# An Inequality with Constraint XX

### Statement

### Solution 1

With the Rearrangement inequality,

$\displaystyle\begin{align} \sum_{cycl}(x+y)^2 &= 2\sum_{cycl}x^2+2\sum_{cycl}xy\\ &\ge 4\sum_{cycl}xy\ge 4\sqrt{3xyz(x+y+1)}=4\sqrt{3xyz}. \end{align}$

### Solution 2

Let $a=y+z,\,$ $b=z+x,\,$ $c=x+y.\,$ Then $a,b,c\,$ are the side lengths of a triangle, with the semiperimeter $s=x+y+z=1,\,$ the area $S=\sqrt{(x+y+z)xyz}=\sqrt{xyz}.\,$ Thus, by the Ionescu-Weitzenbök inequality,

$\displaystyle\sum_{cycl}(x+y)^2=\sum_{cycl}a^2\ge 4\sqrt{3}S=4\sqrt{3xyz}.$

### Solution 3

By the AM-QM inequality

$\displaystyle\begin{align} \sum_{cycl}(x+y)^2 &\ge \frac{\displaystyle\left(\sum_{cycl}(x+y)\right)^2}{3}\\ &=\frac{4(x+y+z)^2}{3}, \end{align}$

so that

$\displaystyle\begin{align} \left(\sum_{cycl}(x+y)^2\right)^2 &\ge \frac{16}{9}\left(x+y+z\right)^4= \frac{16}{9}(x+y+z)^3\\ &\ge\frac{16}{9}\cdot (3\sqrt[3]{xyz})=16\cdot 3\cdot xyz, \end{align}$

implying $\displaystyle\sum_{cycl}(x+y)^2\ge 4\sqrt{3xyz}.$

### Solution 4

By the AM-QM inequality

$\displaystyle\begin{align} \sum_{cycl}(x+y)^2 &\ge \frac{\displaystyle\left(\sum_{cycl}(x+y)\right)^2}{3}\\ &=\frac{4}{3}(x+y+z)^2=\frac{4}{3}(x+y+z)^{\frac{3}{2}}\\ &=4\sqrt{3}\sqrt{\frac{x+y+z}{3}}\ge 4\sqrt{3}\sqrt{xyz}. \end{align}$

### Acknowledgment

The problem, with several solutions, was kindly posted at the CutTheKnotMath facebook page by Dan Sitaru. Dan has previously published the problem at the Romanian Mathematical Magazine. Solution 1 is by Boris Colakovic; Solution 2 is by Dan Sitaru; Solution 3 is by Tri Nitrotoluen; Solution 4 is by Abdelhak Maoukouf.

### 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
- Dan Sitaru's Cyclic Inequality in Three Variables VI
- 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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