be positive real numbers satisfying 
.![\[
3(x^{\frac{2}{3}} + y^{\frac{2}{3}}) \geq 4 + 2x^{\frac{1}{3}}y^{\frac{1}{3}}.
\]](http://latex.artofproblemsolving.com/f/3/c/f3cd114f909f388eacee142d3f90d07f6b149131.png)
with another positive integer 
,
is a perfect square. For example, if the number on the blackboard is 
,
,
,
,
.
,
and 
,
to be the unique polynomial 
such that we can write 
,
is a polynomial such that either 
or the degree of 
with real coefficients such that 
be non-negative real numbers such that 
Prove that
Y by
Given a triangle 
, in which 
. A tangent to the circumcircle of the triangle is drawn through the point 
. This tangent intersects the line 
at the point 
. On the extension of the side 
for the point mark the point 
so that 
. Let 
and 
be the midpoints of the segments 
and 
, respectively, and let 
belong to the segment , and 
. Prove that 
.
(Vyacheslav Yasinsky)
, in which 
. A tangent to the circumcircle of the triangle is drawn through the point 
. This tangent intersects the line 
at the point 
. On the extension of the side 
for the point mark the point 
so that 
. Let 
and 
be the midpoints of the segments 
and 
, respectively, and let 
belong to the segment , and 
. Prove that 
.(Vyacheslav Yasinsky)
