Take the last question as an example. Let $a=b=c=d$ , then $a^2+b^2+c^2+d^2=4\,$
is of course fulfilled, because we know that $a,b,c,d\ge 0$ , if and only if $a=b=c=d=1$ .
Nobody would be surprised (?) if $\;f(a,b,c,d)=a/(b+3)+b/(c+3)+c/(d+3)+d/(a+3)$ assumes
its one and only maximum $f(a,b,c,d)=1/4+1/4+1/4+1/4=1$ precisely for these equal values
of $(a,b,c,d)$ .
Very much the same phenomenon is observed with the other of the above questions.
The optimizing parameters turn out to be all equal, which often may be suspected beforehand: "due to symmetry".
It's frustrating that there seems not to exist a theorem somewhere that guarantees
a maximum or a minimum of a function when all variables in a problem with such high symmetry are just equal.
I mean: sort of formalization of "by symmetry" that helps us to find such solutions immediately.
Now it is supposed that group theory is the discipline that should teach us a lot about symmetries.
So the question is: why doesn't group theory routinely come in here?
Update. Unfortunately - and I think it's against the spirit of MSE as well - it often happens that the better answers are actually given as a comment . Indeed, a key reference for this sort of problems is:
It's only with help of this reference that I could (try to) answer another such question .Update. It's a continuing story: