10.3

Part of 2021 Ukraine National Mathematical Olympiad

Problems(1)

sum (a^2+b^2)/(a+b) >= (a+b+c)+ (a-c)^2/(a+b+c) for a>= b>= c >0

Source: 2021 Ukraine NMO 10.3

For arbitrary positive reals

a\ge b \ge c

prove the inequality:

\frac{a^2+b^2}{a+b}+\frac{a^2+c^2}{a+c}+\frac{c^2+b^2}{c+b}\ge (a+b+c)+ \frac{(a-c)^2}{a+b+c}

algebrainequalities