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 
is written on the blackboard. Every time, Carmela can erase the number 
on the black board and replace it with a new number 
,
is a perfect square. Prove or disprove that all positive integers 
can be written exactly once on the blackboard.
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)
Y by
Let
be the midpoint of side 
of triangle 
. Draw from the midpoint of the segment 
perpendicular on the side BC, and from the midpoint of the segment 
perpendicular on the side 
. Let's denote the intersection point of these perpendiculars by 
. Similarly, construct points 
and 
. Prove that triangles 
and are similar.
be the midpoint of side 
of triangle 
. Draw from the midpoint of the segment 
perpendicular on the side BC, and from the midpoint of the segment 
perpendicular on the side 
. Let's denote the intersection point of these perpendiculars by 
. Similarly, construct points 
and 
. Prove that triangles 
and are similar.This post has been edited 1 time. Last edited by parmenides51, Jun 21, 2022, 2:03 AM
