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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:
be circumcircles of the cyclic pentagons 
The 
are oppositely similar, having equal angles. Let 
be circumcenter of the 
Because of the opposite similarity, 
is isogonal of 
with respect to the angle 
hence 
and similarly, 
Thus is the intersection of the C-altitude of the 
with the perpendicular bisector of 
passing through 
on the perpendicular bisector of 
(both 
), 
(both 
), 
is a parallelogram, so that the circumradius of the 
is equal to 
In exactly the same way, we can find that the circumradius of the 
with circumcenter 
is equal to 
If the bases 
and the circumradii 
of the 
are equal, the angles 
are also equal.
![\[
\begin{cases}
x - y z = 1,\\[2pt]
y - z x = 2,\\[2pt]
z - x y = 4.
\end{cases}
\]](http://latex.artofproblemsolving.com/b/2/d/b2d94e49491a71f86109103f985fe3f238cf4d95.png)
and 
Prove that
be real numbers such that 
Prove that
Find 
and 
lie on sides 
and 
of the convex quadrangle 
Let 
and 
be inscribed pentagons and 
Prove that 
then points 
are collinear and it is not difficult to verify that 
also holds true. 
then