[/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 
,
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 
.
be the points on the circumcircle of triangle 
is a diameter, and 
and 
are parallel. Let 
be the midpoint of 
.
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] $](http://latex.artofproblemsolving.com/a/8/d/a8d4c8205bac0e34129f582a8f255355eda12e0b.png)
.
satisfy that: 
and 
.
and 
Prove that
Y by Adventure10
Given 
a point m inside a triangle acute-angled 
and 
intersections of lines with that 
, 
, 
with![$\left[ BC \right],\left[ CA \right],$](//latex.artofproblemsolving.com/3/c/7/3c72c703a3d03049236bf46efc2e41c8ec3f1ef3.png)
respective ![$\left[ AB \right]$](//latex.artofproblemsolving.com/b/9/e/b9ed2f32ca3f2f1fb12b28a2ff80555bd05529cf.png)
a) Show that the area of the triangle is at most a quarter of the area of the triangle
b) Show that the radius of the circle inscribed in the triangle is at most a quarter of the radius of the circle circumscribed on triangle 
a point m inside a triangle acute-angled 
and 
intersections of lines with that 
, 
, 
with![$\left[ BC \right],\left[ CA \right],$](http://latex.artofproblemsolving.com/3/c/7/3c72c703a3d03049236bf46efc2e41c8ec3f1ef3.png)
respective ![$\left[ AB \right]$](http://latex.artofproblemsolving.com/b/9/e/b9ed2f32ca3f2f1fb12b28a2ff80555bd05529cf.png)
a) Show that the area of the triangle
is at most a quarter of the area of the triangle b) Show that the radius of the circle inscribed in the triangle
is at most a quarter of the radius of the circle circumscribed on triangle 
and 
the inradius and semiperimeter of 
and 
the feet of the altitudes on 
If 
represent the barycentric coordinates of 
then the area of its cevian triangle 
is given by![$\frac{[DEF]}{[ABC]}=\frac{2uvw}{(v+w)(w+u)(u+v)}$](http://latex.artofproblemsolving.com/6/6/2/6628f8543d5d6dd0503a8dd19e0071e53185e52b.png)
Thus by AM-GM we have
![$(v+w)(w+u)(u+v) \ge 8 uvw \Longrightarrow \frac{1}{4} \ge \frac{2uvw}{(v+w)(w+u)(u+v)}=\frac{[DEF]}{[ABC]}.$](http://latex.artofproblemsolving.com/0/f/9/0f90af37df75aa650cc54e4a2cd497fb85188299.png)
is the inscribed triangle with the least perimeter (celebrated Fagnano's problem), for a proof see problem 4 in the thread 
together with the previous ![$\varrho \cdot p(\triangle DEF)=[DEF] \le \frac{_1}{^4}[ABC]$](http://latex.artofproblemsolving.com/2/9/5/29523006670a1d48f2a6466d31549344d253f9d0.png)
gives![$\varrho \le \frac{[ABC]}{4 \cdot p(\triangle XYZ)}=\frac{R}{4}.$](http://latex.artofproblemsolving.com/e/0/8/e088440309c14875903d3aee8ca848faacf98fb9.png)
