Algebraic inequality

Algebra Level 3

For distinct positive real numbers a , b , c a,b,c the inequality ( a + b ) ( b + c ) ( c + a ) > 8 a b c (a+b)(b+c)(c+a)>8abc is

False Neither true nor false True

A Former Brilliant Member by A Former Brilliant Member

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1 solution

Consider the fraction ( a + b ) ( b + c ) ( c + a ) a b c \dfrac{(a+b)(b+c)(c+a)}{abc} . Whence we have ( 1 + b a ) ( 1 + c b ) ( 1 + a c ) > 2 b a 2 c b 2 a c = 8 b a c b a c = 8 \Big(1+\dfrac ba\Big)\Big(1+\dfrac cb\Big)\Big(1+\dfrac ac\Big)>2\sqrt{\dfrac ba}2\sqrt{\dfrac cb}2\sqrt{\dfrac ac}=8\sqrt{\dfrac ba\dfrac cb\dfrac ac}=8 .

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But... Isnt it greater than or equal to?

Md Zuhair - 3 years, 5 months ago

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No, it would be so, if a , b , c a,b,c weren't distinct!

A Former Brilliant Member - 3 years, 5 months ago

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Oh yes sir :). Thanks

Md Zuhair - 3 years, 5 months ago

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