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
A circle that passes through the vertex 
of a rectangle 
intersects the side 
at a second point 
different from 
A line passing through 
is tangent to this circle at a point 
and the circle with center and passing through 
intersects the side 
at the point 
Show that if 
then 
of a rectangle 
intersects the side 
at a second point 
different from 
A line passing through 
is tangent to this circle at a point 
and the circle with center and passing through 
intersects the side 
at the point 
Show that if 
then 
is the midpoint of 
,
.
.
.
.
.
and 
are complementary, so 
.
and 
to each other. Let 
and 
.
and 
,
.
.
,
.
.
gives 
is tangent to 
.
is cyclic. Hence, 
.
