Why not AM-GM? It's obvious!

Algebra Level 4

Let a , b , c , d a,b,c,d be positive real numbers. Find the minimum value of

a + b + c d + b + c + d a + c + d + a b + d + a + b c + a b + c + d + b c + d + a + c d + a + b + d a + b + c \small \frac { a+b+c }{ d } +\frac { b+c+d }{ a } +\frac { c+d+a }{ b } +\frac { d+a+b }{ c } +\frac { a }{ b+c+d } +\frac { b }{ c+d+a } +\frac { c }{ d+a+b } +\frac { d }{ a+b+c }

Write your answer to 3 decimal places.

Hint : When does equality hold?


The answer is 13.333.

Steven Jim by Steven Jim , 17, Vietnam

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1 solution

4 Helpful 0 Interesting 0 Brilliant 0 Confused

why the answer not 8?

I Gede Arya Raditya Parameswara - 3 years, 6 months ago

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When you directly apply AM-GM, you get 8 but you can't find when equality holds. That is, there is no solution for a + b + c d = b + c + d a = c + d + a b = d + a + b c = d a + b + c = a b + c + d = b c + d + a = c d + a + b \dfrac{a+b+c}{d}=\dfrac{b+c+d}{a}=\dfrac{c+d+a}{b}=\dfrac{d+a+b}{c}=\dfrac{d}{a+b+c}=\dfrac{a}{b+c+d}=\dfrac{b}{c+d+a}=\dfrac{c}{d+a+b}

Dexter Woo Teng Koon - 3 years, 6 months ago

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