MathDB

Functional Equation on Z

Source: Pan African Maths Olympiad

August 11, 2005

functioninduction

Problem Statement

Let

f: \mathbb{Z} \rightarrow \mathbb{Z}

be a function such that: For all

a

and

b

in

\mathbb{Z} - \{0\}

,

f(ab) \geq f(a) + f(b)

. Show that for all

a \in \mathbb{Z} - \{0\}

we have

f(a^n) = nf(a)

for all

n \in \mathbb{N}

if and only if

f(a^2) = 2f(a)

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