#18 Measure Your Calibre

Algebra Level 5

Positive reals a a , b b and c c are such that a b c = 1 abc=1 . Find the infimum value of

a 3 ( a − b ) ( a − c ) + b 3 ( b − c ) ( b − a ) + c 3 ( c − a ) ( c − b ) \frac {a^3}{(a-b)(a-c)} + \frac {b^3}{(b-c)(b-a)} + \frac {c^3}{(c-a)(c-b)}

Other problems: Check your Calibre


The answer is 3.000.

Md Zuhair by Md Zuhair , 20, India

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1 solution

Spandan Senapati
Mar 18, 2017

Easy.It's easy to show that the sum equals a + b + c a+b+c using the given constraint a b c = 1 abc=1 .Then use A M − G M AM-GM inequality to get a + b + c > = 3 ( a b c ) ( 1 / 3 ) = 3 a+b+c>=3(abc)^(1/3)=3 .So the Minimum is 3 3

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Yes, Correct , But show

Md Zuhair - 4 years, 2 months ago

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Oh let it.I will take abt 30 mins to latex.......

Spandan Senapati - 4 years, 2 months ago

Is this a RMO problem?

Tapas Mazumdar - 4 years, 2 months ago

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Ya CBSE GMO question

Spandan Senapati - 4 years, 2 months ago

Yes boss.. Why}?

Md Zuhair - 4 years, 2 months ago

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