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and 
, where is not the zero polynomial, we define 
to be the unique polynomial 
such that we can write 
, where 
is a polynomial such that either 
or the degree of is less than the degree of . Find all polynomials 
with real coefficients such that 
be a triangle.Find the maximum of 
is written on the blackboard. Every time, Carmela can erase the number 
on the black board and replace it with a new number 
, if and only if 
is a perfect square. Prove or disprove that all positive integers 
can be written exactly once on the blackboard.
be positive real numbers satisfying 
. Prove that![\[
3(x^{\frac{2}{3}} + y^{\frac{2}{3}}) \geq 4 + 2x^{\frac{1}{3}}y^{\frac{1}{3}}.
\]](http://latex.artofproblemsolving.com/f/3/c/f3cd114f909f388eacee142d3f90d07f6b149131.png)
and 
lie on the hypotenuse 
of a right triangle . Prove that orthocenters the triangles 
and 
coincide if and only if 
and lie on the hypotenuse of a right triangle . Prove that orthocenters the triangles and coincide if and only if 
