Asahanov $H . X$.

Let $f(x)=x^{2}+a x+b \cos x$. Find all values of the parameters $a$ and $b$ for which the equations $f(x)=0$ and $f(f(x))=0$ have coinciding non-empty sets of real roots.
Let $x_{0}$ be a common root, then $f\left(x_{0}\right)=0$ and $f\left(f\left(x_{0}\right)\right)=0$. Substituting the first equality into the second, we get $f(0)=0$, i.e., $b=0$. Thus, $f(x)=x^{2}+a x$, and the equation $f(x)=0$ has roots 0 and $-a$.

The condition of the problem is satisfied if each root of the equation $f(x)=-a$ is equal to 0 or $-a$. Therefore, the cases $a \in(0,4), a=0$ are suitable, since the equation $f(x)=-a$ either has no roots or has the root 0. For other values of $a$, the equation $f(x)=-a$ has at least one root that does not coincide with 0 and - $a$.

## Answer

$b=0,0 \leq a<4$.