15.7. (NBR, 68). Inside the tetrahedron $A B C D$ is a point $O$ such that the lines $A O, B O, C O, D O$ intersect the faces $B C D, A C D, A B D, A B C$ of the tetrahedron at points $A_{1}, B_{1}$, $C_{1}, D_{1}$ respectively, and the ratios

$$
\frac{A O}{A_{1} O}, \frac{B O}{B_{1} O}, \frac{C O}{C_{1} O}, \frac{D O}{D_{1} O}
$$

are all equal to the same number. Find all possible values that this number can take.
15.7. Let \( V \) be the volume of the tetrahedron \( ABCD \), and \( k \) be the desired number. Then we have

\[
\frac{V}{V_{OBCD}} = \frac{AA_1}{OA_1} = \frac{AO}{A_1O} + \frac{OA_1}{OA_1} = k + 1
\]

\[
\frac{V}{V_{OACD}} = \frac{V}{V_{OABD}} = \frac{V}{V_{OABC}} = k + 1,
\]

from which

\[
k + 1 = \frac{4V}{V_{OBCD} + V_{OACD} + V_{OABD} + V_{OABC}} = \frac{4V}{V} = 4, \text{ i.e., } k = 3
\]