Determine 
such that 
Determine 
be circumcenter, orthocenter and A-HM point of triangle 
.
intersect 
at 
,
be the triangle formed by 
.
is tangent to 
of natural numbers such that for every pair of natural numbers 
and 
,![\[
\frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2
\]](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 
Y by aidan0626
A group of 
children sit in a circle facing inward with 
, and each child starts with an arbitrary even number of pencils. Each minute, each child simultaneously passes exactly half of all of their pencils to the child to their right. Then, all children that have an odd number of pencils receive one more pencil.
Prove that after a finite amount of time, the children will all have the same number of pencils.
I do not know the source.
children sit in a circle facing inward with 
, and each child starts with an arbitrary even number of pencils. Each minute, each child simultaneously passes exactly half of all of their pencils to the child to their right. Then, all children that have an odd number of pencils receive one more pencil.Prove that after a finite amount of time, the children will all have the same number of pencils.
I do not know the source.
