AoPS Academy
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 
of a tetrahedron 
are tangent to the edge 
in the same point. Prove that the points of tangency of these incircles to the edges 
are concyclic.
and of a tetrahedron are tangent to the edge in the same point. Prove that the points of tangency of these incircles to the edges are concyclic.
at 
respectively. Then the lines 
and 
pass through the point on the line for which 
. Thus 
and the points 
are concyclic.
and 
: Prove that :
and 
Prove that
be real numbers such that 
Prove that
Find 
