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 positive integer. Find, in terms of , all polynomials 
with integer coefficients such that for every integer 
, there exists an integer 
such that 
.
.
, angle 
. Point 
lies on 
such that 
. Line parallel to 
passing trough intersects 
at 
. Prove that 
.
such that
, prove/disprove the following claim: :maybe:
be a convex cyclic quadrilateral which is not a kite, but whose diagonals are perpendicular and meet at 
. Denote by 
and 
the midpoints of 
and 
. Rays 
and 
meet 
and 
at 
and 
, respectively. Prove that there exists a point 
, lying outside quadrilateral , such that
bisects both angles 
, 
, and
.
be the circumcircle of triangle 
, and let 
be the tangent line of 
passing 
. Let 
be the points each on side 
such that 
. Line 
meets at points 
. The line parallel to passing 
meets at , the line parallel to 
passing meets at 
. Prove that there exists a circle passing four points 
and tangent to line 
.
be an integer, and let 
be positive integers. Show that there exist positive integers 
satisfying the following three conditions:
for 
the remainders of on division by 
are pairwise different; and
denotes the integer part of real number 
, that is, the largest integer that does not exceed .)
touch 
at 
, respectively .
touch at 
, respectively .
the midpoint of 
is the Incenter 
of 
.
is the perpendicular bisector of 
, is the image of under homothety 
the radius of is twice the radius of .
be a circumcenter of an acute-angled triangle . Consider two circles 
and 
inscribed in the angle 
in such way that is tangent from the outside to the arc 
of a circle circumscribed about the triangle , and the circle is tangent internally to a circumcircle of triangle . Prove that the radius of is twice the radius .
![\[
\text{Find all prime numbers } (p, q) \text{ such that } p^2 + 3pq + q^2 \text{ is a fifth power of an integer.}
\]](http://latex.artofproblemsolving.com/c/f/8/cf855cb087102024a50d41f49c67e7fd1ef22767.png)
