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existence of bijective function

Source: Indonesia IMO 2010 TST, Stage 1, Test 5, Problem 3

November 12, 2009

functioninductionnumber theory proposednumber theory

Problem Statement

Let

\mathbb{Z}

be the set of all integers. Define the set

\mathbb{H}

as follows: (1).

\dfrac{1}{2} \in \mathbb{H}

, (2). if

x \in \mathbb{H}

, then \dfrac{1}{1\plus{}x} \in \mathbb{H} and also \dfrac{x}{1\plus{}x} \in \mathbb{H}. Prove that there exists a bijective function

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