[/quote]
,
is a positive integer. Call all positive divisors of 
and 
beautiful divisors and difference of any 
such that 
.
is defined by 
and for all positive integers 
,
.
,
such that 
is a multiple of 
.
be a subset of 
.
.
be a positive integer. Determine all sequences 
of positive integers such that
for all positive integers 
.
be an odd prime. Define 
over the positive integers as follows:
where 
such that 
.
such that 
and 
is a perfect square.
be the nonnegative integers in 
where 
.
.
.
where 
,
?
where 
,
satisfying :![\[n^{2p}=m^2+n^2+p+1\]](http://latex.artofproblemsolving.com/7/4/e/74ee592cb8b3f3a3996ea34f40b78fde9d3672fc.png)
and 
such that 
divides 
.
Y by
In the right triangle 
(
) on the sides 
, 
and 
, the points 
and 
different from the vertices are chosen, respectively, so that the equality 
takes place. Let the points 
and 
be the feet of the perpendiculars drawn from the points 
and 
respectively on the line 
. Prove that 
.
(
) on the sides 
, 
and 
, the points 
and 
different from the vertices are chosen, respectively, so that the equality 
takes place. Let the points 
and 
be the feet of the perpendiculars drawn from the points 
and 
respectively on the line 
. Prove that 
.
