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 an equilateral triangle, and point 
on its circumcircle. Let 
and 
intersect at 
, 
and 
intersect at 
, and 
and 
intersect at 
. Prove that the area of 
is twice the area of 
.
+1 w
and 
be the roots of the equation 
where 
is an odd integer. Let 
is an integer
) = 1 .
denote the sum of all integers 
such that 
and exactly one of 
and 
is divisible by 
. Compute the remainder when is divided by 
.
, where and are relatively prime positive integers. Find 
.
is given in the plane. The incircle of triangle has center 
and is tangent to diagonal at 
. Let 
be the center of parallelogram . Show that 
and 
are parallel.
is valid while 
is not)
of positive integers such that 
is an integer.
are positive integers such that for every natural number the set of prime factors of 
are same. prove that 
.
square is dissected in to 9 rectangles by lines parallel to its sides such that all these rectangles have integer sides. Prove that there are always two congruent rectangles.
such that![\[f(x(x + f(y))) = (x + y)f(x),\]](http://latex.artofproblemsolving.com/b/0/6/b0668edacd78de23f7611f47a7972a2dd148a4a0.png)
for all 
.
.
All I can do now is hope and pray for HCSSiM. So far I'm 0 for 3 for summer programs lol
Prove that
Where 
