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inequality

Source: 48th Austrian Mathematical Olympiad Beginners' Competition(2017)

February 27, 2018
inequalitiesalgebraAustria

Problem Statement

The nonnegative real numbers aa and bb satisfy a+b=1a + b = 1. Prove that:
12≤a3+b3a2+b2≤1\frac{1}{2} \leq \frac{a^3+b^3}{a^2+b^2} \leq 1
When do we have equality in the right inequality and when in the left inequality?
Proposed by Walther Janous
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