Provide a threshold number $\mu_{100}$, or $\mu_{1} 000000$, such that

$$
F=m n \sin \frac{\pi}{m} \sqrt{\frac{1}{n^{2}}+\sin ^{4} \frac{\pi}{m}}
$$

is definitely greater than 100 and 1000 000, respectively, for all $m$ greater than $\mu_{100}$, or $\mu_{1} 000000$, assuming that $n=m^{3}$. (It is not necessary to provide the smallest such $\mu$ numbers. - See the article about Geő́cze Zoár in our No. VIII. 1.)
Let's substitute $n=m^{3}-t$, we get

$$
\begin{gathered}
F=m^{4} \sin \frac{\pi}{m} \sqrt{\frac{1}{m^{6}}+\sin ^{4} \frac{\pi}{m}}= \\
=m \sin \frac{\pi}{m} \sqrt{1+m^{6} \sin ^{4} \frac{\pi}{m}}>m \sin \frac{\pi}{m} m^{3} \sin ^{2} \frac{\pi}{m}
\end{gathered}
$$

which means

$$
F>m\left(m \sin \frac{\pi}{m}\right)^{3}
$$

![](https://cdn.mathpix.com/cropped/2024_05_02_9222c3e8503001632050g-1.jpg?height=583&width=810&top_left_y=614&top_left_x=690)

As in the article, we should now replace $\frac{\pi}{m}$ with a multiple of $\frac{\pi}{m}$ that is less than $\sin \frac{\pi}{m}$, at least if $m$ is sufficiently large. If we draw the chord in the curve $y=\sin x$ that passes through the points with abscissae $0$ and $\frac{\pi}{4}$, i.e., the points $(0,0)$ and $\left(\frac{\pi}{4}, \frac{\sqrt{2}}{2}\right)$ (Figure 1), then, considering that the sine curve is concave from below in the interval $\left(0, \frac{\pi}{4}\right)$, the points of the chord given by the equation $y=\frac{2 \sqrt{2}}{\pi} x$ are below the points of the sine curve in the interval $\left(0, \frac{\pi}{4}\right)$, i.e., if $0<\frac{\pi}{m}<\frac{\pi}{4}$, then $\sin \frac{\pi}{m}>\frac{2 \sqrt{2}}{\pi} \frac{\pi}{m}=\frac{2 \sqrt{2}}{m}$. Therefore, based on (1),

$$
F>m(2 \sqrt{2})^{3}=16 \sqrt{2} m
$$

assuming that $m>4$.

Since $16 \sqrt{2} m>100$ if $m>\frac{100}{16 \sqrt{2}} \sim 4.42$ and $16 \sqrt{2} m>1000000$ if $m>10000 \cdot 4.42=44200$, the sought threshold values are

$$
\mu_{100}=5 \quad \text { and } \quad \mu_{1000000}=44200 .
$$

Remark: A smaller value can be obtained for the latter threshold if we choose a smaller interval instead of $\left(0, \frac{\pi}{4}\right)$. For example, for the interval $\left(0, \frac{\pi}{12}\right)$, we get $\mu_{1000000}=33445$.

Pál Bártfai (Budapest, I., Petőfi g. III. o. t.)