Problem
ALG-B3-M01-P014 Another quadratic extra term
#14
★★★★★ Level 5 of 5
Let \(f:\mathbb Q\to\mathbb Q\), \(f(x+y)=f(x)+f(y)+xy\), and \(f(1)=0\). Find \(f\).
Hint. Subtract \(\frac{x^2}{2}\).
Let \(g(x)=f(x)-\frac{x^2}{2}\). Then \(g(x+y)=g(x)+g(y)\). Also, \(g(1)=f(1)-1/2=-1/2\). On \(\mathbb Q\), \(g(q)=-q/2\), hence \(f(q)=\frac{q^2-q}{2}\).
Goal: separate guessing the answer from a complete proof and domain check.