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 
be positive real numbers such that 
. Find the minimum value of 
and 
share circumcircle 
and incircle 
so that points 
and 
occur in this order along . Let 
be the triangle formed by lines 
and 
and define triangles 
similarly. Furthermore, let 
and 
be the circumcircle and incircle of triangle , respectively, and define circles 
similarly. and 
and the two common external tangents to and 
are either concurrent or pairwise parallel.
, and define points 
and 
similarly. Prove that points 
, and are collinear.
be an acute-angled triangle with 
. Let 
be the orthocenter of triangle , and let 
be the midpoint of the side 
. Let 
be a point on the side 
and a point on the side 
such that 
and the points , , are on the same line. Prove that the line 
is perpendicular to the common chord of the circumscribed circles of triangle and triangle 
.
as the set of all numbers of the form 
for some nonnegative 
and 
. Find (with proof) all pairs 
such that 
and 
.
over 
, and prove that your solution(s) is/are the only one(s).
. At a maths camp, there are 
children, each of whom selects
tables, with children tables, with children
and 
.Prove that ![$$ a^2-ab+b^2 \geq \sqrt[3]{2} $$](http://latex.artofproblemsolving.com/7/9/8/798484c6b985ebbace2a3dde816f11a46635feba.png)
Let and 
.Prove that ![$$ 3\geq a^2+ab+b^2 \geq \sqrt[3]{4} $$](http://latex.artofproblemsolving.com/d/7/7/d77fbdc50a88395259e58d31413b9ac453e5dfd0.png)
of odd degree with real coefficients such that 
.
be non-negative real numbers such that 
Prove that
be an acute triangle with 
and 
. Let be the midpoint of the side . The circumcircle of triangle 
intersects the side again at 
and the circumcircle of triangle 
intersects the side again at 
. The point 
lies on the perpendicular bisector of the segment so that the points and 
lie on the same side of 
and 
. Prove that the circumcircles of triangles 
and 
are tangent.
Define a cell with 
property if after remove that cell from the board The board can be tile with 
tetromino. cell 
Note: tetromino can be rotated but not flipped
, 
is parallel to one of the sides of a polygon. We have 
diagonals, and only 
sides, which means that there is a side which is parallel to at least 
diagonals, thus these digonals are pairwise parallel, which means that they cannot share a common end. So these diagonals give us 
different enpoints. Together with our chosen side, this gives us different endpionts (obvoiusly these diagonals cannot share a common endpoint with our side), which are all vertex of our polygon. Thus one of our diagonals must be a side of the polygon ( impossible because of assumption about convexity)
and 
.
Prove that
Prove that
be a convex polygon with an even number of corners. Prove that there exists a diagonal 
which is not parallel to any side of the polygon.
