Prove that (a+b)(a+c) 2abc(a+b+c) for all positive real numbers a, b and c.

By AM-GM, (a+b)(a+c) = bc + a(a+b+c) 2bc a(a+b+c)

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XXIII OBM

Brazil algebra

Problem

Prove that

(a + b) (a + c) \geq 2 ab c (a + b + c)

for all positive real numbers

a

b

and

c

Solution

By AM-GM,

(a + b) (a + c) = b c + a (a + b + c) \geq 2 b c \cdot a (a + b + c)

Techniques

QM-AM-GM-HM / Power Mean