Select Page

# Understand the problem

Let $a,b,c,d$ be positive real numbers such that $abcd=1$.Find with proof that $x=3$ is the minimal value for which the following inequality holds:
$$a^x+b^x+c^x+d^x\ge\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{d}$$

##### Source of the problem

Albania IMO TST 2013

Inequalities
Medium
##### Suggested Book
Inequalities by BJ Venkatachala

Do you really need a hint? Try it first!

Choose specific values of $a,b,c,d$ to show that no value of $x$ less than 3 works.
Choosing $a=b=c=t$, the inequality becomes $3t^x+\frac{1}{t^{3x}}\ge \frac{3}{t}+t^3$. Note that, as $t\to\infty$, the LHS grows as $O(t^x)$ and the RHS grows as $O(t^3)$. For the LHS to be always greater than the RHS, it should be of a higher order. Hence, $x\ge 3$. Now show that the inequality is true for $x=3$.
Note that the RHS can be rewritten as $abc+bcd+cda+bda$. This reminds us of the AM-GM inequality.
Indeed, $abc+bcd+cda+bda\le \frac{a^3+b^3+c^3}{3}+\frac{d^3+b^3+c^3}{3}+\frac{a^3+d^3+c^3}{3}+\frac{a^3+b^3+d^3}{3}=a^3+b^3+c^3+d^3$.

# Connected Program at Cheenta

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.

# Similar Problems

## 2016 AMC 8 Problem 24 Number Theory

This beautiful application is from 2016 AMC 8 Problem 24 based on Number Theory . Sequential hints are given to understand and solve the problem .

## 2017 AMC 8 Problem 21 Number Theory

This beautiful application from 2017 AMC 8 Problem 21 is based on Number Theory . Sequential hints are provided to study and solve the problem .

## SMO(senior)-2014 Problem 2 Number Theory

This beautiful application from SMO(senior)-2014 is based on the concepts of Number Theory . Sequential hints are provided to understand and solve the problem .

## SMO (senior) -2014/problem-4 Number Theory

This beautiful application from SMO(senior)-2014/Problem 4 is based Number Theory . Sequential hints are provided to understand and solve the problem .

## The best exponent for an inequality

Understand the problemLet be positive real numbers such that .Find with proof that is the minimal value for which the following inequality holds:Albania IMO TST 2013 Inequalities Medium Inequalities by BJ Venkatachala Start with hintsDo you really need a hint?...

## 2018 AMC 10A Problem 25 Number Theory

This beautiful application from AMC 2018 is based on Number Theory. Sequential hints are given to understand and solve the problem .

## A functional inequation

Understand the problemFind all functions such thatholds for all . Benelux MO 2013 Functional Equations Easy Functional Equations by BJ Venkatachala Start with hintsDo you really need a hint? Try it first!Note that the RHS does not contain $latex y$. Thus it should...

## Mathematical Circles Inequality Problem

A beautiful inequality problem from Mathematical Circles Russian Experience . we provide sequential hints . key idea is to use arithmetic mean , geometric mean inequality.

## RMO 2019

Regional Math Olympiad (RMO) 2019 is the second level Math Olympiad Program in India involving Number Theory, Geometry, Algebra and Combinatorics.

## AMC 2019 12A Problem 15 Diophantine Equation

Beautiful application of Logarithm and Diophantine Equation in American Mathematics Competition (2019) 12A