Problem
ALG-B3-M10-P019 Parameter and Square of Composition
#19
★★★★★ Level 5 of 5
Find all pairs \((a,f)\), where \(f\) is continuous, \(f(x+y)=f(x)+f(y)+a xy\), and \(f(f(x))=x^2\).
Use \(f(x)=\frac a2x^2+cx\) and compare degrees.
We have \(f(x)=\frac a2x^2+cx\). If \(a\ne0\), then \(f(f(x))\) has degree \(4\), while the right-hand side has degree \(2\), impossible. Hence \(a=0\), \(f(x)=cx\). Then \(f(f(x))=c^2x\), which cannot equal \(x^2\) for all \(x\). Therefore there are no solutions.
Strong problem on a parameter and degree of composition.