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.

## Definition

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 $[x]$ represents the coefficient of $x$. In general, a degree $d$ expression is represented by a triangle with side length $d+1$ and the coefficient of $a^{\alpha}b^{\beta}c^{\gamma}$ is placed at Barycentric coordinates $(\alpha, \beta, \gamma)$.

## AM-GM and Schur Inequalities

• The AM-GM inequality for two variables states that $a^2 + b^2 \geq 2ab$. In other words: $% $

• The Schur’s Inequality states that $\sum_{sym} a^r(a-b)(a-c) \geq 0$ for all non-negative numbers $a, b, c, r$. 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 $\sum_{cyc}a^r(a^s-b^s)(a^s-c^2) \geq 0$

## Example

Prove that $\frac{bc}{2a+b+c} + \frac{ac}{a+2b+c} + \frac{ab}{a+b+2c} \leq \frac{1}{4}(a+b+c)$

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.

Well, all the information above is thanks to Mr. Brian Hamrick, this post is just to take a look at CDN. So, if you want to expand the knowledge about it, please visit: The Art of Dumbassing.

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