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such that for any 
:
such that 
for all 
be a triangle with circumcircle 
and let 
be the 
-excircle. Let 
and 
be the intersection points of and . Let 
and 
be the projections of onto the tangent lines to at and respectively. The tangent line at to the circumcircle of the triangle 
intersects the tangent line at to the circumcircle of the triangle 
at a point 
. Prove that 
.
be reals such that 
. Prove that
Let be reals such that 
. Prove that
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)
. Find the minimum and maximum values of 
+ 
+ 
Find the maximum positive integer 
such that for every subset 
with n elements, there always exist two elements a, b in T such that:
and 
meet in and 
and their common external tangents meet in 
. The line 
through meets and in the points and closer to . Let 
, 
, and let 
be such that 
. Show that 
remains constant when varies.
and meet in and and their common external tangents meet in . The line through meets and in the points and closer to . Let , , and let be such that . Show that remains constant when varies.
, midpoint 
and radius 
.
, midpoint 
and radius 
.
lies on the external tangent line.
and 
with 
and 
.
cuts in the point 
with 
. cuts in the point 
.
and 
.
cuts the x-axis in the fixed point 
.
is the midpoint of a circle, going through the point 
; radius of this circle 
.
if 
. to the line equals 
.
.
![\[
\begin{cases}
x - y z = 1,\\[2pt]
y - z x = 2,\\[2pt]
z - x y = 4.
\end{cases}
\]](http://latex.artofproblemsolving.com/b/2/d/b2d94e49491a71f86109103f985fe3f238cf4d95.png)
and 
Prove that
be real numbers such that 
Prove that
Find 
