AoPS Academy
and 
and the radius of the inscribed circle is 
. If and are natural numbers and 
, find the value of 
and .
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)
+1 w
be a rhombus. 
is a point on side 
and 
is a point on side 
such that 
. Prove that centroid of triangle 
lies on the segment 
on segment 
such that 
hence 
is parallelogram.
and 
.
and the problem is done.
and equal 
and 
coordinates for the rhombus. The problem then becomes trivial.
like in first proof; 
is common median of 
, hence they share the same centroid, which lies onto the median of 
, done.
such that : 
and 
be the midpoint of 
.
will be equidistant to 
.
