2017 IMO Problems/Problem 2


Let $\mathbb{R}$ be the set of real numbers , determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for any real numbers $x$ and $y$

\[f(f(x)f(y)) + f(x+y)=f(xy)\]


Looking at the equation one can deduce that the functions that will work will be linear. That is, a polynomial of at most a degree of 1.

Thus, $f$ is in the form $f(x)=mx+b$








$m(m^2-1)=0$ [Equation 1]

$m(bm+1)=0$ [Equation 2]

$b(bm+1)$ [Equation 3]

From [Equation 1] we have, $m=0,\pm 1$

From [Equation 2] we have, $m=0, bm=-1\pm 1$

From [Equation 3] we have, $b=0, bm=-1\pm 1$

When $m=0$, $b=0$, then $f(x)=0$

When $bm=-1$, $b=\frac{-1}{m}$, then since $m=\pm 1$, then $b=\mp 1$

When $bm=-1$, $b=\frac{-1}{m}$, then since $m=\pm 1$, then $b=\mp 1$ then $f(x)=\pm x \mp1$ which gives these two functions:

$f(x)=x-1$ and $f(x)=1-x$, which with $f(x)=0$ provide all the three functions for this problem.

~Tomas Diaz. orders@tomasdiaz.com

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

See Also

2017 IMO (Problems) • Resources
Preceded by
Problem 1
1 2 3 4 5 6 Followed by
Problem 3
All IMO Problems and Solutions