## SUBJECT III

On the sides $A_{1} A_{2}, A_{2} A_{3}, \ldots, A_{n-1} A_{n}, A_{n} A_{1}$ of a regular polygon with side length $a$, consider the points $B_{1}, B_{2}, \ldots, B_{n}$, respectively, in the same direction and such that $\left[A_{1} B_{1}\right] \equiv\left[A_{2} B_{2}\right] \equiv \ldots \equiv\left[A_{n} B_{n}\right]=x, \quad x \in(0, a)$. Determine $x$ such that the area of the polygon $B_{1} B_{2} \ldots B_{n}$ is minimized.
## SUBJECT III

We denote $B_{1} B_{2}=l_{n}$. In $\Delta B_{1} A_{2} B_{2}$ we have $B_{1} A_{2}=a-x ; A_{2} B_{2}=x, m\left(B_{1} \hat{A}_{2} B_{2}\right)=\frac{\pi(n-2)}{n}$.

$l_{n}^{2}=4 x^{2} \sin ^{2} \frac{\pi}{n}-4 a x \sin ^{2} \frac{\pi}{n}+a^{2}$.

If $O M$ is the apothem of the polygon $O M=\frac{l_{n}}{2} \operatorname{ctg} \frac{\pi}{n}$.

Let $f:(0, a) \rightarrow R, f(x)=A_{\left[B_{1} B_{2} . \ldots B_{n}\right]}$

$f(x)=n \sin \frac{\pi}{n} \cos \frac{\pi}{n} x^{2}-n a \sin \frac{\pi}{n} \cos \frac{\pi}{n} x+\frac{a^{2} n}{4} \operatorname{ctg} \frac{\pi}{n}$

That is, a second-degree function which admits a minimum for $x=\frac{n a \sin \frac{\pi}{n} \cos \frac{\pi}{n}}{2 n \sin \frac{\pi}{n} \cos \frac{\pi}{n}}=\frac{a}{2}$.

| $2 p$ |
| :--- |
| $1 p$ |
| $2 p$ |
| $2 p$ |

$2 \mathrm{p}$