137. Two identical triangular pyramids of volume $V$ are arranged in space symmetrically relative to point $O$. Find the volume of their common part if point $O$ lies on the segment connecting the vertex of the pyramid with the centroid of the base and divides this segment in the ratio 1) $1: 1$, 2) $3: 1$, 3) $2: 1$, 4) $4: 1$, counting from the vertex.
137. In figures $22, a-\varepsilon$ the common parts of these two pyramids are depicted for all four cases.
1) The common part is a parallelepiped (Fig. $22, a$ ). To determine the volume, one needs to subtract the volumes of three pyramids similar to the original pyramid with a similarity coefficient of $2 / 3$, and add the volumes of three pyramids, also similar to the original with a similarity coefficient of $1 / 3$. Thus, the volume is:

$$
V\left[1-3\left(\frac{2}{3}\right)^{3}+3\left(\frac{1}{3}\right)^{3}\right]=\frac{2}{3} V_{0}
$$

2) The common part is an octahedron (Fig. 22, b), the volume of which is

$$
V\left[1-4\left(\frac{1}{2}\right)^{3}\right]=\frac{V}{2}
$$

3) The common part is shown in Fig. 22, c. To determine its volume, one needs to subtract the volume of a pyramid similar to the original with a similarity coefficient of $1 / 3$ (this pyramid is at the top in the figure), then subtract the volumes of three pyramids, also similar to the original with a similarity coefficient of $5 / 9$, and add the volumes of three pyramids, with a similarity coefficient of $1 / 9$. Thus, the volume of the common part is

$$
V\left[1-\left(\frac{1}{3}\right)^{3}-3\left(\frac{5}{9}\right)^{3}+3\left(\frac{1}{9}\right)^{3}\right]=\frac{110}{243} v
$$

4) The common part is shown in Fig. 22, d. Its volume is

$$
V\left[1-\left(\frac{3}{5}\right)^{3}-3\left(\frac{7}{15}\right)^{3}+3\left(\frac{1}{15}\right)^{3}\right]=\frac{12}{25} v
$$