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x/(y + 1) + y/(x + 1) >= 2/3 when x + y = 1, x,y>--1

Source: 2022 Austrian Mathematical Olympiad Junior Regional Competition , Problem 1

October 4, 2022

algebrainequalities

Problem Statement

Show that for all real numbers

x

and

y

with

x > -1

and

y > -1

and

x + y = 1

the inequality

\frac{x}{y + 1} +\frac{y}{x + 1} \ge \frac23

holds. When does equality apply?
(Walther Janous)