Library / Estonian Math Competitions /22 of 148

Estonian Math Competitions

Algebra Difficulty 4.9 AIME Prove it Estonia

A function $f: \mathbb{R} \to \mathbb{R}$ satisfies for all real $x$ and $y$ the equation $2f(x)(f(y))^2 + y^2f(-x|y|) = f(xy^2)$ . Find all possible values of $f(1)$ .

Solution

Denote $f(1) = a$ and $f(-1) = b$ . Choosing $x = y = 1$ and $x = y = -1$ yields the system of equations
$\begin{cases} 2a^3 + b = a \\ 2b^3 + a = b \end{cases}$
Adding the equations and cancelling equal terms, we get $2b^3 = -2a^3$ . Therefore $b = -a$ . Substituting this into the first equation yields $2a^3 - 2a = 0$ . So $f(1)$ is one of the numbers $-1$ , $0$ , $1$ . These values can be obtained by the functions $f(x) = -x|x|$ , $f(x) = 0$ and $f(x) = x|x|$ respectively, which satisfy the equations.

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.