I(0) is proportional to Σ(iΔSLDi Vi)2,
where ΔSLDi = SLDi - SLDsolvent is the contrast (or excess scattering length density) of subunit i, and Vi is the volume of subunit i (the subunits of a core-shell particles are the the core and the shell).
In this problem, ΔSLDcore = -ΔSLDshell, so the volumes of the core and shell must be equal to get I(0)=0. Volume of the shell: 4π(Router3-Rcore3)/3
Set this equal to the volume of the core, 4πRcore3/3, and find a relation between the radii, i.e. determine the ratio = Router/Rcore=?