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min max , nested absolute values (IV Soros Olympiad 1997-98 R3 9.4)

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June 1, 2024
absolute valueinequalitiesalgebra

Problem Statement

Find the smallest and largest values of the expression ∣...∣∣x−1∣−1∣...−1∣+1∣∣x−2∣−1∣+1\frac{ \left| ...\left| |x-1|-1\right| ... -1\right| +1}{\left| |x-2|-1 \right|+1} (The number of units in the numerator of a fraction, including the last one, is eleven, of which ten are under the absolute value sign.)
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