B5. We place a red ribbon at an angle of 45 degrees over a cylindrical white stick (see figure). The ribbon is then tightly wrapped around the stick (without folds), creating a red spiral around the stick. Between the red spiral runs a white spiral; this is the part of the stick not covered by the ribbon. The radius of the stick is $2 \mathrm{~cm}$.

It turns out that the white and red spirals are of equal width. What is the width of the ribbon?

![](https://cdn.mathpix.com/cropped/2024_04_17_cc4b7043530fb5206fd0g-1.jpg?height=523&width=511&top_left_y=2157&top_left_x=1318)

## C-Problems

For the C-Problems, not only the answer is important; your reasoning and the method of solving must also be clearly written. Solve each C-Problem on a separate sheet of paper. Each correctly worked-out C-Problem is worth 10 points. Partial solutions can also earn points. Therefore, write everything clearly and submit your scratch paper (for each problem!) as well.
B5.

$\pi \sqrt{2} \mathrm{~cm}$

Imagine the stick as a paper cylinder. Cut it open along its length and unroll it to form a rectangular strip. Points $A$ and $D$ belong to the same point on the cylinder, as do points $B$ and $C$. The width of the strip is equal to the circumference of the cylinder, so $|A D|=|B C|=2 \pi \cdot 2 \mathrm{~cm}=4 \pi \mathrm{cm}$.

Since the red ribbon makes a $45^{\circ}$ angle with the cut line, $A B C D$ is a square. The length of the diagonal $B D$ is equal to $\sqrt{2} \cdot 4 \pi \mathrm{cm}$ and also equal to four times the width of the red ribbon, because the white and red strips are of equal width. The red ribbon is thus $\pi \sqrt{2} \mathrm{~cm}$ wide.

![](https://cdn.mathpix.com/cropped/2024_04_17_913d083fb7ef425a6a93g-2.jpg?height=520&width=417&top_left_y=194&top_left_x=1413)

## C-problems