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
Suppose 
is a convex quadrilateral.Points 
and 
are four points on the line segments 
and 
respectively.The line segments 
and 
meet at 
.Suppose that each of the quadrilaterals 
and 
have an incircle.Prove that the quadrilateral also has an incircle.
is a convex quadrilateral.Points 
and 
are four points on the line segments 
and 
respectively.The line segments 
and 
meet at 
.Suppose that each of the quadrilaterals 
and 
have an incircle.Prove that the quadrilateral also has an incircle.
)
