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.
and two fixed points 
on that circle, let 
be a moving point on such that 
is acute and scalene. Let 
be the midpoint of 
and let 
be the three heights of . In two rays 
, we pick respectively 
such that 
. Let 
be the intersection of 
and 
, and let 
be the second intersection of 
and 
.
always goes through a fixed point.
intersects at 
. In the tangent line through 
of 
, we pick 
such that 
. Let 
be the center of 
. Show that 
always goes through a fixed point.
where 
+1 w
where 
.
cannot be a perfect cube of a positive integer.
is given. Let be set of all 
segments of real line of type ![$[i, j]$](http://latex.artofproblemsolving.com/a/6/4/a644edd67caa59eaded765c417b6d9822f782ec2.png)
, where 
and 
are integers, 
. A subset 
is said to be valuable if the intersection of any two segments from 
is either empty, or is a segment of nonzero length belonging to . Find the number of valuable subsets of set .
with real coefficients obeying![\[P(x) P(x+1) = P(x^2 + x + 1)\]](http://latex.artofproblemsolving.com/b/e/f/beff3420fa8394f4f98c031d81eea6a5a3b8d28c.png)
for all real numbers 
.
, such that 
and 
.
such that 
is prime, 
with 
and ?
?
be the circumcircle of isosceles triangle 
(
). Points and 
lie on and respectively such that 
.
and intersect at 
. Prove that the tangents from and 
to the incircle of 
(different from ) are concurrent on .
, 
is the Gergon point and the incircle 
touch , 
, 
at , 
, 
respectively.
, 
, 
be 
, 
, 
respectively. in 
and then we get the circle 
in 
and then the circle 
in 
and then the circle 
, , are coaxial.
be non-negative real numbers such that 
Prove that
and![\[(x-y)[f(x)+f(y)]\leqslant f(x^2-y^2), \forall x,y \in \mathbb{R}\]](http://latex.artofproblemsolving.com/8/0/b/80bb1d7a992c9683a0c2a07b2febe6271f4e60f5.png)
such that 
such that:
.
.
