= Solution
The mantle density decreases outward when $\rho_m\geq0$. Requiring a nonnegative surface density gives the immediate physical bound
$$
\boxed{0\leq\rho_m\leq\frac{\rho_cR_c}{R_p-R_c}}.
$$
The measured planetary mass also fixes
$$
M_p=4\pi\left[
\frac{(\rho_c+\rho_m)R_p^3}{3}
-\frac{\rho_mR_p^4}{4R_c}
-\frac{\rho_mR_c^3}{12}
\right].
$$
Equivalently,
$$
\rho_m=
\frac{\rho_cR_p^3/3-M_p/(4\pi)}
{R_p^4/(4R_c)-R_p^3/3+R_c^3/12},
$$
and a viable model requires this value to obey the boxed bound.
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