Well, today is the last day of 2014 and we’re just going take a look at non-standard notation called Chinese Dumbass Notation. Actually this post is only a little description about it, since I have never used this one to prove a problem in math contest.
Chinese Dumbass Notation (CDN), is a way to write down three variable homogeneous expressions. It allows to keep like terms combined while not requiring any sort of symmetry from the expression.
CDN uses a triangle to hold the coefficients of a three variable symmetric inequality in a way that is easy to visualize. By example:
Here represents the coefficient of . In general, a degree expression is represented by a triangle with side length and the coefficient of is placed at Barycentric coordinates .
AM-GM and Schur Inequalities
The AM-GM inequality for two variables states that . In other words:
The Schur’s Inequality states that for all non-negative numbers . The prototypical application of Schur’s Inequality looks as follows:
As with AM-GM, the variables can be tweaked so that Schur’s Inequality gives us
Proof. We clear denominators to obtain that this is equivalent to
We’ll expand the left hand side as
We then expand the right hand side as
The inequality is thus equivalent to showing that the difference in non-negative. But the difference is
by Schur and AM-GM inequalities.
I wonder, how many people have used this one in math contest? it’s just curiosity. If you did it and want to share with me, please send me a mail. Maybe we can resolve some problems together. Anyway. The best wishes for everyone in 2015