Question 1
(BUL 6) Let a tetrahedron $ABCD$ be inscribed in a sphere $S$. Find the locus of points $P$ inside the sphere $S$ for which the equality
$$\frac{AP}{PA_1}+\frac{BP}{PB_1}+\frac{CP}{PC_1}+\frac{DP}{PD_1}=4$$
holds, where $A_1$, $B_1$, $C_1$, and $D_1$ are the intersection points of $S$ with the lines $AP$, $BP$, $CP$, and $DP$, respectively.
$$\frac{AP}{PA_1}+\frac{BP}{PB_1}+\frac{CP}{PC_1}+\frac{DP}{PD_1}=4$$
holds, where $A_1$, $B_1$, $C_1$, and $D_1$ are the intersection points of $S$ with the lines $AP$, $BP$, $CP$, and $DP$, respectively.