Problem
ALG-B3-M05-P017 A Cubic Correction
#17
★★★★★ Level 5 of 5
Find all \(f:\mathbb Q\to\mathbb Q\) satisfying \(f(x+y)=f(x)+f(y)+xy(x+y)\).
Use the identity \((x+y)^3-x^3-y^3=3xy(x+y)\).
Let \(g(x)=f(x)-\frac{x^3}{3}\). Then \(g(x+y)=f(x+y)-\frac{(x+y)^3}{3}=f(x)+f(y)+xy(x+y)-\frac{x^3+y^3+3xy(x+y)}{3}=g(x)+g(y)\). On \(\mathbb Q\), \(g(x)=ax\). The answer is \(f(x)=\frac{x^3}{3}+ax\), where \(a\in\mathbb Q\). The check follows from the cubic identity.
Olympiad version: first recognise the correction, then ordinary Cauchy.