In the pyramid $A B C D$, points $M, F$, and $K$ are the midpoints of edges $B C, A D$, and $C D$ respectively. On the lines $A M$ and $C F$, points $P$ and $Q$ are taken such that

$P Q \| B K$. Find the ratio $P Q: B K$.
Let $E$ be the midpoint of $C K$. Then $M E \| B K$. Let the lines $A E$ and $C F$ intersect at point $Q^{\prime}$. In the plane $A E M$, draw a line through point $Q^{\prime}$ parallel to $E M$. Let this line intersect $A M$ at point $P^{\prime}$. Then points $Q^{\prime}$ and $P^{\prime}$ satisfy the condition of the problem. Indeed, they lie on the lines $C F$ and $A M$, and $P^{\prime} Q^{\prime}\|M E\| B K$. Therefore, points $P^{\prime}$ and $Q^{\prime}$ coincide with points $P$ and $Q$ respectively.

Consider the plane $A D C$. Through point $A$, draw a line parallel to $D C$, and extend $C F$ to intersect this line at point $T$. Then triangle $A T F$ is equal to triangle $D C F$, and triangle $A T Q$ is similar to triangle $E C Q$ with a ratio of ${ }^{A Q} /{ }_{E Q}={ }^{A T} / E C=C D / E C=4$.

Therefore, $P Q=M{ }^{A Q}{ }^{A E}=M E \cdot 1 / 5=2 / 5 B K$.

## Answer

$2: 5$.