smc

Note that the first equation represents a circle and the second equation represents a line. If a line is tangent to a circle, then it only hits at one point, so there will only be one solution to the system of equations. In the second equation, $y=-x+\sqrt{2m}$. Substitution results in $$x^2+x^2-2x\sqrt{2m}+2m=m$$ $$2x^2-2x\sqrt{2m}+m=0$$ In order for the system to have one solution, the discriminant must equal $0$. $$4(2m)-4\cdot 2\cdot m=0$$ $$0=0$$ Thus, $m$ can be a non-negative real number, so the answer is $\boxed{\text{m may be any non-negative real number}}$.