2005 AMC 12B Problem 16

Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres?

\mathrm {(A)}\ \sqrt{2} \qquad \mathrm {(B)}\ \sqrt{3} \qquad \mathrm {(C)}\ 1+\sqrt{2}\qquad \mathrm {(D)}\ 1+\sqrt{3}\qquad \mathrm {(E)}\ 3

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.