and 
.
be positive real numbers such that : 
. Prove that : 
.
be the feet of the altitudes from 
and from 
to 
respectively.
and 
as the reflections of 
and 
be the circumcircle of 
.
the second intersection of line 
with 
the intersection of ray 
with 
is the circumcenter of 
can be inscribed within a circle.
be the set of positive integers. Find all functions 
such that![\[ m^2 + f(n) \mid mf(m) +n \]](http://latex.artofproblemsolving.com/2/f/4/2f409d1de993f1af8fd839bb8e9f87a57e1b8608.png)
and 
.
Y by sameer_chahar12
Given an acute 
with incenter 
. Let 
be the symmetric point with respect to the midpoint of 
and 
is the foot of 
. If 
and the circumcircle of vartriangle 
intersect at 
and 
intersects the circumcircle of 
at 
. Furthermore, 
are tangent points of the incircle and 
is the another intersection of the circumcircles of 
. Show that 
.
with incenter 
. Let 
be the symmetric point with respect to the midpoint of 
and 
is the foot of 
. If 
and the circumcircle of vartriangle 
intersect at 
and 
intersects the circumcircle of 
at 
. Furthermore, 
are tangent points of the incircle and 
is the another intersection of the circumcircles of 
. Show that 
.
be the 
be the reflection of 
are collinear. Moreover, 
,
,
,
are all collinear and 
.
and notice that 
is cyclic as 
.
we see that 
lies on the radical axis of 
and 
.
by Power of a Point from 
and 
.
and therefore
over 
be the midpoint of arc 
and let 
be the foot from 
are collinear, and now we have that 
by Spiral Sim so if 
is the midpoint of ![\[
\angle PIA = \angle PNM = \angle D_0NM = \angle D_1NM
\]](http://latex.artofproblemsolving.com/c/c/a/ccad96702781c56fff3970c14ed05b2ced3afad2.png)
where 
meet 
at ![\[
\angle ND_1Q = \angle MND_1 = \angle PIA = \angle QIN,
\]](http://latex.artofproblemsolving.com/e/7/e/e7ef1d6eb0cc9be283936c6e99945928dde79e72.png)
so 
is cyclic. In particular, ![\[
\angle IQT = \angle IND_1 = \angle AIT = \angle AXT,
\]](http://latex.artofproblemsolving.com/5/b/2/5b2893c206e31ac21cffc85a84ad37eebf46c05d.png)
so 
.
by homothety centered at 
.
