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.
such that the relation ![\[|z^{2n}+z^nw^n+w^{2n} | = 2^{2n}+2^n+1 \]](http://latex.artofproblemsolving.com/1/2/2/1229a609403904449c447225094b3d97e063efd4.png)
holds for all positive integers 
.
such that 
for reals 
.
such that 
. Prove that
with 
. Point 
lies inside the rectangle such that 
is equilateral. Given that 
and the midpoint of 
are on the same line, compute the length of 
.
, let 
denote the number of positive divisors of . Determine whether there exists a positive integer triple 
such that there are exactly 
positive integers 
not greater than 
that satisfies the following: the equation![\[ \tau(x) = \tau(y) = \tau(z) = \tau(ax + by + cz) = K \]](http://latex.artofproblemsolving.com/a/8/6/a86ab0dd1aacf681d5d6761a4da73face1647f48.png)
holds for some positive integers 
.
has an incenter 
Suppose 
. Let 
be the midpoint of 
. Suppose that 
. 
meets 
again at point 
. Let points 
and 
be such that is the midpoint of 
and is the midpoint of 
. Point 
lies on the plane such that 
is a parallelogram, and suppose the angle bisectors of 
and 
concur on 
.
and 
meet 
at 
and 
. Prove that 
.
children sit in a circle facing inward with 
, and each child starts with an arbitrary even number of pencils. Each minute, each child simultaneously passes exactly half of all of their pencils to the child to their right. Then, all children that have an odd number of pencils receive one more pencil.
checkerboard satisfy the following conditions:
checkers have been placed on the board.
, 
is the Gergon point and the incircle 
touch , 
, 
at 
, , 
respectively.
, 
, 
be 
, 
, 
respectively. in 
and then we get the circle 
in 
and then the circle 
in 
and then the circle 
, , are coaxial.
the power of with respect to 
(an circle). Prove that 
and 
are conjugate points with respect to 
and lie inside of 
then WLOG we assume that is outside of 
are the tangents from to 
has equal power WRT and 
is on their radical axis 
the polar of WRT and are conjugate points WRT
and are conjugate points with respect to 
lies on polar of with respect to and are conjugate points with respect to 
the circle of diameter is orthogonal to 
(Any user, too)
the radius of and 
the center. If 
denotes the vector from . and are conjugate points with respect to the circle of diameter is orthogonal to i.e. 
and 
Prove that
and 
.
Equality holds when 
or 
or 
and 
: Prove that :
such that ![\[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]](http://latex.artofproblemsolving.com/0/6/2/0623b7c328af6d088cfc5dd288afe3e17a94ff29.png)
for all reals 
and 
.
