[/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 
such that 
.
and 
be the points on the circumcircle of triangle 
is a diameter, and 
and 
are parallel. Let 
be the midpoint of 
.
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
Let 
be a triangle in which 
is obtuse.Let 
be its circumcircle and 
be the centre of .Let the tangent to at 
intersect the line 
in 
.Let 
be the circumcentre of the circumcircle 
of 
.Take any point 
on the segment 
different from 
.Let 
be the point of contact of the tangent from to which is closer to .Let 
be the circumcentre of 
.Prove that 
are concyclic if 
.
be a triangle in which 
is obtuse.Let 
be its circumcircle and 
be the centre of .Let the tangent to at 
intersect the line 
in 
.Let 
be the circumcentre of the circumcircle 
of 
.Take any point 
on the segment 
different from 
.Let 
be the point of contact of the tangent from to which is closer to .Let 
be the circumcentre of 
.Prove that 
are concyclic if 
.
is the perpendicular bisector of 
,
we get 
are collinear .
are concyclic . ... 
,
,
,
.
,
and 
.
are concyclic ... 
are concyclic.
