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Suppose that \[|a - b| + |b - c| + |c - d| + \dots + |m-n| + |n-o| + \cdots+ |x - y| + |y - z| + |z - a| = 20.\] What is the maximum possible value of $|a - n|$?

Guest Apr 24, 2018
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Consider $a$ and $b$ as points on the real number line. Then $|a - b|$ represents the distance between the two points. Similarly, $|b - c|$ represents the distance between the points $b$ and $c$, and so on. Hence, consider a path that starts at $a$, then jumps to $b$, then jumps to $c$, and then back to $a$. Then the total distance covered by the path is \[|a - b| + |b - c| + |c - a| = 20.\] We claim that $|a - b|$ is at most 10. Consider the portion of the path that goes from $b$ to $c$ to $a$. The sum of these segments must add up to at least the distance from $b$ to $a$. (Think of the maxim, "The shortest distance between two points is a straight line." This is a case of the triangle inequality.) In other words, \[|b - c| + |c - a| \ge |b-a| = |a - b|.\] Hence, \[|a - b| + |b - c| + |c - a| \ge 2|a - b|,\] so $2|a - b| \le 20$, or $|a - b| \le 10$. To show that the maximum possible value of $|a - b|$ is 10, we must provide a set of values in which $|a - b|$ is equal to 10. If we set $a = 10$ and all 2 other variables equal to 0, then the given equation is satisfied and $|a - b| = 10$. Hence, the maximum possible value of $|a - b|$ is \(\boxed{10}\).

Lightning  Aug 19, 2018

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