[/quote]
,
natural numbers around the perimeter of a circle. She wants the 
Can she accomplish this?
be a positive integer. There are two numbers written on a blackboard, one red and one blue. Initially, both are 0. We define the following procedure: at each step, we choose a nonnegative integer 
(not necessarily distinct from the previously chosen ones), and, if the red and blue numbers are 
and 
respectively, we replace them with 
and 
,
.
,
with incenter 
,
,
.
be the orthocenter of 
and let 
.
be the point on 
and the line 
meets 
.
and parallel to 
bisects 
.
intersects 
at 
.
and 
are defined similarly. Line 
intersects 
at points 
,
.
,
.
,
,
,
are defined similarly. Prove that points 
and 
. Prove that
. Prove that 
and 
. Prove that
Where 
![$ a,b\in [0,1] $](http://latex.artofproblemsolving.com/9/7/5/975edfc0cf5316d78283b313c6b5cfbdd7b92b6e.png)
. Prove that
Where 
Where 
where 
are lengths of parallel sides and 
are other two sides.
be real numbers satisfying 
and ![\[a_{i+1} - 2a_i + a_{i-1} = a^2_i\]](http://latex.artofproblemsolving.com/c/d/8/cd878e4ac230853182da75c54dcbf7be05b305bc.png)
for 
Prove that 
for 
Y by Mango247
Let 
be an inscribed quadrilateral , such that the diameter of the circumcircle 
has length 
units. The diagonals and 
intersects at point 
and the length of 
is 
units. Let the line 
be the tangent line of the circle at point 
. Extensions of sides 
and 
intersect the line at the points 
and 
respectively. Find the value of 
.
be an inscribed quadrilateral , such that the diameter of the circumcircle 
has length 
units. The diagonals and 
intersects at point 
and the length of 
is 
units. Let the line 
be the tangent line of the circle at point 
. Extensions of sides 
and 
intersect the line at the points 
and 
respectively. Find the value of 
.
