Track / Stage 5 / 224 of 400 #824 of 1964

Problem 824

AIME late

Geometry Difficulty 5.6 Find the answer

## Task $29 / 66$

In a cube with edge length $a$ , nine identical largest possible spheres are to be arranged such that one of them has its center at the intersection of the body diagonals, while the other eight are placed in the corners of the cube.

How large is the diameter $d$ of the spheres, expressed in terms of the cube side $a$ ?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

The centers of the central sphere and the two spheres filling opposite corners must lie on a body diagonal of the cube. In the figure, a section through the cube containing two body diagonals and a top view of the cube are shown.

The length of the body diagonal is $a \sqrt{3}$ . From this figure, the validity of the following equations can be read:

$r: x=a: a \sqrt{3} \quad(1) \quad 4 r+2 x=a \sqrt{3}$

Solving this system of equations for $r$ by substituting $x=r \sqrt{3}$ (from Equation I), we obtain

$r=\frac{a}{2} \cdot \frac{\sqrt{3}}{2+\sqrt{3}}=\frac{a}{2}(2 \sqrt{3}-3) \approx 0.232 a$

From this, it immediately follows that $d=a(2 \sqrt{3}-3) \approx 0.464 a$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.