be an continuous function defined in ![$\text{[0,2015]},f(0)=f(2015)$](http://latex.artofproblemsolving.com/c/9/9/c99f63e726b504aea8c1a65ba2926155a873effa.png)
pairs of 
such that 
![$ a,b\in [0,1] $](http://latex.artofproblemsolving.com/9/7/5/975edfc0cf5316d78283b313c6b5cfbdd7b92b6e.png)
. Prove that
Where 
such that ![\[f(f(x)) + xf(xy) = x + f(y)\]](http://latex.artofproblemsolving.com/a/6/f/a6f84dd53eec985b90bad3df6580eeaf29d73195.png)
for all positive real numbers 
and 
.
Y by Adventure10, Mango247
In triangle 
, 
, 
, 
. The incircle touches 
at 
and 
is a diameter of the incircle. Radical 
intersect at 
. 
equals to the circumradius of 
. Find and 
.
, 
, 
, 
. The incircle touches 
at 
and 
is a diameter of the incircle. Radical 
intersect at 
. 
equals to the circumradius of 
. Find and 
.
so AC is the greatest side
