![$$(\sqrt{x}+1)[2-(x-6)\sqrt{x-3}]=x+8$$](http://latex.artofproblemsolving.com/b/1/d/b1d283f2e7b8e574b6c0ff72236c52a39d3da9e8.png)
and 
Prove that
passing through 
and 
at points 
and 
respectively. Let 
be the point such that 
is a parallelogram. Suppose that 
.
such that![\[f(x+y)+f(y+z)+f(z+x)\ge 3f(x+2y+3z)\]](http://latex.artofproblemsolving.com/9/8/a/98ab64d17bc7d6bf7533b0874177fc963405c1d8.png)
.
Y by Adventure10, Mango247
Let 
and 
be the altitudes of an acute triangle 
with 
on 
and 
on 
. Let 
be the point of intersection of and . Take any line 
through with 
on and 
on . Suppose 
and 
are located on and respectively. such that 
is perpendicular to and 
is perpendicular to . Prove that 
is parallel to 
.
and 
be the altitudes of an acute triangle 
with 
on 
and 
on 
. Let 
be the point of intersection of and . Take any line 
through with 
on and 
on . Suppose 
and 
are located on and respectively. such that 
is perpendicular to and 
is perpendicular to . Prove that 
is parallel to 
.
and 
we have 
.
.
and 
are cyclic we have 
and 
.
we have 
.
.
.
and 
are similar so :
and 
are similar so:
,
and 
are cyclic quadrilaterals
