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be positive real numbers such that 
. Prove 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 
, we have 
.
of triangle 
and the midpoint of the arc of its circumscribed circle is not less than 
by 
Applying Apollonius theorem on 
and 
,we get that 
where 
is the midpoint of and 
is the midpoint of 
.So the problem gets converted and we need to prove that 
Let 
be a point on 
such that 
is a parallelogram It needs to be proven that lies between and Now we consider 3 cases.
:In that case 
and 
Since 
, we get that 
: In that case we get the equality condition.
: Working similar to the first case 
be reals such that 
P
b
a
by 
.
on 
