[/quote]
,
?
(ans- 
)
,
with center 
.
the midpoint of side 
intersects the circle 
.
be a point on the circle 
and 
.
.
satisfying 
Prove that 
When does equality hold?
where 
.
and 
are defined similarly. Line 
intersects 
at points 
,
.
,
.
,
,
,
are defined similarly. Prove that points 
.![\[\frac{1}{a^2+155}+\frac{1}{b^2+155}+\frac{1}{c^2+155}\le \frac{11}{780}\]](http://latex.artofproblemsolving.com/4/2/7/427e12cbfdbbba6f26407dbb5c2430aee26c15eb.png)
When does the equality hold ?
and 
Prove that
,
,
be non-negative real numbers, no two of which are zero. Prove that :
satisfying: 
particles on a circle situated at the vertices of a regular 
-
be the smallest number of collisions after which all particles return to their original positions. 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)
Y by Mango247
Let 
be a triangle and let 
be a point in the interior of the side 
. Let 
and 
be the incenters of the triangles 
and 
, respectively. Let 
be the closest point to 
on the line 
such that 
is perpendicular to 
. Prove that the distance 
is independent of the choice of .
be a triangle and let 
be a point in the interior of the side 
. Let 
and 
be the incenters of the triangles 
and 
, respectively. Let 
be the closest point to 
on the line 
such that 
is perpendicular to 
. Prove that the distance 
is independent of the choice of .
and 
the two incircles centered at 
,
the points of tangency from 
respectively and 
the points where 
the to contact points of the two circles with 
is a common tangent of 
, 
and 
which is evidently indepedent of 
