Overview

Advanced functional equations often mix algebra and number theory. Expect to find hidden constraints via special inputs and symmetry.

Key Ideas

• If $f$ is defined on integers, parity and modular arguments can restrict it.
• Look for fixed points: solve $f(x)=x$ or $f(x)=c$.
• Iteration (apply the equation repeatedly) can uncover structure.

Worked Example

If $f(x)=2x$ satisfies $f(x+y)=f(x)+f(y)$ for all $x,y$, then $f$ is additive. Many contest problems ask you to prove that linear functions are the only solutions.

Practice Problems