Track / Stage 5 / 5 of 400 #605 of 1964

Problem 605

AIME late

Geometry Difficulty 5.0 Prove it China Mathematical Competition · China

Suppose four solid iron balls are placed in a cylinder with the radius of $1$ cm, such that every two of the four balls are tangent to each other, and the two balls in the lower layer are tangent to the cylinder base. Now put water into the cylinder. Then, to just submerge all the balls, we need a volume of ______ cm³ water.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Let points $O_1$ , $O_2$ , $O_3$ , $O_4$ be the centers of the four solid iron balls respectively, with $O_1$ , $O_2$ belonging to the two balls in the lower layer, and $A$ , $B$ , $C$ , $D$ be the projective points of $O_1$ , $O_2$ , $O_3$ , $O_4$ on the base of the cylinder. $ABCD$ constitute a square with the side of $\frac{\sqrt{2}}{2}$ . So the height of the water in the cylinder must be $1 + \frac{\sqrt{2}}{2}$ , so that all the balls are just immersed. Hence the volume of water we need is

$\pi \left(1 + \frac{\sqrt{2}}{2}\right) - 4 \times \frac{4}{3} \pi \left(\frac{1}{2}\right)^3 = \left(\frac{1}{3} + \frac{\sqrt{2}}{2}\right) \pi.$

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.