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 : 
, 
. Extend segment 
through 
to 
. Points 
and 
lie on lines and , respectively, such that 
and 
. Point 
lies on the circumcircle of triangle such that 
. Prove that 
.
, but only 1 works, the midpoint of arc . So we define to be the midpoint of arc instead, and show that 
. Let 
. Then 
is a straight line perpendicular to 
and 
. Now 
, thus 
, which implies that .
and 
. Prove that 
and 
.
