Magnitude of Sum / Sum of Magnitudes

Note that $Q(A,B) \; = \; \frac{\sqrt{A^2 + AB\sqrt{2} + B^2}}{A+B}$ is homogeneous in $A.B$ , so we can restrict our attention to $Q_1(t) \; = \; Q(A,tA) \; = \; \frac{\sqrt{1 + t\sqrt{2} + t^2}}{1+t} \; = \; Q_2\big(\tfrac{1}{t+1}\big)^2$ where $Q_2(u) \; = \; 1 - (2-\sqrt{2})u + (2-\sqrt{2})u^2$ Simple calculus tells us that $Q_2$ is minimized at $u=\tfrac12$ , so that $Q_1$ is minimized at $t=1$ , so that $Q$ is minimized when $A=B$ , and so that $Q_\mathrm{min} \; = \; \frac{\sqrt{2+\sqrt{2}}}{2} \; = \; 0.9238795325...$ making the answer $\boxed{923}$ .