+0

0
338
1

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|\$?

Apr 24, 2018

#1
+902
+1

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 \(\boxed{10}\).

Aug 19, 2018