and 
Find the maximum value of 
and 
on the segment 
such that 
. Consider point 
on 
such that 
. Point 
is the intersection of tangent line through point 
respectively . If the tangent through point 
intersect 
at 
. Line 
intersect 
. Prove that 
bisect 
and 
Find the maximum value of
,
,
,
.
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For 
represents 
.
Y by
Let 
be an acute-angled triangle with orthocenter 
. A line passing through intersects 
at points 
respectively so that 
. The bisector of 
intersects the circumcircle of the triangle 
at point 
. Show that 
bisects segment 
.
be an acute-angled triangle with orthocenter 
. A line passing through intersects 
at points 
respectively so that 
. The bisector of 
intersects the circumcircle of the triangle 
at point 
. Show that 
bisects segment 
.This post has been edited 1 time. Last edited by parmenides51, May 23, 2020, 7:37 AM
Reason: typo: replaced ''perimeter'' with ''the circumcircle of the triangle''
Reason: typo: replaced ''perimeter'' with ''the circumcircle of the triangle''
bisector and 
.
is diameter of 
,
and 
.
and 
from the parallels we have that 
and 
and 
parallelogram. Now see that 
and since 
which means 
and so 
.
on 
then 
is a parallelogram and the conclusion follows.
