Switched... again
⎩ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎧ a 1 ≤ a 2 ≤ ⋯ ≤ a 1 7 5 ≤ a 1 7 6 a 1 + a 2 + ⋯ + a 1 7 5 + a 1 7 6 ∣ a 1 ∣ + ∣ a 2 ∣ + ⋯ + ∣ a 1 7 5 ∣ + ∣ a 1 7 6 ∣ ∈ R = 0 = 1 6 7 7 2 8
What is the minimum value of a 1 7 6 − a 1 ?
The answer is 1906.
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The distance between the smallest and largest number will be smallest if the other numbers contribute the most to the sum of the absolute values. This happens if there are 88 negative numbers equal to each other followed by 88 positive numbers equal to them in absolute value. These numbers are 1 6 7 7 2 8 / 1 7 6 = 9 5 3 for the positive one, and − 9 5 3 for the negative, or difference of 9 5 3 ⋅ 2 = 1 9 0 6 .