AoPS Academy
be an acute-angled triangle with 
. Let 
be the orthocenter of triangle 
, and let 
be the midpoint of the side 
. Let 
be a point on the side 
and 
a point on the side 
such that 
and the points , , are on the same line. Prove that the line 
is perpendicular to the common chord of the circumscribed circles of triangle and triangle 
.
is differentiable and 
for all 
. Show that for some 
, 
.
such that 
Edit: Proposed by Marian Vasile (congrats!).
.
be a continuous function defined on ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
with 
and 
. We define the point 
to be the projection of point 
on the x-axis. Prove that there exist points 
such that 
is a square.
![$f : [0,1] \longrightarrow \mathbb{R}$](http://latex.artofproblemsolving.com/9/a/4/9a4b7997c35990f839742d29866122791ade8b7c.png)
is differentiable with 
. If 
for all ![$x \in [0,1]$](http://latex.artofproblemsolving.com/d/3/4/d3423a3200d0ad7660fc4265d59d5598a53d551d.png)
, then show that 
for all 
.
increasing with
? increasing with
?
. Prove that
and 
. Prove that
Thank lbh_qys.
then number of positive integral divisors of 
,
,
,
,
