Problem
ALG-B3-M06-P015 Shift and Involution
#15
★★★★★ Level 5 of 5
Find all \(f:\mathbb Z\to\mathbb Z\) such that \(f(m+n)=f(m)+f(n)+2\) and \(f(f(n))=n\).
Put \(g(n)=f(n)+2\).
For \(g(n)=f(n)+2\), we get \(g(m+n)=g(m)+g(n)\). Thus \(g(n)=an\), where \(a\in\mathbb Z\), and \(f(n)=an-2\). Then \(f(f(n))=a(an-2)-2=a^2n-2a-2\). This equals \(n\) for all \(n\), so \(a^2=1\) and \(-2a-2=0\). Therefore \(a=-1\). The answer is \(f(n)=-n-2\).
A good task on the correct sign of the shift.