Question 90
Does there exist a function $f$, $f: \mathbb{R} \to \mathbb{R}$, such that
$$
\begin{cases}
\{f(x)\} \sin^2 x + \{x\} \cos f(x) \cos x = f(x), \\
f(f(x)) = f(x),
\end{cases}
$$
for all real $x$.
(Here $\{y\}$ stands for the fractional part of $y$.)
$$
\begin{cases}
\{f(x)\} \sin^2 x + \{x\} \cos f(x) \cos x = f(x), \\
f(f(x)) = f(x),
\end{cases}
$$
for all real $x$.
(Here $\{y\}$ stands for the fractional part of $y$.)