1. On the cube $A B C D A_{1} B_{1} C_{1} D_{1}$, points $P$ and $Q$ are chosen, where $P$ is the midpoint of edge $B C$, and $Q$ is the intersection of the diagonals in the square $C C_{1} D D_{1}$. The plane $A P Q$ divides the cube into two parts. Find the ratio of the volumes of the resulting parts of the cube?
Solution. Let's denote the side of the cube by $a$. Let $T$ be the foot of the perpendicular dropped from point $Q$ to edge $CD$, let the intersection points of plane $APQ$ with edges $DD_1$ and $CC_1$ be $N$ and $M$ respectively, and the intersection point with line $CD$ be point $R$. From the condition of the problem, the following equalities hold: $\overline{BP}=\overline{PC}=\frac{a}{2}$ and $\overline{QT}=\frac{a}{2}$.

![](https://cdn.mathpix.com/cropped/2024_06_05_d0c212a3407f50323a50g-05.jpg?height=353&width=603&top_left_y=1799&top_left_x=588)

Furthermore, $\triangle ARD \sim \triangle PRC$ from which we get $\overline{CR}: \overline{DR}=\overline{CP}: \overline{DA}=\frac{1}{2}$. Then $\overline{CD}=\overline{CR}=a$. Similarly, $\triangle CMR \sim \triangle DNR$ from which $\overline{CM}: \overline{DN}=\overline{CR}: \overline{DR}=\frac{1}{2}$, so we have $\overline{CM}=\frac{1}{2} \overline{DN}$. $QT$ is the midline in trapezoid $DCMN$ and the following holds:

$$
\overline{DN}+\overline{CM}=2 \overline{QT}=a \text { or } \overline{DN}+\frac{\overline{DN}}{2}=a
$$

from which we get $\overline{DN}=\frac{2}{3} a$ and $\overline{CM}=\frac{1}{3} a$.

The plane $APQ$ divides the cube into two parts, one of which is the truncated pyramid ADNPCM, with height the side of the cube and bases right-angled triangles $PCM$ and AND. For the volume of the truncated pyramid, using the two pyramids ANDR and PCMR, we get:

$$
V_{1}=V_{ADNR}-V_{PCMR}=\frac{\overline{AD} \cdot \overline{DN}}{2} \cdot \frac{\overline{DR}}{2}-\frac{\overline{CP} \cdot \overline{CM}}{2} \cdot \frac{\overline{CR}}{2}=\frac{7 a^{3}}{36}
$$

For the volume of the remaining part, we have $V_{2}=a^{3}-V_{1}=\frac{29 a^{3}}{36}$. The required ratio of the volumes is $\frac{V_{1}}{V_{2}}=\frac{7}{29}$.