IMO Shortlist 1983 problem 12
Dodao/la:
arhiva2. travnja 2012. Find all functions
![f](/media/m/9/9/8/99891073047c7d6941fc8c6a39a75cf2.png)
defined on the set of positive reals which take positive real values and satisfy:
![f(xf(y))=yf(x)](/media/m/3/5/0/3501b2d2f67e89ef37b32900d46f47ba.png)
for all
![x,y](/media/m/f/b/6/fb60533620f22cd699e5b58ce9a646a4.png)
; and
![f(x)\to0](/media/m/4/8/e/48e3c5b18b5a6c9813d9bc7678f9ce2d.png)
as
![x\to\infty](/media/m/b/4/8/b48da958f4f3ee4622c4aaeb87d5a1df.png)
.
%V0
Find all functions $f$ defined on the set of positive reals which take positive real values and satisfy: $f(xf(y))=yf(x)$ for all $x,y$; and $f(x)\to0$ as $x\to\infty$.
Izvor: Međunarodna matematička olimpijada, shortlist 1983