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Prove that f(x) = x for all real numbers x

Source: IMO Longlist 1989, Problem 52

September 18, 2008

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Problem Statement

Let

f

be a function from the real numbers to the real numbers such that f(1) \equal{} 1, f(a\plus{}b) \equal{} f(a)\plus{}f(b) for all

a, b,

and f(x)f \left( \frac{1}{x} \right) \equal{} 1 for all

x \neq 0.

Prove that f(x) \equal{} x for all real numbers

x.