AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
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)
matrix whose entries come from the set 
is called a silver matrix if, for each 
, the 
-th row and the -th column together contain all elements of 
. Show that:
;
.
such that: 
and 
be on sides 
and 
respectively in 
such that 
. Let 
be the circumcenter of 
and let 
. Prove that 
![\[ a_n=2^n+3^n+6^n -1,\ n\geq 1. \]](http://latex.artofproblemsolving.com/6/4/9/6499d8f992f3e72765cab1f3afe15df99d01b12a.png)
that satisfy the equation
be a triangle with its 9-point center 
and excentral triangle 
. Denote the tangency points of the 
-excircle with sides 
, 
, and 
as 
, respectively. Similarly, define 
and 
for the 
- and 
-excircles.
, 
, and 
. Let 
be the radical center of the circles 
, 
, and 
.
, 
, 
are concurrent. be a triangle with circumcenter 
, incenter 
and incentral triangle 
. Let the line 
intersect 
again at 
. Similarly, define 
and 
.
and 
be the 9-point centers of and 
, respectively.
, 
are collinear. be a triangle, and let 
be an isogonal conjugate pair. Suppose the line through 
and perpendicular to 
intersects 
again at 
. Similarly, define 
. Suppose the line through 
and perpendicular to 
intersects 
again at 
. Similarly, define 
.
and 
be the orthocenters of 
and 
, respectively. Define 
and 
. Let 
and 
be the isogonal conjugates of and 
with respect to and , respectively.
are concurrent.
with 
. Prove that a point 
lies on Euler line of iff 
be a right triangle at 
, and let 
be the angle bisector of 
. From point , draw 
at 
. Extend to meet 
at point 
. Prove that:
be positive real numbers such that : 
. Prove that : 
is a inscribed quadrilateral.
is the intersection point of its diagonals.
is its circumcenter.
is the circumcircle of 
.
is the circumcircle of 
.
is the midpoint of arc 
on who doesn't contain .
is the midpoint of arc 
on who doesn't contain .
are collinear.
is a inscribed quadrilateral. is the intersection point of its diagonals. is its circumcenter. is the circumcircle of . is the circumcircle of . is the midpoint of arc on who doesn't contain . is the midpoint of arc on who doesn't contain . are collinear.
cuts 
at 
.
is the intersection of the two tangencies of (
) through 
.We will show that 
are colinear.
cuts 
at 
respectively and let be the intersection of .
are concurrent so by Menelaus theorem, we get (
)=-1 so 
=-1, which means (
)=-1.So 
are conjugate. is the polar of wrt (
) and it passes 
so the polar of wrt (), which passes 
, also passes .It leads to the fact that are colinear. wrt (), it is easy to get lies on the radical axis of 
and (
).
is perpendicular to 
.
is perpendicular to , so 
are colinear.(Q.E.D) 
ia a tangential quadrilateral and, as per Newton, its diagonals 
and the lines connecting the tangency points of the opposite sides 
and 
, i.e. 
are concurrent.
with an inversion of center and then with an inversion of center 
and you get the classic figure for harmonic points 
and 
, notice that is the intersection of the tangents from 
to and 
is the intersection of the tangents from 
to , now take polars w.r.t. so we want 
to be concurrenr but this holds by Brokard since 
and 
so they meet at and by brokard 
so they are concurrent at , thus we are done 
and 
. Prove that 
and 
.
