AoPS Academy
. Let 
be an interior point such that 
, 
and 
. Find 
.
be real numbers satisfying 
. Find the maximum value of![\[P = \dfrac{a(b+c)}{a^2+2bc+3} + \dfrac{b(a+c) }{b^2+2ca +3 } + \dfrac{c(a+b)}{c^2+2ab+3}.\]](http://latex.artofproblemsolving.com/7/b/2/7b2ec0a87ca05d3cb29da888a4dfa73e10f238f8.png)
touches the sides 
at 
respectively. A circle through 
and 
encloses and intersects the line 
at points and 
. Prove that the midpoint of 
lies on the circumircle of 
.
touches the sides at respectively. A circle through and encloses and intersects the line at points and . Prove that the midpoint of lies on the circumircle of .
-centered):
touches the sides at 
respectively. Suppose that is a point on 
and let 
Prove that 
and animate on line 
Define 
Consider the following series of projective maps:
to 
through 
(
) to 
through the point at infinity along the pencil of lines perpendicular to 
(
) to 
through any point. (
) swapping 
and 
(
)
to through a well-known lemma. In order to show that this composition of maps is the identity map, it suffices to check that 
for 3 cases of 
Pick 
the first two are easy to check by the Iran Lemma, and the last is trivial.
. As 
, it suffices to show that 
. Let 
and 
; then because 
, 
which can be checked through algebraic expansion.
and now since 
we have by McLaurin that 
where 
is the midpoint of , hence 
is cyclic thus we are done 
Does EV theorem work?
be real numbers . Prove that
![$$-\frac{1}{18}\leq \frac{ab+a+b+3}{(a^2+3)(b^2+3)}\leq \frac{2+\sqrt[3] 2+\sqrt[3] {4}}{12} $$](http://latex.artofproblemsolving.com/9/9/9/9997397eb38cd4bceec2ff8c85951c78e2797d0f.png)
a
,
