Not only AM-GM 3 (The Other AM-GM Part 5)

Algebra Level 4

Find the minimum value of the expression below, for real \(a,b,c,d,e >0 \)

\[\begin{eqnarray} \dfrac{b+c+d+e}{a} + \dfrac{a+c+d+e}{b} + \dfrac{a+b+d+e}{c} + \dfrac{a+b+c+e}{d} + \dfrac{a+b+c+d}{e} \end{eqnarray} \]

Bonus: For real \( x_1 , x_2 , x_3 , x_4 , \cdots , x_n >0 \), let \( x_1 + x_2 + x_3 + x_4 + \cdots + x_n = S\). Find the minimum value of \( \dfrac{S- x_1}{x_1} + \dfrac{S-x_2}{x_2} + \dfrac{S-x_3}{x_3} + \cdots + \dfrac{S-x_n}{x_n} \) in term of \(n\).

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