[/quote]
,
be vertices of regular pentagon inscribed in a circle whose radius is 
and center is at 
.
Prove that
are integers whose greatest common divisor is 1. Let 
be a set of integers with the following properties:
,
.
(not necessarily distinct), 
.
,
,
.
.
,
such that 
is divisible by 
,
.
be a quadrilateral circumscribed about a circle, whose interior and exterior angles are at least 60 degrees. Prove that![\[
\frac{1}{3}|AB^3 - AD^3| \le |BC^3 - CD^3| \le 3|AB^3 - AD^3|.
\]](http://latex.artofproblemsolving.com/5/5/e/55ee0fba8392ee81b7b37da45d99755ea6d6754a.png)
such that 
for all 
with parallel sides 
and 
,
on line 
outside segment 
inside segment 
such that 
.
the point of intersection of 
,
the point of intersection of 
be the midpoint of segment 
if and only if 
.
with an area of 
,
.
,
.
.
.
.
such that 
.![$$2 \sqrt[4]{\frac{a}{b+c-a}}\ge 2 +\frac{2a^2-b^2-c^2}{(a+b)(a+c)}.$$](http://latex.artofproblemsolving.com/f/0/7/f0731fbe5c83aedc8474975f930f7194c08cc070.png)
Y by Tumon2001, Adventure10, and 1 other user
Let 
, 
and 
be three points on a line (in this order).
For each circle
through the points and , let 
be one point of intersection of the perpendicular bisector of 
with the circle . Further, let 
be the second point of intersection of the line 
with .
Show that for each circle, the ratio of lengths 
is the same.
, 
and 
be three points on a line (in this order).For each circle
through the points and , let 
be one point of intersection of the perpendicular bisector of 
with the circle . Further, let 
be the second point of intersection of the line 
with .Show that for each circle
, the ratio of lengths 
is the same.
is tangent to 
and 
(
and from this also 
) So,by similarity of 
, 
,we get 
and 
As 
.Therefore 
Hence the ratio is fixed.
and 
be two circles passing through 
and 
be the perpendicular bisector of 
.
= 
and 
.
= 
and 
,
= 
and let 
= 
.
and let 
= 
.
and 
.
is cyclic.
+ 
= 
°
= 
°
is also cyclic.
= 
is a diameter of 
passes through 
= 
(angle bisector theorem) = a constant.
