be a function such that for every regular 
-
we have 
.
for all reals 
.
be a triangle 
)
are the altitudes of the triangle. The bisector of 
intersects 
at 
be a point such that 
and 
.
passes through the midpoint of 
.
& 
.
with 
and at most one of them being zero, show that![\[\frac{1}{\max(a^2,b)}+\frac{1}{\max(b^2,c)}+\frac{1}{\max(c^2,a)}\geq\frac{27}{4}\]](http://latex.artofproblemsolving.com/9/e/1/9e14a9ef2d1a0a98006bcf716fa33c3d9ea29bd5.png)
Y by Adventure10, Mango247
Consider a trihedral angle 
with 
and 
. Let 
denote the dihedral angles at edges 
respectively.
Prove that 
Show that 
if and only if 
Assume that 
. Let 
be a fixed point such that 
and let 
be variable points on 
respectively. Prove that 
is constant and find the locus of the incenter of 
.
with 
and 
. Let 
denote the dihedral angles at edges 
respectively.
Prove that 
Show that 
if and only if 
Assume that 
. Let 
be a fixed point such that 
and let 
be variable points on 
respectively. Prove that 
is constant and find the locus of the incenter of 
.
with 
,
,
,
,
be the dihedral angles at 
,
,
respectively.
on 
and let 
be the projection of 
is perpendicular to 
is perpendicular to 
