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China Mathematical Olympiad P3

Source: China Nanjing 21 Dec 2013

December 21, 2013
functioninductionalgebra proposedalgebra

Problem Statement

Prove that: there exists only one function f:N∗→N∗f:\mathbb{N^*}\to\mathbb{N^*} satisfying: i) f(1)=f(2)=1f(1)=f(2)=1; ii)f(n)=f(f(n−1))+f(n−f(n−1))f(n)=f(f(n-1))+f(n-f(n-1)) for n≥3n\ge 3. For each integer m≥2m\ge 2, find the value of f(2m)f(2^m).
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