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 
,
.
Y by Adventure10, buddyram
Let 
with 
be an acute angled triangle and 
its circumcircle. Let 
be the point diametrically opposite to 
.
Point
is on 
such that 
. The circle 
intersects 
at point 
. Prove that the circle 
is tangent to .
with 
be an acute angled triangle and 
its circumcircle. Let 
be the point diametrically opposite to 
.Point
is on 
such that 
. The circle 
intersects 
at point 
. Prove that the circle 
is tangent to .
be tangent to 
at 
,Now 
Now since 
also 
so 
so 
with 
= 
+ 
= 
+ 
)
