AoPS Academy
is polynomial with minimum degree such that 
for 
, find 
and 
such that 
and is to the left of , and replaces the pair 
by either 
or 
. Prove that she can perform only finitely many such iterations.
be non-negative real numbers such that 
. Find the maximum and minimum values of the expression![\[
P = \frac{a}{a^2 + 2} + \frac{b}{b^2 + 2} + \frac{c}{c^2 + 2}.
\]](http://latex.artofproblemsolving.com/d/b/e/dbea7db769c4996fc081ee75837e177b298cffcb.png)
be reals such that 
. Prove that
Let be reals such that 
. Prove that
be a permutation of 
such that there are exactly 
ordered pairs of integers 
satisfying 
and 
. How many possible exist?
The vertices of an 
gon are placed on the edges of a regular 
gon so that the perimeter of the gon is divided into equal parts. How does one choose these 
points in order to obtain the gon with
maximal area;
minimal area?
with side length 1, then clearly exactly one edge will have exactly 2 vertices of the n+1-gon, and WLOG let that side be 
. Note that all other edges have exactly one vertex. Suppose the vertices 
are on with 
closer to 
and let 
. Finding the min or max of the area is equivalent to finding the max or min of the set of triangles outside the n+1-gon but inside the n-gon, and the area of those set of triangles is exactly![\[(\frac12 \sin\alpha)(a(\frac{n}{n+1}-a)+(a-\frac{1}{n+1})(\frac{n-1}{n+1}-a)+\]](http://latex.artofproblemsolving.com/f/c/4/fc43be53503c6e8c56a0b59d7b410ca4d3b5fc24.png)
![\[(a-\frac{2}{n+1})(\frac{n-2}{n+1}-a)+...+(a-\frac{n-1}{n+1})(\frac{1}{n+1}-a)))
\]](http://latex.artofproblemsolving.com/c/1/0/c10441e65e4368c6cacbe6944b36b48a95123052.png)
where 
is the angle of the n-gon. Now we remove any constant terms or multiples since we are only looking for min and max, so this is equivalent to![\[\sum_{i=0}^{n-1}{(a-\frac{i}{n+1})(\frac{n-i}{n+1}-a)}
\]](http://latex.artofproblemsolving.com/b/f/d/bfd0e6fde1202d2018a0d4cd96ec94e49125675b.png)
![\[-na^2+\frac{a}{n+1}\sum_{i=0}^{n-1}{\frac{n-2i}{n+1}}
\]](http://latex.artofproblemsolving.com/a/a/f/aaf9ff29cf64fcf1293215a04384004efa9d3871.png)
![\[-na^2+\frac{n}{n+1}a
\]](http://latex.artofproblemsolving.com/a/e/a/aea2fa8af2de2f3f2e1fdcd87d4698476c19daab.png)
![\[-a^2+\frac{1}{n+1}a
\]](http://latex.artofproblemsolving.com/f/a/7/fa7f08be323407719ccc14fb1e6340998f34b989.png)
and this attains max when 
and min when 
since 
. Thus the minimum area is when 
(i.e. One of the edge lies on the center of the other edge) and maximum is when one of the vertex coincides with the other polygon's vertex.
such that 
.
Prove that :
Where 
and 
.
