Problem
ALG-B3-M11-P020 Parameter, Midpoints, and Composition
#20
★★★★★ Level 5 of 5
Find all pairs \((a,f)\), where \(f\) is continuous, \(f(x+y)+f(x-y)=2f(x)+2a y^2\), and \(f(f(x))=x\).
First find the general form from the Jensen-type equation.
From the first condition, after subtracting \(a x^2\), we get \(f(x)=a x^2+cx+d\). If \(a\ne0\), then the composition \(f(f(x))\) has degree \(4\), impossible. Hence \(a=0\), \(f(x)=cx+d\). The condition \(f(f(x))=x\) gives \(c^2=1\), \(d(c+1)=0\). The answer is \(a=0\), \(f(x)=x\), and \(f(x)=-x+d\) for any \(d\in\mathbb R\).
Final mixed problem: parameter, Jensen, degree, and composition.