(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Given n real numbers [tex]x_1, x_2, \dotsb , x_n[/tex] which satisfy [tex]0 \leq x_1 \leq x_2 \leq \dotsb \leq x_n[/tex],

show that

[tex]\frac{x_1 + x_2 + \dotsb + x_k}{x_1 + x_2 + \dotsb + x_n} \leq \frac{k}{n}, \forall 1 \leq k \leq n[/tex].

2. Relevant equations

3. The attempt at a solution

If [tex]x_1 = x_2 = \dotsb = x_n[/tex], then

[tex]\frac{x_1 + x_2 + \dotsb + x_k}{x_1 + x_2 + \dotsb + x_n} = \frac{k(x_1)}{n(x_1)} = \frac{k}{n}[/tex].

If they are not all equal, suppose [tex]x_1 \neq x_2[/tex], then everything except [tex]x_1[/tex] would be strictly positive. Then I don't know how to continue. I can't seem to get a nice inequality coming out.

Edit: I just realised that by moving the terms around in the above inequality, I get

[tex]\frac{x_1 + x_2 + \dotsb + x_k}{k} \leq \frac{x_1 + x_2 + \dotsb + x_n}{n}[/tex]

This is like saying that the mean of a set of numbers increases when even bigger numbers are added to the set. This seems intuitive enough, but I haven't figured out how to prove this.

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# Homework Help: Ratio of partial sum to total sum

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