A number theory problem by dewita sonya

Notice that it is not necessary to find all the values of $k$ .

If there exist $k$ such that $x^2+y^2+z^2=kxyz$ with $(x,y,z)$ as its solution, it is easy to see that $-k$ is also satisfies with $(-x,y,z)$ is its solution. So if $k$ satisfies, so does $-k$ . Thus it is clear that the sum of all $k$ must be $\boxed0$

As it stands, any integer $k$ permits the integer solution $(0,0,0)$ . In which case, the sum of all relevant integers $k$ does not exist.

Let us suppose that the author intends that the integers $x,y,z$ are to be nonzero . Then the only possible values of $k$ which permit nonzero integer solutions are $\pm1$ and $\pm3$ , making the sum $\boxed{0}$ . The solutions $(3,3,3)$ , $(3,3,-3)$ , $(1,1,1)$ and $1,1,-1)$ show that each of these is possible. A lengthy proof by infinite descent shows the impossibility of all other values of $k$ .