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f(xy) = f(x)f(y) - f(x+y) + 1

Source: IMO 1980 Luxembourg, problem 1

May 6, 2004

functionalgebrafunctional equationIMO Shortlist

Problem Statement

The function

f

is defined on the set

\mathbb{Q}

of all rational numbers and has values in

\mathbb{Q}

. It satisfies the conditions

f(1) = 2

and

f(xy) = f(x)f(y) - f(x+y) + 1

for all

x,y \in \mathbb{Q}

. Determine

f

.

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