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.
, define 
to be the least positive integer for which the following holds:
into 
disjoint subsets 
, all of the same size, let 
. Then there exist 
for which![\[\deg(P_i(x)-P_j(x))\geq \frac{n}{k}-f(n)\]](http://latex.artofproblemsolving.com/2/0/9/2092eb944c746d1449e7865f9da4e0065b75bdef.png)
so that 
for all ., one has 
.
+1 w
be an acute triangle with orthocenter 
and circumcircle 
. A line through intersects segments 
and 
at 
and 
, respectively. Let 
be the circumcenter of 
, and suppose line 
intersects again at a point 
. Prove that line 
and the line through perpendicular to 
meet on .
and the domain 
passing through the origin touch 
.
be a mobile point on the parabola . Let denote 
the distances between the point and the lines respectively. Find the coordinate of the point giving the minimum value of 
on a coordinate plane such that for all points 
over the domain , the inequality 
holds.
which are perfect squares and for which the equality holds:
.
be an acute triangle with 
. Let 
be its circumcircle, its orthocenter, and the foot of the altitude from . Let 
be the midpoint of 
. Let 
be the point on such that 
and let be the point on such that 
. Assume that the points , 
, , and are all different and lie on in this order.
and 
are tangent to each other. with 
, denote its incenter and midpoint of by 
, , respectively. Let 
be the reflected point of wrt 
. Let the lines 
and 
meet at 
. Suppose that 
. Prove that 
.
is a triangle in the plane. Find the locus of point 
for which 
form a triangle whose area is equal to one third of the area of triangle .
has circumcenter 
. The lines 
and 
intersect at 
, while 
and 
intersect at 
and 
and 
intersect at 
. Show that if the triangles and 
are similar(with vertices in that order), than is equilateral.
, where 
is a positive integer.
with exactly 
elements such that the sum of its elements is a prime number.
help in one of my classrooms, and one of the Halpers posted. Is this a glitch or a bug(see screenshot)?
