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, lian_the_noob12, Rounak_iitr
Two circles 
intersect each other at points 
. Let 
be a common tangent line of these two circles with 
and 
. An arbitrary point 
lies on 
. Line 
intersects 
for the second time at 
. Point 
lies on 
such that 
. Line 
intersects 
for the second time at 
. Prove that 
.
Proposed by Morteza Saghafian
intersect each other at points 
. Let 
be a common tangent line of these two circles with 
and 
. An arbitrary point 
lies on 
. Line 
intersects 
for the second time at 
. Point 
lies on 
such that 
. Line 
intersects 
for the second time at 
. Prove that 
.Proposed by Morteza Saghafian
This post has been edited 1 time. Last edited by parmenides51, Sep 20, 2018, 9:24 AM
Reason: Proposed
Reason: Proposed
.
.
hence 
.
and 
,
.
and it needs to be shown that 
but 
is a simple consequence of reim's theorem.
