[/quote]
,
prove that![$\frac{a}{b} + \sqrt {\frac{b}{c}} + \sqrt[3]{{\frac{c}{a}}} > \frac{5}{2}$](http://latex.artofproblemsolving.com/6/a/6/6a657cb647e279b0745e3189c3f9d54aa25bf976.png)
with circumcenter 
and orthocenter 
,
be an arbitrary point on the circumcircle of triangle 
and that line 
is not parallel to line 
.
be the point on the circumcircle of triangle 
is perpendicular to 
be the point on line 
such that 
.
and 
be the points on the circumcircle of triangle 
is a diameter, and 
and 
are parallel. Let 
be the midpoint of 
.
and 
are perpendicular.
and 
are perpendicular.
with 
,
intersects the extension of 
at point 
.
is the midpoint of 
,
intersects the circumcircle of 
.
.
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
,
,
is __
satisfying 
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 Adventure10, Mango247
Let 
be a convex octagon such that all of its sides are equal and its opposite sides are parallel. For each 
, define 
as the intersection between segments 
and 
, where 
and 
for all 
. Show some number 
, amongst 1, 2, 3, and 4 satisfies
![\[\frac{A_iA_{i+4}}{B_iB_{i+4}} \leq \frac{3}{2}\]](//latex.artofproblemsolving.com/a/4/6/a461fa27c5faa2833abb5dcc290c98af156b8726.png)
be a convex octagon such that all of its sides are equal and its opposite sides are parallel. For each 
, define 
as the intersection between segments 
and 
, where 
and 
for all 
. Show some number 
, amongst 1, 2, 3, and 4 satisfies![\[\frac{A_iA_{i+4}}{B_iB_{i+4}} \leq \frac{3}{2}\]](http://latex.artofproblemsolving.com/a/4/6/a461fa27c5faa2833abb5dcc290c98af156b8726.png)
is a parallelogram. Assume, without loss of generality, that this parallelogram is the one with smallest area out of all the parallelograms formed by two adjacent sides of the polygon. Now construct parallelograms 
,
,
,
,
with 
,
like in the figure.
such that 
,
,
,
,
,
,
share a height,![\[\frac{A_1A_5}{B_1B_5}=\frac{(WZYX)}{(A_2A_4A_6A_8)},\]](http://latex.artofproblemsolving.com/3/a/d/3add948e712a492ab02b46e578e895145c58798a.png)
where parentheses denote area.
So if we consider the area of the entire octagon, we have:
.
has to be greater or equal than the area of one of the parallelograms with sides 
)
)
or 
to 
to 
,
.
where the last inequality comes from defining 
