100 followers problem

Algebra Level 5

4 ( a 6 + b 6 + c 6 ) + 5 a b c ( a 3 + b 3 + c 3 ) \large 4(a^6+b^6+c^6)+5abc(a^3+b^3+c^3)

Given that a a , b b and c c are reals satisfying a b + b c + c a = 100 3 ab+bc+ca=\sqrt[3]{100} .

Find the minimum value of the expression above.

Source : Israel winter camp 2015 test.


The answer is 100.

Son Nguyen by Son Nguyen , 20, Vietnam

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1 solution

Parth Sankhe
Oct 20, 2018

Symmetric expressions always attain minimum values when its variables are all equal.

Thus for the minimum value we can take a = b = c a=b=c giving us 3 a 2 = 10 0 ⅓ 3a^2=100^⅓ and the minimum value of the expression as 100 .

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it is not the rigorous method

Mohammed Imran - 1 year, 5 months ago

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