6. On the faces $BCD, ACD, ABD$, and $ABC$ of the tetrahedron $ABCD$, points $A_{1}, B_{1}, C_{1}$, and $D_{1}$ are marked, respectively. It is known that the lines $AA_{1}, BB_{1}, CC_{1}$, and $DD_{1}$ intersect at point $P$, and $\frac{AP}{A_{1}P}=\frac{BP}{B_{1}P}=\frac{CP}{C_{1}P}=\frac{DP}{D_{1}P}=r$. Find all possible values of $r$.
Solution. Let V be the volume* of tetrahedron ABCD. We introduce the consideration of the partition of the original tetrahedron ABCD into tetrahedra PBCD, PACD, PABD, and PABC. Then for the volumes of the specified tetrahedra, the following is true:

$$
V=V_{\mathrm{PBCD}}+V_{\mathrm{PACD}}+V_{\mathrm{PABD}}+V_{\mathrm{PABC}}
$$

Note that if $\mathrm{H}$ is the height of tetrahedron $\mathrm{ABCD}$, dropped from vertex A to the base BCD, and $\mathrm{h}$ is the height of tetrahedron PBCD, dropped from vertex P to the base BCD, then $\frac{H}{h}=\frac{A A_{1}}{A_{1} P}$. Due to this observation and the condition $r=\frac{A P}{A_{1} P}=\frac{B P}{B_{1} P}=\frac{C P}{C_{1} P}=\frac{D P}{D_{1} P}$, we have

$$
\frac{V}{V_{\mathrm{PBCD}}}=\frac{\mathrm{A} A_{1}}{A_{1} P}=\frac{A P+A_{1} P}{A_{1} P}=1+\frac{A P}{A_{1} P}=r+1
$$

Similar relations are valid for the other tetrahedra, so

$$
\frac{v}{V_{\mathrm{PBCD}}}=\frac{v}{V_{\mathrm{PACD}}}=\frac{v}{V_{\mathrm{PABD}}}=\frac{v}{V_{\mathrm{PABC}}}=r+1 \text { or }
$$

$V=(r+1) V_{\mathrm{PBCD}}=(r+1) V_{\mathrm{PACD}}=(r+1) V_{\mathrm{PABD}}=(r+1) V_{\mathrm{PABC}}$.

Then

$$
r+1=\frac{(r+1) V_{\mathrm{PBCD}}+(r+1) V_{\mathrm{PACD}}+(r+1) V_{\mathrm{PABD}}+(r+1) V_{\mathrm{PABC}}}{V_{\mathrm{PBCD}}+V_{\mathrm{PACD}}+V_{\mathrm{PABD}}+V_{\mathrm{PABC}}}
$$

or

$$
r+1=\frac{4 V}{V_{\mathrm{PBCD}}+V_{\mathrm{PACD}}+V_{\mathrm{PABD}}+V_{\mathrm{PABC}}}=\frac{4 V}{V}=4
$$

Thus, $r=3$.

*Note that for solving this problem, it is not necessary to know the formula for the volume of a tetrahedron. It is sufficient to introduce a conditional quantity $V=S h$, and use its additivity (a known fact from mathematics and physics courses).

## Recommendations for evaluating solutions.

A correct solution for a specific case - 2 points.

The partition of the original tetrahedron ABCD into tetrahedra PBCD, PACD, PABD, and PABC is presented, and relations of the form $\frac{V}{V_{\text {PBCD }}}=\frac{\mathrm{AA}_{1}}{A_{1} P}=\frac{A P+A_{1} P}{A_{1} P}=1+\frac{A P}{A_{1} P}=r+1$ are obtained - no less than 3 points.

Note that the problem can be solved using the vector method.