[/quote]
,
and 
.
?
be a sequence of positive real numbers that satisfies 
and 
for every natural number 
.
for every natural number 
be positive real numbers such that : 
. Prove that : 
.
be the feet of the altitudes from 
to 
respectively.
and 
as the reflections of 
be the circumcircle of 
.
the second intersection of line 
with 
the intersection of ray 
with 
is the circumcenter of 
can be inscribed within a circle.
be the set of positive integers. Find all functions 
such that![\[ m^2 + f(n) \mid mf(m) +n \]](http://latex.artofproblemsolving.com/2/f/4/2f409d1de993f1af8fd839bb8e9f87a57e1b8608.png)
and 
(ans- 
)
,
with center 
intersects the circle 
and 
.
.
satisfying 
Prove that 
When does equality hold?
where 
intersects 
.
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 ?
Y by Adventure10, Mango247
Let 
be the midpoint of side 
of triangle 
(
), and let 
be the bisector of the angle 
. The line passing through perpendicular to intersects the side 
at the point 
. Prove that 
is equal to half the perimeter of triangle .
be the midpoint of side 
of triangle 
(
), and let 
be the bisector of the angle 
. The line passing through perpendicular to intersects the side 
at the point 
. Prove that 
is equal to half the perimeter of triangle .
so we need to prove that 
.
as the point on side 
(and hence 
)
i.e. that 
.
.
.
.
so 
hence 
is isosceles.
as required.
cut resp. 
at 
it s clear that 
are isoceles and 
where 
pass through the midpoint of 
are the midpoints of 
hence 
Then, by LoS and 
isosceles, 
Hence, 
