Problem

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

Tags: inequalities, algebra, Austria



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