Two unit-radius spheres intersect in such a way that the resulting 3 parts have equal volume. How far apart are the centers of the spheres?
The intersection of the two spheres in the resulting 3 parts consists of two congruent spherical caps. If the height of the spherical caps is $m$, and the radius of their base circle is $r$, then their combined volume is $2 \cdot \frac{\pi}{6} m\left(3 r^{2}+m^{2}\right)$.

If we now examine one of the unit-radius spheres, the volume of the part that intersects with the other sphere and the part that does not intersect are equal. The sum of these two volumes is exactly the volume of the sphere, so each one is half of it. The volume of half of a unit-radius sphere is $\frac{2 \pi}{3}$, so we get the equation

$$
\frac{2 \pi}{3}=\frac{\pi}{3} \cdot m\left(3 r^{2}+m^{2}\right)
$$

Let the distance between the centers of the two spheres be $2a$. Based on the diagram, which shows the intersection of the spheres with a plane that contains both centers, it is clear that

$$
m=1-a \quad \text { and } \quad r=\sqrt{1-a^{2}}
$$

Substituting these into equation (1) and simplifying, we get:

$$
a^{3}-3 a+1=0
$$

We need to find the roots of this equation in the interval $(0 ; 1)$. If $0<a<1$, then there is exactly one acute angle $\gamma$ such that $\cos \gamma=\frac{a}{2}$. Substituting $a=2 \cos \gamma$ into equation (2), we get:

$$
8 \cos ^{3} \gamma-6 \cos \gamma+1=0
$$

It is known that $\cos 3 x=4 \cos ^{3} x-3 \cos x$, so equation (3) can be written as:

$$
2 \cos 3 \gamma+1=0
$$

Thus, $\cos 3 \gamma=-\frac{1}{2}$. From this equation, $\gamma=\frac{4 \pi}{9}$ (considering that $\frac{\pi}{3}<\gamma<\frac{\pi}{2}$).

Therefore, the desired distance, $O_{1} O_{2}=2 a=4 \cos \frac{4 \pi}{9} \approx 0.695$.

Note. Since solving cubic equations is not part of the high school curriculum, we accepted any approximate or graphical solution to equation (2) as a complete solution.

![](https://cdn.mathpix.com/cropped/2024_05_02_54e0a0df65360b121847g-1.jpg?height=429&width=629&top_left_y=1542&top_left_x=737)