[/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 
.
be the points on the circumcircle of triangle 
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
On the sides 
and 
of a rhombus 
we consider the points 
and 
respectively. The line 
intersects the segment 
in the point 
, and the line 
intersects the segment in the point 
. We denote by 
the intersection between the line 
and the side 
and with 
the intersection point between the line 
and the side 
.
Prove that the triangles
and 
have the same area.
and 
of a rhombus 
we consider the points 
and 
respectively. The line 
intersects the segment 
in the point 
, and the line 
intersects the segment in the point 
. We denote by 
the intersection between the line 
and the side 
and with 
the intersection point between the line 
and the side 
.Prove that the triangles
and 
have the same area.This post has been edited 1 time. Last edited by Valentin Vornicu, Apr 2, 2005, 2:09 PM
have the same altitudes from the vertices 
to the sides 
,
.
,
be the side of the square and 
,
,
.
regardless of 
.
,
.
,
.
,
.
,
.
.
or 
