+4 Let \(a_1, a_2, a_3, \dots, a_{10}\)be real numbers such that \(|a_1 - a_2| + |a_2 - a_3| + |a_3 - a_4| + |a_4 - a_5| + \dots + |a_9 - a_{10}| + |a_{10} - a_1| = 500.\)
What is the largest possible value of \(|a_1 - a_6|? \)
+1725 By the Triangle Inequality,
[|a_1 - a_6| = |a_1 - a_2 + a_2 - a_3 + a_3 - a_4 + a_4 - a_5 + a_5 - a_6|
\le |a_1 - a_2| + |a_2 - a_3| + |a_3 - a_4| + |a_4 - a_5| + |a_5 - a_6|.]
Equality occurs when a2, a3, a4, and a5 all lie on the line segment joining a1 and a6.
Also by the Triangle Inequality,
[|a_1 - a_2| + |a_2 - a_3| + |a_3 - a_4| + |a_4 - a_5| + |a_5 - a_6| + |a_6 - a_{10}| + |a_{10} - a_1|
\ge |a_1 - a_{10}| + |a_{10} - a_1| = 2|a_1 - a_{10}|.]
Equality occurs when a2, a3, a4, a5, and a6 all lie on the line segment joining a1 and a10.
Thus, if a2, a3, a4, a5, and a6 all lie on the line segment joining a1 and a10, then
[|a_1 - a_6| = 500 - |a_6 - a_{10}| - |a_9 - a_{10}| - |a_8 - a_9| - |a_7 - a_8|.]
To maximize ∣a1−a6∣, we minimize ∣a6−a10∣+∣a9−a10∣+∣a8−a9∣+∣a7−a8∣.
The minimum value is 0, which occurs when a6=a7=a8=a9=a10.
Thus, the largest possible value of ∣a1−a6∣ is 500.
