5. The value of the parameter $a$ is chosen so that the number of roots of the first equation $4^{x}-4^{-x}=2 \cos a x, 4^{x}+4^{-x}=2 \cos a x+4$ is 2021. How many roots does the second equation have for the same $a$?
Solution. Transform the second equation:

$\left.4^{x}+4^{-x}=2 \cos a x+4 \Leftrightarrow 4^{x}-2+4^{-x}=2(1+\cos a x) \Leftrightarrow\left(2^{x}-2^{-x}\right)^{2}=4 \cos ^{2 a x} / 2\right) \Leftrightarrow$

$$
\left[\begin{array} { l } 
{ 4 ^ { x / 2 } - 4 ^ { - x / 2 } = 2 \operatorname { cos } \frac { a x } { 2 } , } \\
{ 4 ^ { x / 2 } - 4 ^ { - x / 2 } = - 2 \operatorname { cos } \frac { a x } { 2 } }
\end{array} \Leftrightarrow \left[\begin{array}{l}
4^{x / 2}-4^{-x / 2}=2 \cos \frac{a x}{2} \\
4^{-x / 2}-4^{x / 2}=2 \cos \frac{a x}{2}
\end{array}\right.\right.
$$

Both equations in this system reduce to the first equation from the problem's condition by the substitutions $x=2 y$ and $x=-2 z$ respectively. Therefore, each of these two equations has 2021 roots. If these equations have a common root $x=x_{0}$, then $4^{x_{0}{ }^{\prime 2}}-4^{-x_{0} / 2}=0$ and $\cos ^{a x_{0} / 2}=0$, which is impossible. Consequently, these equations do not have common roots, and the second equation from the condition has $2 \cdot 2021=4042$ roots.

Answer: 4042.

## General grading criteria:

| Points | Correctness (incorrectness) of the solution |
| :---: | :---: |
| 7 | Complete correct solution. |
| $6-7$ | Correct solution. There are minor flaws, which do not significantly affect the solution. |
| $5-6$ | The solution is generally correct. However, it contains several errors, or the omission of individual cases, but can become correct after minor corrections or additions. |
| 4 | One of the two (more complex) significant cases is correctly considered, or in a problem of the type "estimate + example" the estimate is correctly obtained. |
| $2-3$ | Auxiliary statements that help in solving the problem are proved. |
| $0-1$ | Individual important cases are considered in the absence of a solution (or in the case of an incorrect solution). |
| 0 | The solution is incorrect, no progress is made. |
| 0 | The solution is absent. |