Prove inequality with positive numbers: Given n positive numbers a1,a2,..an (n>=2).

[mashiro.shiina]

Prove inequality with positive numbers: Given n positive numbers a1,a2,..an (n>=2).

Given n positive numbers a₁,a₂,..a_n (n>=2). Let S=a₁ⁿ+a₂ⁿ+...+a_nⁿ and P=a₁a_2..a_n. Prove that 1/(S-a₁ⁿ+P)+1/(S-a₂ⁿ+P)+...+1/(S-a_nⁿ+P)<=1/P

 

[stapel]

Given n positive numbers a₁,a₂,..a_n (n>=2). Let S=a₁ⁿ+a₂ⁿ+...+a_nⁿ and P=a₁a_2..a_n. Prove that 1/(S-a₁ⁿ+P)+1/(S-a₂ⁿ+P)+...+1/(S-a_nⁿ+P)<=1/P

Just to be sure we're "on the same page", does the above mean the following?

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Given n positive numbers, a₁, a₂, ..., a_n, with n > 2. Let:

. . .\(\displaystyle S\, =\, a_1^n\, +\, a_2^n\, +\, ...\, +\, a_n^n\)

and

. . .\(\displaystyle P\, =\, a_1\, \cdot\, a_2\, \cdot\, ...\, \cdot\, a_n\)

Prove the following:

. . .\(\displaystyle \dfrac{1}{S\, -\, a_1^n\, +\, P}\, +\, \dfrac{1}{S\, -\, a_2^n\, +\, P}\, +\, ...\, +\, \dfrac{1}{S\, -\, a_n^n\, +\, P}\, \leq\, \dfrac{1}{P}\)

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What are your thoughts? What have you tried? How far have you gotten? Where are you stuck?

Please be complete. Thank you!