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 a triangle with incenter 
and intouch triangle 
. Let 
be the foot of the perpendicular from 
onto 
. Assume that 
, 
intersect the sides 
, 
in 
respectively. Finally, let the rays 
, 
meet the circumcircle of 
in 
, 
respectively. Prove that the tangent from to the incircle and the line 
meet on the circumcircle of . be an acute, non-isosceles triangle with orthocenter 
. Let 
be the reflections of over 
, respectively, and let 
be the reflections of 
over , respectively. Let 
be the circumcenter of triangle 
, and let 
be the orthocenter of triangle . Define 
as the center of the circle passing through the three projections of onto the lines 
. Prove that 
is parallel to 
.
![$P \in \mathbb R[x,y]$](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.
is a prime number such that its remainder divided by 8 is 3. Find all pairs of rational numbers 
that satisfy the following equation.
Note: consider 
and 
I doubt that the person still visits AoPS :sob: @Dhruv Jha I'm sry but I don't think that they're gonna respond
and 
Prove that
Where 
and 
satisfies the following recursion formulas: 
,
.
,![$P_{i}(x)\in \mathbb{R}[x], \forall i=\overline{1,n}.$](http://latex.artofproblemsolving.com/0/5/1/0514f85850830af9a9e1a4dd5e7a4e8e68b2dff9.png)
.
polynomial with only one root.
Prove that
Where 
Prove that
