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>0a,b,c,d,e >0

b+c+d+ea+a+c+d+eb+a+b+d+ec+a+b+c+ed+a+b+c+de\begin{aligned} \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{aligned}

Bonus: For real x1,x2,x3,x4,,xn>0 x_1 , x_2 , x_3 , x_4 , \cdots , x_n >0 , let x1+x2+x3+x4++xn=S x_1 + x_2 + x_3 + x_4 + \cdots + x_n = S. Find the minimum value of Sx1x1+Sx2x2+Sx3x3++Sxnxn \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 nn.

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