# Cubes Constrained

### Problem

### Solution

Let $a^3=x,b^3=y,\,$ then $x+y=2.\,$ Consider function $f(t)=t^{\frac{1}{3}},\,$ $t\ge 0.\,$ The function is concave such that

$\displaystyle f(x)-f(1)\le (x-1)f'(1),$

or, $\displaystyle x(f(x)-f(1))\le x\frac{x-1}{3}.\,$ Similarly,

$\displaystyle\begin{align}y(f(y)-f(1)\le y\frac{y-1}{3}\;\text{and}\\ xy(f(xy)-f(1)\le xy\frac{xy-1}{3}, \end{align}$

from which

$\displaystyle \begin{align} 3(a^4+b^4)+2a^4b^4&=3\left(x^{\frac{4}{3}}+y^{\frac{4}{3}}\right)+2x^{\frac{4}{3}}y^{\frac{4}{3}}\\ &=3(xf(x)+yf(y)]+2xyf(xy)\\ &\le 3(x+y)f(1)+x(x-1)+y(y-1)\\ &\qquad\qquad+2xyf(1)+2xy\frac{xy-1}{3}\\ &=6+x^2+y^2-2+2xy+2xy\frac{xy-1}{3}\\ &=4+(x+y)^2+2xy\frac{xy-1}{3}\\ &=8+2xy\frac{xy-1}{3}\le 8, \end{align}$

because, by the AM-GM inequality, $xy-1\le 0.$

### Acknowledgment

Marian Dinca has kindly communicated to me his solution to an year-old post by Leo Giugiuc at the The School Yard Olympiad facebook group. Leo credits the problem to Andrei Eckstein.

- 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| |Up| |Front page| |Contents| |Algebra|

Copyright © 1996-2017 Alexander Bogomolny62681844 |