IMO Shortlist 1969 problem 8
Dodao/la:
arhiva2. travnja 2012. ![(BUL 2)](/media/m/b/c/2/bc26c2f54307fef81d280ca4562e9d7e.png)
Find all functions
![f](/media/m/9/9/8/99891073047c7d6941fc8c6a39a75cf2.png)
defined for all
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
that satisfy the condition
![xf(y) + yf(x) = (x + y)f(x)f(y),](/media/m/a/c/b/acbb04bd1d86a823a83d1d61d5200127.png)
for all
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
and
![y.](/media/m/9/2/3/923fc0c0f0d7a54598bbd1f90c33b74f.png)
Prove that exactly two of them are continuous.
%V0
$(BUL 2)$ Find all functions $f$ defined for all $x$ that satisfy the condition $xf(y) + yf(x) = (x + y)f(x)f(y),$ for all $x$ and $y.$ Prove that exactly two of them are continuous.
Izvor: Međunarodna matematička olimpijada, shortlist 1969