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 positive integers and let 
be a non-increasing list of positive real numbers. Prove that there exists 
sets 
which partition the set 
such that 
, 
be the orthocenter of it and 
be any point on the side 
. The points 
are on the segments 
, respectively, such that the points 
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
. Prove that the points 
and lie on the same line.
+1 w
, 
, and 
, are marked on the plane. You have a pencil and a double-edged ruler of width 
. Using them, you may perform the following operations:
and 
are marked, draw the line connecting and .
and 
are drawn, mark the intersection of and .
is drawn, draw a line parallel to that is at distance away from (note that two such lines may be drawn).
using these operations.
such that for all 
,![\[f(x+yf(x))+y = xy + f(x+y).\]](http://latex.artofproblemsolving.com/c/f/3/cf3d20a041c27244e90876119b4568b7a3e13c03.png)
, define 
to be the set of all triplets 
of natural numbers, with 
odd and 
, such that 
and divides 
. Find all values of for which is finite.
be positive real numbers . Prove that
and 
in the plane, draws the perpendicular bisector of 
. Show that if three lines forming a triangle are drawn, Janabel can mark the orthocenter of the triangle using this device, a pencil, and no other tools. with centroid 
. Let 
be the power of w.r.t circumcircle of and 
be the power of w.r.t incircle of . prove that 
.
satisfy the equality
Find the maximum possible value of 
.
, be positive integers. Julia and Florian play a game on a 
board. Julia cells. such that Florian has a strategy to always deduce the entire tiling.
be a triangle with 
, and let 
be its -excenter. Let 
be the projection of to 
. Let 
be the intersection of 
and , and let 
be the points on 
, respectively, such that 
are on a line perpendicular to . Let the circumcircle of 
intersect again at . Suppose that the tangent of the circumcircle of at intersects at 
, and the segment 
intersects the circumcircle of at . Show that 
.
in prime and 
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lies on the perpendicular bisector of 
, triangle 
is isosceles. Therefore, 
. Since 
, lies on 
. It is well known that 
lies on this circle too and 
is a diameter of this circle, so 
. Because 
and are perpendicular to the same line, they must be parallel.
