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
Let 
and 
be points on the sides 
and 
of triangle 
respectively. Segments 
and 
intersect at 
and segments 
and 
intersect at 
. The circumscribed circles of triangles 
and 
intersect side 
at 
and 
respectively. Lines 
and 
intersect at . Show that the lines 
, 
, 
are parallel or concurrent.
and 
be points on the sides 
and 
of triangle 
respectively. Segments 
and 
intersect at 
and segments 
and 
intersect at 
. The circumscribed circles of triangles 
and 
intersect side 
at 
and 
respectively. Lines 
and 
intersect at . Show that the lines 
, 
, 
are parallel or concurrent.
