[/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)
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 
.
, such that 
Prove that :
equality cases : ?
and 
Prove that
.
is in the interior of ![\[\angle PAD:\angle PBA:\angle DPA=1:2:3=\angle CBP:\angle BAP:\angle BPC\]](http://latex.artofproblemsolving.com/9/3/8/9388a91325842f1ec67a4693f0f10af46ed66c6d.png)
Prove that the following three lines meet in a point: the internal bisectors of angles 
and 
and the perpendicular bisector of segment 
Y by Adventure10, Mango247
Let 
be a isosceles triangle, with 
. A line 
that pass through the incenter 
of touches the sides 
and 
at the points 
and 
, respectively. Let 
and 
be points on 
such that 
and 
. Show that the angle 
is constant when we vary the line .
be a isosceles triangle, with 
. A line 
that pass through the incenter 
of touches the sides 
and 
at the points 
and 
, respectively. Let 
and 
be points on 
such that 
and 
. Show that the angle 
is constant when we vary the line .
followed by symmetry about the midpoint 
of 
followed by symmetry about 
Since 
is a perspectivity between 
then it follows that 
is a homography clearly with no fixed points 
it is induced by two superposed congruent pencils, or the sides of a rigid angle whirling around its vertex 
is reflection of 
on 
is reflection of 
then 
are the limiting points of 
of 
When 
then 
and 
become rhombi, thus by symmetry, we have 
which implies that 
for all 
be the orthogonal projections of 
,
,
.
and 
are congruent, since 
and 
.
,
.
Hence, trinangles 
and 
are similar, which means that 
,
