= Solution
Let $D_c$ be the parent <comet>'s diameter, with the same <mass density> as the fragments, so $m=\pi\rho D_c^3/6$. The required <geometric cross-section> is $\sigma_{\rm tot}=4\pi a^2\sqrt{1-e^2}\,f$, using the corrected <phase-mixed orbit> interpretation.
Substitute this and the spherical <mass> into the endpoint-dominated fragment formula. The <parent-body size from a fragment cross-section> becomes
$$
\boxed{D_c=\left[\frac{16(\alpha-3)}{4-\alpha}\,
f a^2\sqrt{1-e^2}\,D_{\min}^{\alpha-3}D_{\max}^{4-\alpha}\right]^{1/3}}.
$$
The common <mass density> cancels. The <comet>'s radius is $D_c/2$. This expression treats both fragment cutoffs as specified independently and has the same endpoint-approximation restrictions as part (iii).
If a model instead ties the largest fragment to the parent through $D_{\max}=\eta D_c$, with fixed $0<\eta\leq1$, one must solve that dependence too:
$$
\boxed{D_c=\left[\frac{16(\alpha-3)}{4-\alpha}\,
f a^2\sqrt{1-e^2}\,D_{\min}^{\alpha-3}\eta^{4-\alpha}\right]^{1/(\alpha-1)}}.
$$
That extra assumption is not supplied by the PDF; setting $D_{\max}=D_c$ without saying so would change the requested size relation.
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