For which positive integers $n>2$ has the equation $$a^n+b^n=2c^n$$ integer solutions with $0\le a<b$ ?
A small test with PARI/GP using this program :
? for(n=3,10,for(a=0,2000,for(b=a+1,2000,c=round(((a^n+b^n)/2)^(1/n));if(a^n+b^n ==2*c^n,print([a,b,c]))))) ?
shows that for $3\le n\le 10$ and $0\le a<b\le 2000$, no solutions exist. So, they seem to be pretty rare at first glance.