AoPS Academy
of positive integers with the following property: all of the terms of the sequence 
have a greatest common divisor 
, 
, 
be nonzero real numbers such that 
and 
. Find the value of
.
kawaii if it satisfies 
(
is the number of positive factors of , while 
is the number of integers not more than that are relatively prime with ). Find all that is kawaii.
be non-negative real numbers such that ![\[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]](http://latex.artofproblemsolving.com/b/9/e/b9e86e898c0536b7323a03611d5bdbf679caa710.png)
Prove that 
be the orthocenter of an acute-angled triangle 
and 
a point on the circumcenter of triangle . Prove that the Simson line of bisects the segment ![$[P H]$](http://latex.artofproblemsolving.com/5/8/7/587225c92de7ccb1a3837835001032884cf125b8.png)
.
