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.
be the Fermat point of a 
. Prove that the Euler line of the triangles 
, 
, 
are concurrent and the point of concurrence is 
, the centroid of .
such that
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 
-digit number is divisible by 
, then the number that results from moving the ones digit to the beginning of the number is also a multiple of .
-digit number is divisible by , then the number that results from moving the ones digit to the beginning of the number is also a multiple of .
digit number is divisible by , then the number that results from moving the ones digit to the beginning of the number is also a multiple of .
digit number be 
, and let the number itself be 
for some integer 
. Now the new number is the same as 
. We just need to prove that this new expression is 
. Now 
(Since 
).
. So the whole expression is 
, and we're done.
.
and 
Prove that
Prove that
