Cauchy inequalities in systems of equations For positive real numbers solve the system of equations

Ellen Chang

Ellen Chang

Answered question

2022-07-01

Cauchy inequalities in systems of equations
For positive real numbers solve the system of equations
{ x 1 + x 2 + · · · + x n = 1 4 1 x 1 + 4 x 2 + · · · + n 2 x n = n 2 ( n + 1 ) 2

Answer & Explanation

bap1287dg

bap1287dg

Beginner2022-07-02Added 13 answers

Titu's Lemma might be what you're looking for. Applying it yields:
1 x 1 + 4 x 2 + + n 2 x n ( 1 + 2 + + n ) 2 x 1 + x 2 + + x n = n 2 ( n + 1 ) 2 4 1 4 = n 2 ( n + 1 ) 2
This lemma is simply an application of CS Inequality, with a i = k x k and b i = x k . Now CS Inequality holds iff:
a 1 b 1 = a 2 b 2 = = a n b n
In other words:
1 x 1 = 2 x 2 = = n x n Can you continue?

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