[/quote]
,
be positive real numbers such that 
Prove that we have![\[(a+b)(b+c)(c+a)\geq 8.\]](http://latex.artofproblemsolving.com/2/a/6/2a63f19de634fce70ff4ebc02ad13d442a20c378.png)
such that: 
.
such that 
Prove that 
with the property that ![\[f(x-f(y))=f(f(x))-f(y)-1\]](http://latex.artofproblemsolving.com/f/2/5/f25cc1e8ae0be1fd02b347fd94be4fab88af1d46.png)
holds for all 
.
be a function such that
for all 
.
.
be a function such that
for all 
.
.
such that for all 
be the set of real numbers. Find all functions 
where there exist a real constant 
such that 
holds for all reals 
,
,
that satisfy 
.
with its center 
and one moving point 
is 
.
is a fixed point on 
intersect
,
intersects 
at 
.
when 
such that
and circumcenter 
the midpoints of 
,
.
with diameter 
,
at a point 
.
meets line 
at 
.
and 
intersect at a point 
on 
.
such that
for all 
)
Y by HamstPan38825, Adventure10
Point 
lies on one ray and points 
lie on the other ray of an angle with the vertex at 
such that 
lies between and 
. Let 
be the incenter of 
and 
be the center of the excircle of 
touching side 
. Prove that if 
, then the triangle 
is isosceles.
lies on one ray and points 
lie on the other ray of an angle with the vertex at 
such that 
lies between and 
. Let 
be the incenter of 
and 
be the center of the excircle of 
touching side 
. Prove that if 
, then the triangle 
is isosceles.
the 
and observe that 
is cyclic as by well-known angle formulas we have
and 
(and 
)
is angle bisector in 
and thus 
are collinear because they all lie on the angle bisector of 
it suffices to show that 
Then 
It suffices to show that 
Since 
and 
and 
this becomes 
which is true.
