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be a positive integer. Given a sequence of nonnegative real numbers 
we define the transformed sequence 
as follows: the number 
is the greatest possible value of the average of consecutive terms of the sequence that contain 
. For example, the transformed sequence of 
is 
.
, the number of such that 
is less than or equal to 
.
holds.
be positive real numbers. Prove that:![$$\sqrt[3]{a^3+b^3}+\sqrt[3]{b^3+c^3}+\sqrt[3]{c^3+a^3}\geq \sqrt[3]{2}(a+b+c)$$](http://latex.artofproblemsolving.com/2/6/e/26ebcdae35153bee73abbcfaf4a767d3a32b8912.png)
, let 
be all the positive integers smaller than that are coprime to . Find all 
such that
for all 
.
is written on the board, with 
linear factors on each side. What is the least possible value of 
for which it is possible to erase exactly of these 
linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?
and 
be interior points of sides 
and 
of a triangle 
, respectively, sonthat the line 
intersects the extension of 
at 
. Prove that 
is similar to the triangle whose vertices are the circumcenters of 
, 
and 
.
and be interior points of sides and of a triangle , respectively, sonthat the line intersects the extension of at . Prove that is similar to the triangle whose vertices are the circumcenters of , and .
be the miquel point of 
, then using directed angles, 
by radical axis, and symmetric solutions for the angle 
and we are done.
