USA ELMO Shortlist 2013

v_Enhance wrote: Let $a, b, c$ be positive reals with $a^{2014}+b^{2014}+c^{2014}+abc=4$. Prove that \[ \frac{a^{2013}+b^{2013}-c}{c^{2013}} + \frac{b^{2013}+c^{2013}-a}{a^{2013}} + \frac{c^{2013}+a^{2013}-b}{b^{2013}} \ge a^{2012}+b^{2012}+c^{2012}. \]Proposed by David Stoner Dear v_Enhance, $ a^{2013} + b^{2013} + c^{2013} + abc = 4,$ or $ a^{2014}+b^{2014}+c^{2014}+abc=4?$

For those two conditions, the inequality is true.

You are welcome to show us your solution~~

Ok, I post my solution. First, notice that there are two numbers $ a,b $ such that $ a \ge 1, b \le 1 $ ((Otherwise, $ a^{2014}+b^{2014}+c^{2014}+abc \neq 4 $ So, $ (a^{2013}-1)(b^{2013}-1) \le 0 \Rightarrow a^{2013}+b^{2013} \ge (ab)^{2013}+1 $ Similarly, $ \frac{1}{a^{2013}}+\frac{1}{b^{2013}} \ge \frac{1}{(ab)^{2013}}+1 $ Thus, we have the following inequality : $ (a^{2013}+b^{2013}+c^{2013}-1)(\frac{1}{a^{2013}}+\frac{1}{b^{2013}}+\frac{1}{c^{2013}}-1) \ge 4 $ Therefore, $ \frac{b^{2013}+c^{2013}}{a^{2013}}+\frac{c^{2013}+a^{2013}}{b^{2013}} +\frac{a^{2013}+b^{2013}}{c^{2013}} $ $ \ge a^{2013}+b^{2013}+c^{2013}+\frac{1}{a^{2013}}+\frac{1}{b^{2013}}+\frac{1}{c^{2013}} $ $ \ge a^{2012}+b^{2012}+c^{2012}+\frac{1}{a^{2012}}+\frac{1}{b^{2012}}+\frac{1}{c^{2012}} $, as desired. ((Notice that $ f(x)=x+\frac{1}{x} $ is a decreasing function on the interval $ [0,1] $