How many positive integer solutions to $a^x+b^x+c^x=abc$? How many positive integer solutions are there to $a^{x}+b^{x}+c^{x}=abc$? (e.g the solution $x=1$, $a=1$, $b=2$, $c=3$). Are there any solutions with $\gcd(a,b,c)=1$? Any solutions to $a^{x}+b^{x}+c^{x}+d^{x}=abcd$ etc. ?

Unfortunately, I don't know how to make comments here, would be better to move it there. Anyways, consider for example, $a^x+b^x+c^x=abc$ with $a\le b\le c.$ Clearly, $c^x<LHS\le c^3,$ therefor $x<3.$ Case $x=1$ is fairly simple as you can bound $ab\le 3$ (following the same lines). So let's consider the case $x=2.$ Our equation is quadratic with respect to $a,$ so it has an integer solution iff $D=(c^2-4)b^2-4c^2=x^2.$ This equation has infinitely many solutions even for some fixed values of $c.$ Take. for example, $c=3.$ Then,
$x^2-5b^2=-36,$ has infinitely many solutions, since it is Pell type equation with one solution being $x=12,$ $b=6.$ 
By considering equation modulo $3,$ we get $(a,b,c)\ne 1.$  
