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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 \(|a_1 - a_6|? \)

 Dec 22, 2023
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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​.

 Dec 24, 2023

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