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

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SUMMARY

The forum discussion centers on a problem involving the numbers $x_1, x_2, \ldots, x_{1991}$ that satisfy the equation $|x_1-x_2|+|x_2-x_3|+\cdots+|x_{1990}-x_{1991}|=1991$. The objective is to determine the maximum value of the expression $|y_1-y_2|+|y_2-y_3|+\cdots+|y_{1990}-y_{1991}|$, where $y_k=\dfrac{1}{k}(x_1+x_2+\cdots+x_k)$. The discussion emphasizes the relationship between the absolute differences of the $x_k$ values and the derived $y_k$ values, leading to insights on optimizing the expression.

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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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