[/quote]
,
such that: 
.
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 
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 
on 
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
Let 
be an acute nonisosceles triangle with incenter 
and 
is an arbitrary line tangent to 
at 
. The lines passes through , perpendicular to 
cut at 
respectively. Suppose that cuts 
at 
respectively. The lines through 
and respectively parallel to the internal bisectors of 
in triangle meet each other to define a triange 
. Prove that three lines 
are concurrent and 
is tangent to the circle 
be an acute nonisosceles triangle with incenter 
and 
is an arbitrary line tangent to 
at 
. The lines passes through , perpendicular to 
cut at 
respectively. Suppose that cuts 
at 
respectively. The lines through 
and respectively parallel to the internal bisectors of 
in triangle meet each other to define a triange 
. Prove that three lines 
are concurrent and 
is tangent to the circle 
This post has been edited 3 times. Last edited by parmenides51, Sep 13, 2020, 7:20 AM
.
be the reflection of 
are concyclic
at 
at 
.
Which leads to 
is tangent to 
both lies on the Simson line of 
,
.
.
,
,
