While wandering through nature, Professor Bug encountered a new species of beetle. Based on the number of its little legs and the shape of individual segments, he named it the Round-legged Beetle: each segment of this beetle has a spherical shape, and from each segment except the first (the head) grow two pairs of legs. Additionally, each segment has a cross-sectional area that is $21 \%$ larger than the segment behind it. Determine the exact length of the beetle, given that the head has a radius of $0,2662 \mathrm{~cm}$.

(Bednářová)
An example of such a creature is in the picture.

![](https://cdn.mathpix.com/cropped/2024_04_17_a43dcb16461b35366d89g-2.jpg?height=154&width=436&top_left_y=877&top_left_x=796)

Head: radius $r_{H}=0.2662 \mathrm{~cm}$, thus the diameter is $d_{H}=0.5324 \mathrm{~cm}$.

Let's calculate the cross-sectional area of the head:

$$
S_{H}=\pi r_{H}^{2}=0.22251 \mathrm{~cm}^{2} .
$$

First segment:

$$
\begin{aligned}
S_{1} & =\frac{S_{H}}{1.21}=0.18389096 \mathrm{~cm}^{2} \\
r_{1} & =\sqrt{\frac{S_{1}}{\pi}}=0.242 \mathrm{~cm}, \quad d_{1}=0.484 \mathrm{~cm}
\end{aligned}
$$

Second segment:

$$
\begin{aligned}
S_{2} & =\frac{S_{1}}{1.21}=0.151976 \mathrm{~cm}^{2} \\
r_{2} & =\sqrt{\frac{S_{2}}{\pi}}=0.22 \mathrm{~cm}, \quad d_{2}=0.44 \mathrm{~cm}
\end{aligned}
$$

Third segment:

$$
\begin{aligned}
S_{3} & =\frac{S_{2}}{1.21}=0.1256 \mathrm{~cm}^{2}, \\
r_{3} & =\sqrt{\frac{S_{3}}{\pi}}=0.2 \mathrm{~cm}, \quad d_{3}=0.4 \mathrm{~cm} .
\end{aligned}
$$

The total length $d$ is calculated as the sum of the lengths of the individual segments:

$$
d=d_{H}+d_{1}+d_{2}+d_{3}=0.5324+0.484+0.44+0.4=1.8564 \mathrm{~cm}
$$

The pill bug measures $1.8564 \mathrm{~cm}$.

Marta is categorizing triangles whose sides, expressed in centimeters, are whole numbers, into tall, short, and others. A tall triangle is one where the sum of the lengths of any two sides equals four times the length of the third side. A short triangle is one where the product of the lengths of any two sides is twice the length of the third side. Find

a) all tall triangles where one side measures $6 \mathrm{~cm}$,

b) all short triangles where one side measures $6 \mathrm{~cm}$,

c) all triangles where one side measures $4 \mathrm{~cm}$ and are both tall and short at the same time.

(Dillingerová)