2024 DMC Mock 10 Problems/Problem 12

Take out the absolute values one by one. First, we have $|||||x-1|-1|-1|-1|-1|=\frac{1}{2}$ so $||||x-1|-1|-1|-1|=\frac{1}{2}\text{ or } \frac{3}{2}$ Repeating this process, we eventually get $|x-1| = \frac{1}{2} \text{ or } \frac{3}{2} \text{ or } \frac{5}{2} \text{ or } \frac{7}{2} \text{ or } \frac{9}{2}$

There are two solutions for each value of $|x-1|$ , so there are $\boxed{10}$ solutions total.

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2024 DMC Mock 10 Problems/Problem 12