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Prove that F has exactly two elements

Source: Romanian MO 2010 Grade 9

August 6, 2012
functioninductionalgebrafunctional equationnumber theory proposednumber theory

Problem Statement

Consider the set F\mathcal{F} of functions f:N→Nf:\mathbb{N}\to\mathbb{N} (where N\mathbb{N} is the set of non-negative integers) having the property that f(a2−b2)=f(a)2−f(b)2, for all a,b∈N, a≥b.f(a^2-b^2)=f(a)^2-f(b)^2,\ \text{for all }a,b\in\mathbb{N},\ a\ge b. a) Determine the set {f(1)∣f∈F}\{f(1)\mid f\in\mathcal{F}\}. b) Prove that F\mathcal{F} has exactly two elements.
Nelu Chichirim
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