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be an acute triangle with 
and orthocenter 
. Let 
and 
be the feet of the altitudes from 
and 
onto sides 
and 
, respectively. Let 
be the midpoint of 
, and 
be the midpoint of 
. Let the perpendicular line through to 
meet at 
and at 
. The line 
meets at 
and at 
. Let 
be the second intersection point (different from ) of the circumcircles of triangles 
and 
. Prove that the points , , , , and lie on the same circle.
such that
is not a polynomial .
for all 
, 
.
be a positive real number. Find all functions 
such that 
and![\[
f(af(x) - f(y) + y) \leq x + f(y) - y, \quad \forall x, y \in \mathbb{R}^+.
\]](http://latex.artofproblemsolving.com/e/2/9/e296b71ad8b8bd977e2bb2a3630a5bc020b8e03a.png)
. Find all functions 
such that for all 
,![\[
f(xf(x+y)+z) = f(z) + f(x)y + f(xf(x)).
\]](http://latex.artofproblemsolving.com/f/9/3/f93b7526ff4338eea89a1c59bc009aa001ffc15d.png)
be a triangle. The external and internal angle bisectors of 
intersect side at 
and 
, respectively. Let 
be a point on the segment . The circumcircle of triangle 
intersects and at 
and 
, respectively. Let 
be the mid-point of 
and the foot of on 
. Prove that is the incenter of triangle 
, or the center of the excircle.
