Galperin G.A.

Archimedes already knew that a sphere occupies exactly $\frac{2}{3}$ of the volume of the cylinder in which it is inscribed (the sphere touches the walls, bottom, and lid of the cylinder). In a cylindrical package, there are 5 spheres stacked on top of each other. Find the ratio of the empty space to the occupied space in this package.

![](https://cdn.mathpix.com/cropped/2024_05_06_c271881b12104a3b803eg-13.jpg?height=611&width=378&top_left_y=1581&top_left_x=840)
Divide the package into 5 cylinders, each inscribed with a sphere. In each of the cylinders, the ratio of empty space to occupied space is $\frac{1-\frac{2}{3}}{\frac{2}{3}}=\frac{1}{2}$. Therefore, the same ratio applies to the entire package, $1: 2$.

## Answer

$1: 2$. Send a comment

Points $M$ and $N$ are located on side $BC$ of triangle $ABC$, and point $K$ is on side $AC$, such that $BM: MN: NC = 1: 1: 2$ and $CK: AK = 1: 4$. It is known that the area of triangle $ABC$ is 1. Find the area of quadrilateral $AMNK$.

## Solution

Triangles $AMB$ and $ABC$ share a common height from vertex $A$, so their areas are proportional to their bases, hence,

$$
S_{\triangle AMB} = \frac{BM}{BC} \cdot S_{\triangle ABC} = \frac{1}{4} \cdot 1 = \frac{1}{4}
$$

Similarly,

$$
S_{\triangle AVC} = \frac{1}{2}, S_{\triangle QNX} = \frac{CK}{AC} \cdot S_{\triangle ANW} = \frac{1}{5} \cdot \frac{1}{2} = \frac{1}{10}
$$

Therefore,

$$
S_{AMQN} = S_{\triangle ABC} - S_{\triangle AMC} - S_{\triangle QNK} = 1 - \frac{1}{4} - \frac{1}{10} = 1 - \frac{7}{20} = \frac{13}{20}
$$

## Answer

$\frac{13}{20}$