[/quote]
,
be a sequence defined by 
.
if 
.
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 
,
.
that satisfy 
be reals such that 
P
,
,
is __
satisfying 
be a triangle and 
denote its inradius and ex-radii opposite to the vertices 
respectively. If 
and 
,
is obtuse
is acute
,
is semi perimeter
,![$ a,b\in[\frac{1}{2},1] $](http://latex.artofproblemsolving.com/0/d/2/0d24f742e1f7db553b3a8795c061c796f07a897f.png)
.
Let ![$ a,b\in[\frac{1}{2},2] $](http://latex.artofproblemsolving.com/a/8/d/a8d4c8205bac0e34129f582a8f255355eda12e0b.png)
.
satisfy that: 
and 
.
and 
Prove that
Y by Mango247
Point 
lies on side 
of the parallelogram 
, and point 
lies on side 
Segments 
and 
intersect at point 
, and segments 
and 
intersect at point 
. Find the largest value of the ratio of area of the quadrangles 
and .
lies on side 
of the parallelogram 
, and point 
lies on side 
Segments 
and 
intersect at point 
, and segments 
and 
intersect at point 
. Find the largest value of the ratio of area of the quadrangles 
and .This post has been edited 1 time. Last edited by parmenides51, Jul 11, 2020, 5:00 PM
