MHB What is the Greatest Value of an Expression with Given Numbers and Equation?

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The problem involves determining the maximum value of the expression related to the averages of a sequence of numbers constrained by a specific equation involving absolute differences. The equation states that the sum of the absolute differences between consecutive numbers equals 1991. Participants are encouraged to explore the relationship between the original numbers and their averages to find the greatest possible value of the new expression. The discussion invites community engagement and submissions for solutions over the next week. This mathematical challenge emphasizes the interplay between sequences and their statistical properties.
anemone
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Here is this week's POTW:

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The numbers $x_1,\,x_2,\,\cdots,\,x_{1991}$ satisfy the equation $|x_1-x_2|+|x_2-x_3|+\cdots+|x_{1990}-x_{1991}|=1991$.

What is the greatest possible value of the expression $|y_1-y_2|+|y_2-y_3|+\cdots+|y_{1990}-y_{1991}|$, where $y_k=\dfrac{1}{k}(x_1+x_2+\cdots+x_k)$?

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As usual, I will give the community another week's time to attempt at last week's POTW. And I am looking forward to receiving submissions from the members!
 

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