\section*{Problem 5 - 121235}

Determine whether there exist regular n-gons for which the difference between the lengths of the longest and the shortest diagonals is equal to the side length of the n-gon.

If such n-gons exist, provide all natural numbers \(n(n \geq 4)\) for which this is true.
}

O.B.d.A. let the side length of the n-gon be 1. The shortest diagonal spanning two sides is \(d_{2}\). Then,

\[
1n \quad ; \quad d_{m}>\frac{n}{\pi}
\]

If \(n \geq 10\), then \(d_{m}>3\) and thus (1) is not satisfied. By sketching an 8-gon, the diagonals can be calculated quite easily using the Pythagorean theorem. For \(n=8\):

\[
\begin{gathered}
d_{2}=\sqrt{\left(1+\frac{1}{2} \sqrt{2}\right)^{2}+\left(\frac{1}{2} \sqrt{2}\right)^{2}}=\sqrt{2+\sqrt{2}} \approx \sqrt{3.4} \approx 1.8 \\
d_{m}=\sqrt{(1+\sqrt{2})^{2}+1}=\sqrt{4+2 \sqrt{2}} \approx \sqrt{6.8} \approx 2.6
\end{gathered}
\]

For the 8-gon, the difference between the maximum diagonal \(d_{m}\) and the shortest diagonal \(d_{2}\) is too small, so only odd \(n \geq 7\) are possible.

The following is a drawing of an \(n\)-gon with odd \(n\):

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1939.jpg?height=543&width=556&top_left_y=705&top_left_x=750)

It is

\[
\alpha=\frac{\pi}{n}
\]

Thus,

\[
d_{2}=2 \cos \alpha
\]

and

\[
d_{m}=\frac{\frac{1}{2}}{\sin \frac{1}{2} \alpha}=\frac{1}{2 \sin \frac{1}{2} \alpha}
\]

It must hold that:

\[
\frac{1}{2 \sin \frac{1}{2} \alpha}=2 \cos \alpha+1
\]

With \(\cos \alpha=1-2 \sin ^{2} \frac{1}{2} \alpha\), we get

\[
\begin{aligned}
& \frac{1}{2 \sin \frac{1}{2} \alpha}=3-4 \sin ^{2} \frac{1}{2} \alpha \\
& \frac{1}{2}=3 \sin \frac{1}{2} \alpha-4 \sin ^{3} \frac{1}{2} \alpha
\end{aligned}
\]

At this point, the equation for the sine of the triple angle can be used:

\[
\sin 3 x=3 \sin x-4 \sin ^{3} x
\]

Then,

\[
\begin{gathered}
\sin \frac{3}{2} \alpha=\frac{1}{2} \\
\frac{3}{2} \alpha=30^{\circ} \\
\alpha=20^{\circ}
\end{gathered}
\]

We see, therefore, that only the 9-gon satisfies the condition, as \(\alpha=\frac{180^{\circ}}{9}=20^{\circ}\). (The other solutions of the cubic equation yield \(\alpha=100^{\circ}\) and \(\alpha=260^{\circ}\), from which no meaningful \(n\)-gon can be constructed).