5. The base of the pyramid $S A B C D$ is a rectangle $A B C D$, with the height being the edge $S A=25$. Point $P$ lies on the median $D M$ of the face $S C D$, point $Q$ lies on the diagonal $B D$, and the lines $A P$ and $S Q$ intersect. Find the length of $P Q$, if $B Q: Q D=3: 2$.
# Answer: 10.

## Solution:

Since lines $A P$ and $S Q$ intersect, points $A, P, S, Q$ lie in the same plane. Let $R$ be the intersection point of $S P$ and $A Q$. Then

$$
\frac{R Q}{A Q}=\frac{D Q}{B Q}=\frac{2}{3} \Rightarrow \frac{R Q}{R A}=\frac{2}{5}
$$

We will prove that $\frac{R M}{R S}=\frac{2}{5}$. Note that $\frac{D Q}{B A}=\frac{2}{3} \Rightarrow \frac{D Q}{C D}=\frac{2}{3} \Rightarrow \frac{C D}{D Q}=\frac{1}{2}$.

Let $H$ be a point on edge $S C$ such that $R H \| D M$. Then

$$
\frac{C H}{H M}=\frac{C R}{R D}=\frac{1}{2} \Rightarrow \frac{M S}{M H}=\frac{3}{2} \Rightarrow \frac{M H}{S H}=\frac{2}{5} \Rightarrow \frac{R M}{R S}=\frac{2}{5}
$$

From the similarity of triangles $R P Q$ and $R S A$, we get that

$$
\frac{P Q}{S A}=\frac{2}{5} \Rightarrow \frac{P Q}{25}=\frac{2}{5} \Rightarrow P Q=10
$$

## Evaluation criteria are provided in the table:

| Points | Evaluation Criteria |
| :---: | :--- |
| $\mathbf{7}$ | Complete and justified solution. |
| $\mathbf{6}$ | Justified solution with minor flaws. |
| $\mathbf{5 - 6}$ | The solution contains minor errors, gaps in justification, but is generally correct and can become fully correct with minor corrections or additions. |
| $\mathbf{4}$ | The problem is more solved than not, for example, one of the two (more complex) significant cases is correctly considered. |
| $\mathbf{2 - 3}$ | The problem is not solved, but formulas, diagrams, considerations, or auxiliary statements relevant to the solution of the problem are provided. |
| $\mathbf{1}$ | The problem is not solved, but an attempt at a solution has been made, for example, individual (partial) cases have been considered in the absence of a solution or in the case of an incorrect solution. |
| $\mathbf{0}$ | The solution is absent or does not meet any of the criteria listed above. |