[/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)
,
be an arbitrary point on the circumcircle of triangle 
and that line 
is not parallel to line 
be the point on the circumcircle of triangle 
is perpendicular to 
be the point on line 
.
and 
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 
be a triangle,
its circumcenter and 
the radius of its circumcircle.Denote by 
the symmetric of with respect to 
,
the symmetric of with respect to 
and by 
the symmetric of with respect to 
.
(a)Prove that the circles
, 
, 
have a common point.
(b)Denote by
this point.Let 
be an arbitary line passing through which intersects 
at 
, 
at 
and 
at 
.From 
drop perpendiculars to 
respectively.Prove that these perpendiculars pass through a point.
be a triangle,
its circumcenter and 
the radius of its circumcircle.Denote by 
the symmetric of with respect to 
,
the symmetric of with respect to 
and by 
the symmetric of with respect to 
.(a)Prove that the circles
, 
, 
have a common point.(b)Denote by
this point.Let 
be an arbitary line passing through which intersects 
at 
, 
at 
and 
at 
.From 
drop perpendiculars to 
respectively.Prove that these perpendiculars pass through a point.
, 
, 
.
We'll prove that 
.
.
in any acute 
.
.
, 
is the orthocenter of 
be an abitrary line passing through the orthocenter of 
of 
be the reflections of 
about 
respectively and let 
be the concurrency point. (Note that by the lemma, 
)
is the reflection of 
is reflected about 
.
is reflected about 
.
.
perpendiculars from 
.
