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Nice functional equation - ILL 1990 KOR2

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September 18, 2010
functioninductionalgebra proposedalgebra

Problem Statement

Let Q\mathbb Q be the set of all rational numbers and R\mathbb R be the set of real numbers. Function f:Q→Rf: \mathbb Q \to \mathbb R satisfies the following conditions:
(i) f(0)=0f(0) = 0, and for any nonzero a∈Q,f(a)>0.a \in Q, f(a) > 0. (ii) f(x+y)=f(x)f(y)∀x,y∈Q.f(x + y) = f(x)f(y) \qquad \forall x,y \in \mathbb Q. (iii) f(x+y)≤max⁡{f(x),f(y)}∀x,y∈Q,x,y≠0.f(x + y) \leq \max\{f(x), f(y)\} \qquad \forall x,y \in \mathbb Q , x,y \neq 0.
Let xx be an integer and f(x)≠1f(x) \neq 1. Prove that f(1+x+x2+⋯+xn)=1f(1 + x + x^2+ \cdots + x^n) = 1 for any positive integer n.n.
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