be a quadrilateral that has both an incircle and a circumcircle, with radii 
and 
respectively. If the diagonals 
and 
are perpendicular, find the area of 
be a tower of number fields. Let 
be a prime of 
let 
and let 
be the residue class degree of 
in 
that satisfy the following equation for all real numbers 
:![\[
f\big(f(x - y)\big) = f(x) - f(y).
\]](http://latex.artofproblemsolving.com/c/3/c/c3c423f0ceb2fcb91064c5de218ec88f69c95e6d.png)
be a function satisfying![\[
f(f(x)) = x^2 - 6x + 6
\quad \text{for all} \quad x \in \mathbb{Z}.
\]](http://latex.artofproblemsolving.com/8/a/e/8aebed1e260316d675ca7d104d83fabcd8b5717e.png)
Given that![\[
f(i) < f(i+1) \quad \text{for} \quad i = 0, 1, 2, 3, 4, 5,
\]](http://latex.artofproblemsolving.com/9/f/9/9f92574f9fe14e2e30c3e51e194c661fbd4ece07.png)
find the value of![\[
f(0) + f(1) + f(2) + \cdots + f(6).
\]](http://latex.artofproblemsolving.com/4/0/7/4075d7efa78dda4c46a5cf764fb8852e8c0fb09b.png)
Y by Adventure10, Mango247, GeoKing
Two disjoint circles 
and 
are given in the plane. Point 
is on circle 
and 
touch the circle 
at 
respectively. Find the loci of the incenter and orthocenter of triangle 
.
and 
are given in the plane. Point 
is on circle 
and 
touch the circle 
at 
respectively. Find the loci of the incenter and orthocenter of triangle 
.This post has been edited 1 time. Last edited by littletush, Feb 29, 2012, 3:24 AM
for this use inversion with center 
