Forced to a Half

★ 6/10functions Substitution algebra Functional Equations

Problem

A function $f \colon \mathbb{R} \to \mathbb{R}$ satisfies, for all real $x, y, z$ ,

f(xy) + f(xz) - 2f(x)f(yz) \geqslant \frac{1}{2}.

Find

\lfloor 1 \cdot f(1) \rfloor + \lfloor 2 \cdot f(2) \rfloor + \cdots + \lfloor 2022 \cdot f(2022) \rfloor,

where $\lfloor x \rfloor$ denotes the greatest integer not exceeding $x$ .

Answer

Solution

Difficulty	6/10
Topics	functions, Substitution, algebra, Functional Equations

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