AoPS Academy
for which 
divides 
.
and 
, 
cannot be a power of 
.
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)
such that 
for all 
be a triangle with incenter 
. Let 
be the point of intersection between the incircle and the side 
, the points 
and 
are in the rays 
and 
, respectively, such that 
and 
. Prove that 
.
, then the problem is clearly true. Assume otherwise. Let 
be the second intersection of 
with line (otherwise known as the foot of the 
-external bisector). This is our key point:
and 
.
(using directed angles), and 
by using the fact that the original information implies 
.
and 
. However, both fractions on the RHS are equal due to the external angle bisector theorem (similar to the internal angle bisector theorem). Thus , as desired.
be a triangle with incenter . Let be the point of intersection between the incircle and the side , the points is on the ray IB . Prove that: if and only if 
.
