AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
that is not isosceles at 
, let 
be its incircle, which is tangent to 
at 
, respectively. The lines 
and 
intersect the line passing through and parallel to 
at 
and 
, respectively. The lines passing through 
and perpendicular to 
intersect 
and 
at 
and 
, respectively.
are collinear and that 
are concurrent.
be a prime number. Prove that there exists an integer 
with 
such that neither 
nor 
is divisible by 
.
such that: 
be an odd prime number. Prove that there exist sequence such that: 
with 
, let 
denote the incenter and 
the midpoint of side 
. The line through 
perpendicular to 
intersects the tangent from to the incircle (different from line ) at a point 
> Show that is tangent to the circumcircle of triangle 
.![$P \in \mathbb R[x,y]$](http://latex.artofproblemsolving.com/8/a/0/8a0576299e976c51607959fb68c7c742b2a49c6a.png)
with ![$P \not\in \mathbb R[x] \cup \mathbb R[y]$](http://latex.artofproblemsolving.com/a/f/7/af71ba8c643d5bd8a2928fa6714ccd91713bd641.png)
and 
homeomorphismes of 
, 
is an homoemorphisme too.
, such that 
and 
are both integers.
be real numbers such that 
Prove that 
Let be real numbers such that 
Prove that 
is finite (and of course is an infinite set since 
for 
). But then, by Kobayashi, the set of prime divisors of 
is infinite. But 
, whose prime divisors must belong to the original set, a contradiction.
, if 
then 
. Therefore, we must get a new prime divisor for each 
, proving the result. 
, but that's easy.)
Prove that
and 
Prove that
is defined by 
and 
Show that there exist infinitely many prime number that divide at least one number in this sequences
