# Cyclic Inequality with Square Roots And Absolute Values

### Problem

### Solution

By the AM-GM inequality, $\displaystyle a+\frac{1}{3}\ge\frac{2\sqrt{a}}{\sqrt{3}},\,$ so that $\displaystyle \frac{\sqrt{3}}{2}(a+1)\ge\sqrt{a}+\frac{1}{\sqrt{3}},\,$ and, subsequently,

$\displaystyle \prod_{cycl}\frac{1}{2\sqrt{2}}(a+1)\ge\prod_{cycl}\frac{1}{\sqrt{6}}\left(\sqrt{a}+\frac{1}{\sqrt{3}}\right).$

Suffice it to show that

$\displaystyle \prod_{cycl}\left(\sqrt{a-a^2}+\frac{1}{2\sqrt{2}}|3a-1|\right)^2\ge\prod_{cycl}\frac{1}{8}(a+1)^2.$

But

$\displaystyle \begin{align} \left(\sqrt{a-a^2}+\frac{1}{2\sqrt{2}}|3a-1|\right)^2&=\frac{1}{8}(a+1)^2+\frac{1}{\sqrt{2}}|3a-1|\sqrt{a-a^2}\\ &\ge\frac{1}{8}(a+1)^2, \end{align}$

completing the proof.

### Acknowledgment

Leo Guigiuc has kindly posted this problem at the CutTheKnotMath facebook page, with a solution of his. The problem is by Kunihiko Chikaya.

### Inequalities with the Sum of Variables as a Constraint

- An Inequality with Constraint
- An Inequality with Constraints II
- An Inequality with Constraint V
- An Inequality with Constraint VI
- An Inequality with Constraint XI
- Monthly Problem 11199
- Problem 11804 from the AMM
- Sladjan Stankovik's Inequality With Constraint
- Sladjan Stankovik's Inequality With Constraint II
- An Inequality with Constraint V
- An Inequality with Constraint VI
- An Inequality with Constraint XI
- An Inequality with Constraint XII
- An Inequality with Constraint XIII
- Inequalities with Constraint XV and XVI
- An Inequality with Constraint XVII
- An Inequality with Constraint in Four Variables
- An Inequality with Constraint in Four Variables II
- An Inequality with Constraint in Four Variables III
- An Inequality with Constraint in Four Variables IV
- Inequality with Constraint from Dan Sitaru's Math Phenomenon
- An Inequality with a Parameter and a Constraint
- Cyclic Inequality with Square Roots And Absolute Values
- From Six Variables to Four - It's All the Same
- Michael Rozenberg's Inequality in Three Variables with Constraints
- Michael Rozenberg's Inequality in Two Variables
- Dan Sitaru's Cyclic Inequality in Three Variables II
- Dan Sitaru's Cyclic Inequality in Three Variables IV
- An Inequality with Arbitrary Roots
- Inequality 101 from the Cyclic Inequalities Marathon
- Sladjan Stankovik's Inequality With Constraint II
- An Inequality with Constraint in Four Variables
- An Inequality with Constraint in Four Variables IV
- Cyclic Inequality with Square Roots And Absolute Values
- From Six Variables to Four - It's All the Same
- Michael Rozenberg's Inequality in Two Variables
- Dan Sitaru's Cyclic Inequality in Three Variables II
- Inequality 101 from the Cyclic Inequalities Marathon

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