[/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 circumcenter 
and orthocenter 
,
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 and 1 other user
Let 
be a cyclic convex quadrilateral. Let 
be the intersection of lines 
. 
is the intersection of line passing 
and perpendicular to 
, and line passing 
and perpendicular to 
. 
is the intersection of line passing 
and perpendicular to , and line passing 
and perpendicular to 
. Prove that three points 
are collinear.
be a cyclic convex quadrilateral. Let 
be the intersection of lines 
. 
is the intersection of line passing 
and perpendicular to 
, and line passing 
and perpendicular to 
. 
is the intersection of line passing 
and perpendicular to , and line passing 
and perpendicular to 
. Prove that three points 
are collinear.
,
,
.
,
,
.
are collinear.
,
.
and 
are similar, hence 
.
to 
.
.
,
,
and 
.
,
is cyclic. But 
is the power of point 
,
.
)
,
are collinear, as desired.
with center 
to 
.
to 
so it sends 
to 
.
with center 
,
.
to 
so it sends 
to 
.
or![\[{\overline{EM}\over \overline{EA}} ={\overline{EB}\over \overline{EL}}\]](http://latex.artofproblemsolving.com/6/f/6/6f6f2659af985943f0d47f840013121e050e8730.png)
![\[ \overline{EM} \cdot \overline{EL} = \overline{EA}\cdot \overline{EB} (=\overline{EC}\cdot \overline{ED}) \]](http://latex.artofproblemsolving.com/2/f/7/2f7921adb1c46973df00f5022f3fb159ca16ae3f.png)
![\[ \angle BDC = \angle BAC = \alpha \]](http://latex.artofproblemsolving.com/9/4/6/946706eb26dfa666e5d54ecdf492aadbe0b89c4c.png)
and![\[ \angle LDB = \angle ACM = \beta \]](http://latex.artofproblemsolving.com/a/1/8/a18a11c024a8562bc2060f86ea9973614e07e694.png)
![\[\angle AMC = \pi -\alpha-\beta\]](http://latex.artofproblemsolving.com/1/f/7/1f7c8f86cd0c05915273501b03d7bca9abc30040.png)
and finally ![\[\angle LMC +
\angle LDC = \pi\]](http://latex.artofproblemsolving.com/c/5/6/c564b72cceb119e4a593046876e7809fa5983cfb.png)
and 
intersect the line 
and 
intersect 
at 
at 
and 
respectively. 
and 
and 
.
and 
.Since 
and 
they are obviously similar---1 .Similarly we can prove that triangles 
and 
are similar----2.Now consider the two triangles 
and 
.
=Angl
.
is cyclic .similarly 
is cyclic.Angle 
a
angle 
.Hence triangles 
are similar----3 .Combining 1,2,3 we have Quadrilaterals 
and 
maps point 
and 
are perspective from a line (intersection of 
and 
are concurrent.
,
are concurrent.
