This'll be hard

Algebra Level 5

a b + b c + c a + 28 ( a b + b c + a c ) ( a + b + c ) 2 \large \dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}+\dfrac{28(ab+bc+ac)}{(a+b+c)^2} Let a , b a,b and c c be positive reals, find the minimum value of the expression above.

Note : You don't have to try a = b = c a=b=c , because it's not the answer.


The answer is 12.

P C by P C , 20, Vietnam

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1 solution

Jatin Chauhan
Feb 23, 2016

For first half that is a/b +b/c +c/a min. Value is 3.now for next half apply AM>= GM and AM >=HM

4 Helpful 0 Interesting 0 Brilliant 0 Confused

Wouldn't the answer be 12.33333?

Pi Han Goh - 5 years, 3 months ago

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i am also getting the same thing

Racchit Jain - 5 years, 3 months ago

That is what I got, but I rounded down to 12 and it was right.

Mason Maron - 5 years, 3 months ago

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The question didn't ask us to round it to the nearest integer.

Pi Han Goh - 5 years, 3 months ago

a = b = c a=b=c can't be happen.We can't applying AM-GM

Son Nguyen - 5 years, 3 months ago

Even I got 12.33

Nitin Kumar - 1 year, 4 months ago

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