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functional with fractional part , {f(x)} sin^2 x+{x} cos(f(x))cosx=f(x)=f(f(x))

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2015 3.3

September 23, 2021
algebrafractional partfunctional equationfunctional

Problem Statement

Indicate (justifying your answer) if there exists a function f:R→Rf: R \to R such that for all x∈Rx \in R fulfills that
i) {f(x))}sin⁡2x+{x}cos(f(x))cosx=f(x)\{f(x))\} \sin^2 x + \{x\} cos (f(x)) cosx =f (x) ii) f(f(x))=f(x)f (f(x)) = f(x)
where {m}\{m\} denotes the fractional part of mm. That is, {2.657}=0.657\{2.657\} = 0.657, and {−1.75}=0.25\{-1.75\} = 0.25.
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