AoPS Academy
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, 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 non-negative real variables with 
Prove that
Equality holds iff 
or 
where 
be a convex quadrilateral such that diagonals 
and 
intersect at right angles, and let 
be their intersection. Prove that the reflections of across 
are concyclic.
we are given :
and one side length of the triangle equals 
, then under these conditions find the maximum area of the triangle 
.
be a trapezoid with parallel sides 
and 
, where 
. A circle passing through 
and 
intersects 
again at 
respectively. Prove that 
are concurrent.
in it. Prove that there are an infinite number of perfect cubes in this progression. (A perfect cube is an integer of the form 
, where 
is an integer. For example, 
, 
and are perfect cubes.)
there are elements in an arithmetic progression in the order 
such that 
.
days later (but perhaps earlier, as well). Hedvig received the trophy today. Determine all integers 
for which we can be absolutely certain that she cannot receive the trophy again in 
days, given the above information.
such that 
. Prove that 
are positive integers pairwise distinct and is also a positive integer 
.![$\sqrt[n]{p},\sqrt[n]{q},\sqrt[n]{r}$](http://latex.artofproblemsolving.com/3/1/3/3133b60c921168020d149a9d28ef69b756a704d9.png)
form terms of a same arithmetic progression.
and 
. Prove that
Let 
be reals such that 
. Prove that
of the triangle (
) and it's circumscribed circle intersect at point 
. The point 
lies on the segment 
is chosen so that 
. Prove that the lines , 
and 
intersect at one point if and only if 
.
of the triangle () and it's circumscribed circle intersect at point . The point lies on the segment is chosen so that . Prove that the lines , and intersect at one point if and only if .
?
and 
Prove that
Let 
Prove that
Prove that
