Understand the problem

Let $a,b,c\ge-1$ be real numbers with $a^3+b^3+c^3=1$.
Prove that $a+b+c+a^2+b^2+c^2\le4$, and determine the cases of equality.
Source of the problem
Austria MO 2016. Final Round, Problem 4
Topic
Inequality
Difficulty Level
Suggested Book
Challenges and Thrills of Pre-College Mathematics

Start with hints

Do you really need a hint? Try it first!

The idea is that you have to capture the symmetry in the equations and correspondingly find it. Observe that the inequality $a+b+c+a^2+b^2+c^2\le4$ with the constraint $a^3+b^3+c^3=1$ can be written as  \( (a^3 – a ^2 – a +1)  + (b^3 – b ^2 – b +1) + (c^3 – c ^2 – c +1) \ge 0  \) using the constraint.   
Now, \( (a^3 – a ^2 – a +1)  + (b^3 – b ^2 – b +1) + (c^3 – c ^2 – c +1) \ge 0  \) demands you to look into the polynomial  $P(x)=(x+1)(x-1)^2=x^3-x^2-x+1$. Thus, the problem reduces to show that if $a,b,c\ge-1$ are real numbers, then  $P(a)+P(b)+P(c)\ge0$.
What if we can show that individually if  $x\ge-1$ we always have $P(x)\ge0$? Then our problem will be solved right? We have $P(x)=(x+1)(x-1)^2=x^3-x^2-x+1$, observe that it automatically implies that if \( x+1 \geq 0 \) then we will have \( P(x) \geq 0\).  
Equality Cases: For equality we must have $P(a)=P(b)=P(c)=0$, and hence $a,b,c\in\{-1,+1\}$.
Hence equality holds if and only if one of the three variables is $-1$ and the other two are $+1$. QED

Watch video

Connected Program at Cheenta

Math Olympiad Program

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

Sum of Sides of Triangle | PRMO-2018 | Problem No-17

Try this beautiful Problem on Geometry from PRMO -2018.You may use sequential hints to solve the problem.

Recursion Problem | AMC 10A, 2019| Problem No 15

Try this beautiful Problem on Algebra from AMC 10A, 2019. Problem-15, You may use sequential hints to solve the problem.

Roots of Polynomial | AMC 10A, 2019| Problem No 24

Try this beautiful Problem on Algebra from AMC 10A, 2019. Problem-24, You may use sequential hints to solve the problem.

Set of Fractions | AMC 10A, 2015| Problem No 15

Try this beautiful Problem on Algebra from AMC 10A, 2015. Problem-15. You may use sequential hints to solve the problem.

Indian Olympiad Qualifier in Mathematics – IOQM

Due to COVID 19 Pandemic, the Maths Olympiad stages in India has changed. Here is the announcement published by HBCSE: Important Announcement [Updated:14-Sept-2020]The national Olympiad programme in mathematics culminating in the International Mathematical Olympiad...

Positive Integers and Quadrilateral | AMC 10A 2015 | Sum 24

Try this beautiful Problem on Rectangle and triangle from AMC 10A, 2015. Problem-24. You may use sequential hints to solve the problem.

Rectangular Piece of Paper | AMC 10A, 2014| Problem No 22

Try this beautiful Problem on Rectangle and triangle from AMC 10A, 2014. Problem-23. You may use sequential hints to solve the problem.

Probability in Marbles | AMC 10A, 2010| Problem No 23

Try this beautiful Problem on Probability from AMC 10A, 2010. Problem-23. You may use sequential hints to solve the problem.

Points on a circle | AMC 10A, 2010| Problem No 22

Try this beautiful Problem on Number theory based on Triangle and Circle from AMC 10A, 2010. Problem-22. You may use sequential hints to solve the problem.

Circle and Equilateral Triangle | AMC 10A, 2017| Problem No 22

Try this beautiful Problem on Triangle and Circle from AMC 10A, 2017. Problem-22. You may use sequential hints to solve the problem.