[/quote]
,
Prove that
Let 
Prove that
Prove that
be vertices of regular pentagon inscribed in a circle whose radius is 
and center is at 
.
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)
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 
.
![\[
\frac{|a|}{a}+\frac{|b|}{b}+\frac{|c|}{c}
\]](http://latex.artofproblemsolving.com/8/4/a/84a1a990f8bbde7670e9154dd9a46bb18c14d547.png)
For all non zero integers 
with an area of 
,
.
,
.
memorized but I'm sure there are people out there with more!
Y by
Another window has a design which is shown below. The whole window is a square; the
central section is made of plain glass; the outer section is coloured. The curved lines are
arcs of circles of the same radius as the side length of the square and centred on its
corners. What is the ratio of coloured to plain glass in the whole window? Give your
answer in the form 1 : ?, where ? is a decimal correct to 3 sf.
central section is made of plain glass; the outer section is coloured. The curved lines are
arcs of circles of the same radius as the side length of the square and centred on its
corners. What is the ratio of coloured to plain glass in the whole window? Give your
answer in the form 1 : ?, where ? is a decimal correct to 3 sf.
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