What is the minimum of the following expression?

Algebra Level 2

If x 1 , x 2 , x 3 , … , x 2018 x_1,x_2,x_3,\dots,x_{2018} are positive integers, then what is the minimum of the value of the following expression: ( x 1 + x 2 + x 3 + ⋯ + x 2018 ) ( 1 x 1 + 1 x 2 + 1 x 3 + ⋯ + 1 x 2018 ) \Big(x_1+x_2+x_3+\dots+x_{2018}\Big)\Big(\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}+\dots+\frac{1}{x_{2018}}\Big)

2036 2036 2018 2018 1009 1009 ( 2019 ) 2 (2019)^2 ( 2018 ) 2 (2018)^2

Hana Wehbi by Hana Wehbi , 51, USA

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1 solution

Hana Wehbi
Sep 29, 2018

If x 1 , x 2 , x 3 , … , x 2018 x_1,x_2,x_3,\dots,x_{2018} are positive integers, then the minimum of the value of the following expression by Holder Inequality is: ( x 1 + x 2 + x 3 + ⋯ + x 2018 ) ( 1 x 1 + 1 x 2 + 1 x 3 + ⋯ + 1 x 2018 ) ≥ ( 1 + 1 + ⋯ + 1 ⏟ 2018 ) ( 1 + 1 + ⋯ + 1 ⏟ 2018 ) = ( 2018 ) 2 \Big(x_1+x_2+x_3+\dots+x_{2018}\Big)\Big(\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}+\dots+\frac{1}{x_{2018}}\Big)\ge (\underbrace{1+1+\dots+1}_{2018})(\underbrace{1+1+\dots+1}_{2018})=(2018)^2

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