2017 IMO Problems/Problem 2

Problem

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 $y$

$$f(f(x)f(y)) + f(x+y)=f(xy)$$

Solution

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 $f(x)=mx+b$

Therefore,

$f((mx+b)(my+b))+m(x+y)+b=mxy+b$

$f(m^2xy+mb(x+y)+b^2)+m(x+y)+b=mxy+b$

$m(m^2xy+mb(x+y)+b^2)+b+m(x+y)+b=mxy+b$

$m^3xy+m^2b(x+y)+mb^2+m(x+y)+b=mxy$

$m(m^2-1)xy+m(bm+1)(x+y)+b(bm+1)=0$

Therefore,

$m(m^2-1)=0$ [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 byProblem 1 1 • 2 • 3 • 4 • 5 • 6 Followed byProblem 3 All IMO Problems and Solutions