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 reals with sum 1. Show that for any 
-tuple of reals 
, there exists an -tuple of reals 
satisfying 
for all 
, where 
.
, 
, 
are pairwise distinct real numbers satisfying 
Prove that
Let , , are pairwise distinct real numbers satisfying 
Prove that
chessboard such that no two pawns attack each other?
+1 w
be a triangle with incenter 
, and let 
be a point on line 
satisfying 
. Let the excircle of triangle opposite the vertex 
be tangent to 
at 
. Define points 
on 
and 
on 
analogously, using the excircles opposite 
and 
, respectively.
is cyclic, then 
is tangent to the circumcircle of 
.
, consider the following parabola on the coordinate plane.
ranges over all positive real numbers, draw the domain of the set swept out by .
, Find the minimum of 
, Find the minimum of 
. Prove that
be a triangle with circumcircle 
and incenter and let 
be the midpoint of . The points , 
, 
are selected on sides , , such that 
, 
, and 
. Suppose that the circumcircle of 
intersects at a point 
other than . Prove that lines 
and 
meet on .
be pairwise different positive real numbers such that![\[a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}\]](http://latex.artofproblemsolving.com/8/a/e/8ae0bd1f55eb2fb0b7209b62739a9b98f462ee56.png)
is an integer for every 
Prove that 
such that 
is a perfect square.
be the circumcenter of triangle : Points and are chosen at sides 
and 
respectively such that 
and 
is inscribed quadrangle. Prove or disprove that is isosceles triangle.
and 
Prove that
Let 
Prove that
