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be circumcenter, orthocenter and A-HM point of triangle 
. Let 
intersect 
at 
, respectively. Let 
be the triangle formed by 
. Prove that 
is tangent to 
of natural numbers such that for every pair of natural numbers 
and 
, the following inequality holds:![\[
\frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2
\]](http://latex.artofproblemsolving.com/1/6/7/167679c1707b957d87311298ea5b72347a9bdc45.png)
are given. Prove that one can only use straightedge (which draws a straight line passing two drawn points) and compass (which draws a circle with a chosen radius equal to the distance of two drawn points and centered at a drawn points) to construct a line and two points on it with length 
in only 
steps (Namely, the total number of circles and straight lines drawn is at most .)
, let 
denote the sum of squares of digits of . Prove that the sequence 
is eventually periodic.
we have that 
. But when 
, then 
, so from some point all elements in this sequence will be smaller than 
. For each 
sequence will be unique, which means that if some value in our sequence repeat, then whole sequence will have to start to repeat. We know that from some points all elements are in interval ![$[1,1000]$](http://latex.artofproblemsolving.com/a/b/b/abb6453846b59c5e5c88a4dcce4f6a03e915c347.png)
, so problem follows.
Determine 
such that 
Determine 
