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f(1) = 1, f(x + y) = f(x) + f(y),f(1/x)= f(x)/x^2

Source: 2015 Saudi Arabia GMO TST III p1

July 26, 2020
algebrafunctionalfunctional equation

Problem Statement

Find all functions f:R→Rf : R \to R satisfying the following conditions (a) f(1)=1f(1) = 1, (b) f(x+y)=f(x)+f(y)f(x + y) = f(x) + f(y), ∀(x,y)∈R2\forall (x,y) \in R^2 (c) f(1x)=f(x)x2f\left(\frac{1}{x}\right) =\frac{ f(x)}{x^2 }, ∀x∈R−{0}\forall x \in R -\{0\}
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