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 Adventure10, Mango247
Let 
be an acute triangle with the circumcircle 
and incenter 
. The perpendicular through in 
intersects segment ![$[BC]$](//latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
in 
and in 
. In particular and 
are on different sides of 
. The parallel line through to 
intersects 
in 
.
Prove: If
and are perpendicular to each other, then intersects segment ![$[AC]$](//latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
in its midpoint 
.
(Notation:![$[\cdot]$](//latex.artofproblemsolving.com/5/7/9/579a0ec9e030cc17f3b2c9cd1510c9493b1813b2.png)
denotes the line segment.)
be an acute triangle with the circumcircle 
and incenter 
. The perpendicular through in 
intersects segment ![$[BC]$](http://latex.artofproblemsolving.com/e/a/1/ea1d44f3905940ec53e7eebd2aa5e491eb9e3732.png)
in 
and in 
. In particular and 
are on different sides of 
. The parallel line through to 
intersects 
in 
.Prove: If
and are perpendicular to each other, then intersects segment ![$[AC]$](http://latex.artofproblemsolving.com/0/9/3/0936990e6625d65357ca51006c08c9fe3e04ba0c.png)
in its midpoint 
.(Notation:
![$[\cdot]$](http://latex.artofproblemsolving.com/5/7/9/579a0ec9e030cc17f3b2c9cd1510c9493b1813b2.png)
denotes the line segment.)
and 
.
are concyclic ,
is the midpoint of arc 
in 
is the perpendicular bisector of 
,
.
is the midpoint of arc 
(contain 
)
,
,
are collinear 
passes through the midpoint 
of 
.
