3. Find all natural values of $n$ for which

$$
\cos \frac{2 \pi}{9}+\cos \frac{4 \pi}{9}+\cdots+\cos \frac{2 \pi n}{9}=\cos \frac{\pi}{9}, \text { and } \log _{2}^{2} n+45<\log _{2} 8 n^{13}
$$

In your answer, write the sum of the obtained values of $n$.

(6 points)
# Solution.

$$
\begin{aligned}
& \cos \frac{2 \pi}{9}+\cos \frac{4 \pi}{9}+\cdots+\cos \frac{2 \pi n}{9}=\cos \frac{\pi}{9} \quad \Leftrightarrow 2 \sin \frac{\pi}{9} \cos \frac{2 \pi}{9}+2 \sin \frac{\pi}{9} \cos \frac{4 \pi}{9}+\cdots+ \\
& 2 \sin \frac{\pi}{9} \cos \frac{2 \pi n}{9}=2 \sin \frac{\pi}{9} \cos \frac{\pi}{9} \Leftrightarrow
\end{aligned}
$$

$\sin \frac{3 \pi}{9}-\sin \frac{\pi}{9}+\sin \frac{5 \pi}{9}-\sin \frac{3 \pi}{9}+\cdots+\sin \frac{\pi}{9}(1+2 n)-\sin \frac{\pi}{9}(2 n-1)=\sin \frac{2 \pi}{9} \Leftrightarrow$ $\sin \frac{\pi}{9}(1+2 n)-\sin \frac{\pi}{9}=\sin \frac{2 \pi}{9} \Leftrightarrow \sin \frac{\pi}{9}(1+2 n)=\sin \frac{2 \pi}{9}+\sin \frac{\pi}{9} \Leftrightarrow \sin \frac{\pi}{9}(1+2 n)=$ $2 \sin \frac{\pi}{6} \cos \frac{\pi}{18} \Leftrightarrow \sin \frac{\pi}{9}(1+2 n)=\sin \frac{4 \pi}{9} \Leftrightarrow 2 \sin \left(\frac{\pi n}{9}-\frac{\pi}{6}\right) \cos \left(\frac{\pi n}{9}+\frac{5 \pi}{18}\right)=0 \Leftrightarrow \frac{\pi n}{9}-\frac{\pi}{6}=$ $\pi k, k \in \mathbb{Z}, \quad \frac{\pi n}{9}+\frac{5 \pi}{18}=\frac{\pi}{2}+\pi m, m \in \mathbb{Z}, \Leftrightarrow 2 n=3(6 k+1), k \in \mathbb{Z}, \quad n=2+$
$9 m, m \in \mathbb{Z}, m \geq 0$. Since $n$ is a natural number, the first relation does not hold (an even number cannot be equal to an odd number). Therefore, $n=2+9 m, m \in \mathbb{Z}, m \geq 0$.

Solve the inequality $\log _{2}^{2} n+45<\log _{2} 8 n^{13} \Leftrightarrow \log _{2}^{2} n-13 \log _{2} n+42<0$, $\Leftrightarrow$ $\left(\log _{2} n-6\right)\left(\log _{2} n-7\right)<0, \Leftrightarrow 2^{6}<n<2^{7}, \Leftrightarrow 64<2+9 m<128, m \in \mathbb{Z}, m \geq$ $0 \Leftrightarrow 6 \frac{8}{9}<m<14, m \in \mathbb{Z}, m \geq 0 \Leftrightarrow m=7,8,9,10,11,12,13 . S_{7}=\frac{65+119}{2}$. $7=644$.

## Answer: 644