# Understand the problem

Let be real numbers with .

Prove that , and determine the cases of equality.

Prove that , 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 with the constraint 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 . Thus, the problem reduces to show that if are real numbers, then .

What if we can show that individually if we always have ? Then our problem will be solved right? We have , 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 , and hence .

Hence equality holds if and only if one of the three variables is and the other two are . QED

Hence equality holds if and only if one of the three variables is and the other two are . QED

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