 ## Understand the problem

Find all real numbers $a$ for which there exists a non-constant function $f :\Bbb R \to \Bbb R$ satisfying the following two equations for all $x\in \Bbb R:$
i) $f(ax) = a^2f(x)$ and
ii) $f(f(x)) = a f(x).$
Baltic Way 2016

### Functional Equation

Easy
##### Suggested Book
Functional Equations by BJ Venkatachala

Do you really need a hint? Try it first!

Show that the choices $a=0,1$ work.

Show that $af(f(x))=a^2f(f(x))$. As we have already dealt with $a=0$, this gives $af(f(x))=f(f(x))$
Hint 3 gives $(a-1)f(f(x))=0$. As $a=1$ has already been dealt with, we must consider the option $f(f(x))\equiv 0$.
Hint 3 gives $af(x)\equiv 0$. As $a\neq 0$, we have $f(x)\equiv 0$. This contradicts the fact that $f$ is non-constant. Hence, $a=0,1$ are the only options.

## Connected Program at Cheenta

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.

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