3. (12 points) The sequence of functions is defined by the formulas

$$
f_{0}(x)=2 \cos x, f_{n+1}(x)=\frac{4}{2-f_{n}(x)}
$$

for any integer $n \geq 0$. Find $f_{2023}\left(\frac{\pi}{3}\right)$.
Answer: $f_{2023}\left(\frac{\pi}{3}\right)=4$.

Solution. It is easy to compute: $f_{3}(x)=f_{0}(x)$, therefore $f_{2023}(x)=f_{1}(x)=$ $\frac{4}{2-2 \cos x}$. Consequently, $f_{2023}\left(\frac{\pi}{3}\right)=4$.

Remark. One can immediately compute the values of the functions at the given point. This results in a cyclic sequence

$$
f_{0}\left(\frac{\pi}{3}\right)=1 ; f_{1}\left(\frac{\pi}{3}\right)=4 ; f_{2}\left(\frac{\pi}{3}\right)=-2 ; f_{3}\left(\frac{\pi}{3}\right)=1 ; \ldots
$$

Grading criteria. Full solution - 12 points. The relation $f_{3}(x)=f_{0}(x)$ is found - 7 points, the equality $f_{2023}(x)=f_{1}(x)$ is found - another 4 points. Calculation errors - minus 1 point.