3.28. Point $O$ lies on the segment connecting the vertex of a triangular pyramid with volume $V$ to the point of intersection of the medians of the base. Find the volume of the common part of the given pyramid and the pyramid symmetric to it with respect to point $O$, if point $O$ divides the specified segment in the ratio: a) $1: 1$; b) $3: 1$; c) $2: 1$; d) $4: 1$ (counting from the vertex).
3.28. In figures $30, a-d$, the common parts of the pyramids in all four cases are depicted.

a) The common part is a parallelepiped (Fig. 30, a). It is obtained from the original pyramid by cutting off three pyramids similar to it with a coefficient of $2 / 3$; at the same time, three pyramids similar to the original with a coefficient of $1 / 3$ are common for pairs of the cut-off pyramids. Therefore, the volume of the parallelepiped is $V(1 - 3(2 / 3)^{3} + 3(1 / 3)^{3}) = 2V / 9$.

b) The common part is an "octahedron" (Fig. 30, b). The volume of this polyhedron is $V(1 - 4(1 / 2)^{3}) = V / 2$.

c) The common part is depicted in Fig. 30, c. To calculate its volume, one needs to subtract the volume of a pyramid similar to the original with a coefficient of $1 / 3$ (this pyramid is at the top in the figure), then subtract the volume of three pyramids similar to the original with a coefficient of $5 / 9$, and add the volume of three pyramids similar to the original with a coefficient of $1 / 9$. Therefore, the volume of the common part is $V(1 - (1 / 3)^{3} - 3(5 / 9)^{3} + 3(1 / 9)^{3}) = 110V / 243$.

d) The common part is depicted in Fig. 30, d. Its volume is $V(1 - (3 / 5)^{3} - 3(7 / 15)^{3} + 3(1 / 15)^{3}) = 12V / 25$.