[/quote]
,
prove that![$\frac{a}{b} + \sqrt {\frac{b}{c}} + \sqrt[3]{{\frac{c}{a}}} > \frac{5}{2}$](http://latex.artofproblemsolving.com/6/a/6/6a657cb647e279b0745e3189c3f9d54aa25bf976.png)
and orthocenter 
,
and that line 
is not parallel to line 
be the point on the circumcircle of triangle 
is perpendicular to 
be the point on line 
such that 
.
and 
be the points on the circumcircle of triangle 
is a diameter, and 
and 
are parallel. Let 
.
and 
are perpendicular.
and 
are perpendicular.
with 
,
intersects the extension of 
at point 
.
is the midpoint of 
,
intersects the circumcircle of 
.
.
be positive real numbers such that 
.![\[ \frac{a_{1} a_{2} \cdots a_{n} \left[ 1 - (a_{1} + a_{2} + \cdots + a_{n}) \right] }{(a_{1} + a_{2} + \cdots + a_{n})( 1 - a_{1})(1 - a_{2}) \cdots (1 - a_{n})} \leq \frac{1}{ n^{n+1}}. \]](http://latex.artofproblemsolving.com/6/5/5/65558feba82e32266d3d3cbdfea85e079483403f.png)
with integer coefficients such that for all positive number 
and prime 
satisfying 
,
.
be reals such that 
P
,
,
is __
satisfying 
denote its inradius and ex-radii opposite to the vertices 
respectively. If 
and 
,
is obtuse
is acute
,
is semi perimeter
,![$ a,b\in[\frac{1}{2},1] $](http://latex.artofproblemsolving.com/0/d/2/0d24f742e1f7db553b3a8795c061c796f07a897f.png)
.
Let ![$ a,b\in[\frac{1}{2},2] $](http://latex.artofproblemsolving.com/a/8/d/a8d4c8205bac0e34129f582a8f255355eda12e0b.png)
.
satisfy that: 
and 
.
and 
Prove that
Y by Adventure10, Mango247
Let the incircle 
of the triangle 
touch its side 
at 
. Let the line 
intersect at 
and denote the excentre of opposite to 
by 
. Let 
and 
be the midpoints of and 
respectively.
Prove that the points
and 
are concyclic.
of the triangle 
touch its side 
at 
. Let the line 
intersect at 
and denote the excentre of opposite to 
by 
. Let 
and 
be the midpoints of and 
respectively.Prove that the points
and 
are concyclic.
T
e
.
T
at 
, 
.
.
.
, 
.
, 
.
The facts that 
are concurrent and 
and 
a
is cyclic.
.
(
be the reflection of 
with respect to 
.
be the intersection of the polar of 
with respect to 
and 
.
be the reflection of 
with respect to the perpendicular bisector of 
are collinear.
,
.
is the polar of 
is tangent to 
.
and 
,
.
are concyclic
,
are concyclic.
.
.
.
and 
,
.
is a symmedian of the triangle 
so 
,
is a midpoint of 
.
.
,
are concyclic.
makes it clear.
be the perpendicular bisector of 
and 
as the images of 
be a point on line 
and note that 
is tangent to 
.
are concyclic, so 
are also concyclic. By power of point we get 
so 
are concyclic. Thus, 
so 
are collinear.
,
are concyclic. By power of point, 
so 
are concyclic. The result follows as 
is an isosceles trapezoid. 
triangle 
similar triangle 
.
Similarity:
concyclic.
and 
.
thus by MPT in 
we get 
be reflection of 
be other touch point of incircle with 
then 
.
.
and 
is 
cyclic.
by using power of point we get 
Which give us 
.
hence 
.
which yields 
.
cyclic.
