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
Y by doxuanlong15052000, Adventure10, Mango247, Rounak_iitr
Let 
- be a midpoint of side 
in triangle 
. A cricumcircle of 
intersects segment 
at points 
and 
(
). A circumcircle of 
intersects segment 
at points and 
(
). Let 
be a circumcircle of 
. Prove that 
- be a midpoint of side 
in triangle 
. A cricumcircle of 
intersects segment 
at points 
and 
(
). A circumcircle of 
intersects segment 
at points and 
(
). Let 
be a circumcircle of 
. Prove that 
be the radius of 
,
we get 
,
is cyclic 
,
.
,
,
.
.
.
.
.
,
bisects 
.
,
or 
can be dealt in a similar manner with some similar angle chase.
denote the power of 
with respect to 
,
Join 
and 
.
as point 
.
.
is similar t
.
=
.
=
.
is isosceles with 
=
=
.
.
