be a triangle with 
.
be the orthocenter of 
is the midpoint of 
and point 
lies on the side 
such that 
.
.
be a regular polygon, and let 
be its set of vertices. Each point in 
be an integer greater than 
.
,
.
-
such that
;
and 
,
and 
denote the product and sum of the digits of 
is differentiable and 
for all 
.
,
.
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 
.
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 
.
for all ![$x \in [0,1]$](http://latex.artofproblemsolving.com/d/3/4/d3423a3200d0ad7660fc4265d59d5598a53d551d.png)
,
for all 
.
increasing with
?
?
Y by TDVOLIMPTEAM
While I was thinking on some other geometry problem, a NT problem came to my mind. Despite some tries(which were mostly order), I could not find a way to solve the problem. As I searched, this problem has never been posted before. Here is the problem.
Find all positive integers
such that:
Moreover, I wonder if there is a way to solve the question in this variant:
Find all positive integers
such that:
Find all positive integers
such that:
Moreover, I wonder if there is a way to solve the question in this variant:
Find all positive integers
such that:
where 
.
.
.
