AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
, with 
, and the circle 
with center 
and radius 
. Denote by 
the midpoint of side . The line 
intersects the circle for the second time in 
. Let 
be a point on the circle such that 
and 
. Show that 
.
. Let and be the feet of the altitudes from to 
and from 
to respectively.
and 
as the reflections of across lines 
and , respectively. Let 
be the circumcircle of 
. Denote by 
the second intersection of line 
with , and by 
the intersection of ray 
with .
is the circumcenter of , prove that , , and are collinear if and only if quadrilateral 
can be inscribed within a circle.
+1 w![\[\sqrt{a+h_B}+\sqrt{b+h_C}+\sqrt{c+h_A}=\sqrt{a+h_C}+\sqrt{b+h_A}+\sqrt{c+h_B}\]](http://latex.artofproblemsolving.com/5/d/6/5d6881be29327bcdc3e7bd76c221968578c7f50b.png)
is isosceles.
be all the divisors of 
. It is known, that 
are all divisors for some 
(except 
). Find all such 
be a sequence of positive real numbers that satisfies 
and 
for every natural number . Prove that 
for every natural number .
be positive real numbers such that : 
. Prove that : 
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 .
(ans- 
)
satisfying 
Prove that 
When does equality hold?
with 
is inscribed in a circle . Circles 
and 
touch the lines and , and their centres lie on the circumference of . Prove that 
lies on a common external tangent to and .
with is inscribed in a circle . Circles and touch the lines and , and their centres lie on the circumference of . Prove that lies on a common external tangent to and .
and lay on internal and external bisectors of 
, namely they lay on midpoints of arcs and 
, i.e. on perpendicular bisector of , which is symmetry axis of this two circles and points and . So, due to this symmetry if one common tangent () goes through , the other goes through .
such that 
and 
.
?
