banned, too exhausting, if they start I say "do I look like I have any medals to hand out?"
I have a 3 year old like this- I think it's a common trait in 3 year olds tbh. His big sister (6) can't be bothered with his tantrums and just let's him win most things, which I'm sure is not teaching him the right things . However I do think this type of thing is just a learning curve and he'll grow out of it, although he may always be competitive.
What you need to do is get to the bottom of why your 5 year old needs to prove to everyone he is the best/fastest, etc. 'They make everything into a competition' -- does this mean the 3 yo initiates this or the 5 yo?
Your 3 year old is perfectly normal in his response to losing.
Your 5 year old needs more play with children his own age.