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f(f(x) - y) = f(xy) + f(x)f(-y)

Source: Mexico National Olympiad Mock Exam (OMMock) 2020 P6

November 9, 2020
algebrafunctional equation

Problem Statement

Find all functions f ⁣:R→Rf \colon \mathbb{R} \to \mathbb{R} such that f(f(x)−y)=f(xy)+f(x)f(−y)f(f(x) - y) = f(xy) + f(x)f(-y) for any two real numbers x,yx, y.
Proposed by Pablo Valeriano
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