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.
. Prove that
Let 
. Prove that
Let 
. Prove that
Let 
. Prove that
![$P(x) \in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/4/5/3/453a624c3b002c1b0e78e0023b24dd22ddd03557.png)
such that:
with 
is an 
th prime and 
is an arbitrary positive integer.
be a positive integer. We say that a positive integer 
is -good if 
is divisible by 
for all positive integers 
with 
. Suppose is a positive integer such that is -good, but 
is not -good. Prove that 
is prime.
that verfy the following equation :
let 
and 
be the circumcenter and the orthocenter. The point 
is the reflection of 
with respect to 
. Assume that is not on the same side of 
as . Points 
lie on 
respectively such that 
. Let 
be the intersection point of 
. Prove that 
a grid of 9*9, we choose a little square in of this grid three times, we can choose three times the same.
be the incenter of triangle 
. Let 
, 
, and 
be the orthocenters of triangles 
, 
, and 
, respectively. Prove that the lines through , , and , parallel to 
, 
, and 
, respectively, are concurrent.
of acute triangle is 
. The ray drawn from through the circumcenter intersects at 
. Let the midpoint of 
be 
. Define 
, 
, 
, 
, 
, 
similarly. Prove that 
, 
, 
are concurrent.
be real numbers such that 
. Find the value of 
and find at least one triplet 
that satisfy those conditions.
defined in the set of rational numbers and taking their values in the same set such that the equation 
holds for all rational numbers 
and 
.
and 
are unnecessary; we only need points 
on 
such that 
is cyclic and 
tangent to 
also touches 
becomes C-excenter of 
are on circle with diameter 
Angle chase gives 
Likewise 
hence 
are concyclic.
,
.
is the inverse of function 
and thus, they might intersect that 
axis (if they do, we are done). Thus: 
is the only solution.
.
Let 
.
be the altitudes and 
be the angle bissector of acute non-isosceles triangle 
Denote by 
the incentres of triangles 
respectively. Prove that points 
and 
lie on the same circle.
