of degree 
having 
distinct real roots 
)
: 
.
defined on all real numbers and taking real values such that ![\[f(f(y)) + f(x - y) = f(xf(y) - x),\]](http://latex.artofproblemsolving.com/f/5/2/f528fc4047e21247663d4993913eadf346315be7.png)
for all real numbers 
.
such that 
.
![$P\in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/7/4/c/74cb2247069abb8a22666e87fcbd69a55a015004.png)
for which there exists 
such that for all 
we have: ![\[P\left(x+\frac1n\right)+P\left(x-\frac1n\right)=2P(x).\]](http://latex.artofproblemsolving.com/9/0/9/90954cdeade9d13149e1aaa03df2c87dff157041.png)
Y by
In a triangle 
, 
is the orthogonal projection of the incenter 
onto 
. Line 
meets the incircle again at 
. Line 
intersects side at point 
. Suppose that the segment IO is parallel to , where 
is the circumcenter of 
. If 
is the circumradius and 
the inradius of the triangle, prove that 
.
, 
is the orthogonal projection of the incenter 
onto 
. Line 
meets the incircle again at 
. Line 
intersects side at point 
. Suppose that the segment IO is parallel to , where 
is the circumcenter of 
. If 
is the circumradius and 
the inradius of the triangle, prove that 
.
