# An Inequality with Constraint in Four Variables II

### Statement

### Solution

There are $y,z,t\in [0,1]\,$ such that $b=1-y,\;$ $c=1-z,\;$ $d=1-t,\,$ $a=1+y+z+t.\,$ The inequality to prove becomes

$\begin{align} (1-y)^3&+(1-z)^3+(1-t)^3+(1+y+z+t)^3\\ &\qquad\qquad +6(1-y)(1-z)(1-t)(1+y+z+t)-10\\ &=9(y^2z+yz^2+y^2t+yt^2+z^2t+zt^2)\\ &\qquad\qquad +6yzt(1-y)+6yzt(1-z)+6yzt(1-t)\\ &\ge 0. \end{align}$

As to the equality, all the terms in

$9(y^2z+yz^2+y^2t+yt^2+z^2t+zt^2)+6yzt(1-y)+6yzt(1-z)+6yzt(1-t)$

are non-negative. For there to be an equality, all of them should vanish. The expression in the first pair of parentheses is zero provided any two of $y,z,t\,$ vanish. The other three terms vanish automatically. When two of $y,z,t\,$ vanish, the third may be pretty much arbitrary, i.e., within $[0,1].],$ Thus, in terms of $a,b,c,d,\,$ equality is achieved when, two of $b,c,d\,$ are $1\,$ and one is in $[0,1].\,$ Given their ordering, the only possibility is $a=1+k,\;$ $b=c=1,\,$ $d=1-k,\,$ $k\in [0,1].$

### Illustration

### Acknowledgment

The problem and the Solution have been kindly posted by Leo Giugiuc at the CutTheKnotMath facebook page. The solution is by Marian Cucoanes. Illustration by Nassim Nicholas Taleb.

- A Cyclic But Not Symmetric Inequality in Four Variables
- An Inequality with Constraint
- An Inequality with Constraints II
- An Inequality with Constraint III
- An Inequality with Constraint IV
- An Inequality with Constraint VII
- An Inequality with Constraint VIII
- An Inequality with Constraint IX
- An Inequality with Constraint X
- Problem 11804 from the AMM
- Sladjan Stankovik's Inequality With Constraint
- An Inequality with Constraint XII
- An Inequality with Constraint XIV
- An Inequality with Constraint XVII
- An Inequality with Constraint in Four Variables II
- An Inequality with Constraint in Four Variables III
- An Inequality with Constraint in Four Variables V
- An Inequality with Constraint in Four Variables VI
- A Cyclic Inequality in Three Variables with Constraint
- Dorin Marghidanu's Cyclic Inequality with Constraint
- Dan Sitaru's Cyclic Inequality In Three Variables with Constraints
- Dan Sitaru's Cyclic Inequality In Three Variables with Constraints II
- Dan Sitaru's Cyclic Inequality In Three Variables with Constraints III
- Inequality with Constraint from Dan Sitaru's Math Phenomenon
- Another Problem from the 2016 Danubius Contest
- Gireaux's Theorem
- An Inequality with a Parameter and a Constraint
- Unsolved Problem from Crux Solved
- An Inequality With Six Variables and Constraints
- Cubes Constrained
- Dorin Marghidanu's Inequality with Constraint
- Dan Sitaru's Integral Inequality with Powers of a Function
- Michael Rozenberg's Inequality in Three Variables with Constraints
- Dan Sitaru's Cyclic Inequality In Three Variables with Constraints IV
- An Inequality with Arbitrary Roots
- Leo Giugiuc's Inequality with Constraint
- Problem From the 2016 IMO Shortlist
- Dan Sitaru's Cyclic Inequality with a Constraint and Cube Roots
- Dan Sitaru's Cyclic Inequality with a Constraint and Cube Roots II
- A Simplified Version of Leo Giugiuc's Inequality from the AMM
- Kunihiko Chikaya's Inequality $\displaystyle \small{\left(\frac{(a^{10}-b^{10})(b^{10}-c^{10})(c^{10}-a^{10})}{(a^{9}+b^{9})(b^{9}+c^{9})(c^{9}+a^{9})}\ge\frac{125}{3}[(a-b)^3+(b-c)^3+(c-a)^3]\right)}$

|Contact| |Front page| |Contents| |Algebra|

Copyright © 1996-2017 Alexander Bogomolny62608385 |