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 real polonomial such that all roots are real and distinct. Prove that there is a rational number 
that all roots of 
are real numbers
be an acute angled triangle with orthocenter 
and 
. Let 
be any other point on the arc 
of the circumcircle of 
and let line 
intersect line 
at 
and let line 
intersect line 
at 
. Let the circumcircles of 
and intersect again at 
(
). Let the lines 
intersect the circumcircle of 
again at 
(
). Prove that the lines 
concur.
such that:
for any positive integer 
, the numbers 
are all distinct, and for any positive integers and 
, 
?
have altitudes 
circumcenter 
and orthocenter . Let 
be a point on line 
. The points 
are on lines 
respectively such that 
and 
. Prove that 
is perpendicular to 
., consider points 
on line 
such that 
are in that order and 
and 
. Similarly consider points 
on line , and 
on line . Prove that orthocenters of triangles 
and 
are equidistant from the circumcenter of .
be an acute triangle inscribed in its circumcircle 
, and let 
be its incircle. Let 
be the point where the 
incircle of triangle touches 
. Suppose line 
intersects segment 
at 
, and intersects line 
at 
. Let the line through 
perpendicular to intersect line 
at 
. Prove that: ![\[AI \parallel PA'.\]](http://latex.artofproblemsolving.com/0/2/6/0264e64b5edff4aca1e35037c5b0f86763c3b4e7.png)
whose corners are vertices of the 100-gon, so that are pairwise disjoint, and
has three corners of one color and one corner of the other color.
be positive integers and 
be prime. If 
is a perfect square, prove that 
is also a perfect square.
be real numbers satisfying 
Prove that 
and determine all cases of equality.
be the set of rational numbers. A function 
is called aquaesulian if the following property holds: for every 
,![\[ f(x+f(y)) = f(x) + y \quad \text{or} \quad f(f(x)+y) = x + f(y). \]](http://latex.artofproblemsolving.com/4/0/e/40e7d033daf1bd62c2bfd5d022eeaea3f860c5b3.png)
Show that there exists an integer 
such that for any aquaesulian function 
there are at most different rational numbers of the form 
for some rational number 
, and find the smallest possible value of .
lines lines are used as roads by the townsfolk. In the past, the roads in Linetown used to be two-way, but this often led to residents accidentally cycling back to where they started. roads one-way such that it is impossible for any resident to start at a house, follow the roads in the correct directions, and end up back at the original house. In how many ways can Turbo achieve this?
such that for all 
, 
be a fixed integer. There are 
beads on a circular necklace. You wish to paint the beads using 
colors, such that among any 
consecutive beads every color appears at least once. Find the largest value of for which this task is 
possible.
there exists positive integers 
such that
where 