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Bundeswettbewerb Mathematik 1975 Problem 2.1

Source: Bundeswettbewerb Mathematik 1975 Round 2

October 22, 2022

inequalitiesRealalgebra

Problem Statement

Let

a, b, c, d

be distinct positive real numbers. Prove that if one of the numbers

c, d

lies between

a

and

b

, or one of

a, b

lies between

c

and

d

, then

\sqrt{(a+b)(c+d)} >\sqrt{ab} +\sqrt{cd}

and that otherwise, one can choose

a, b, c, d

so that this inequality is false.