8.1. Let all numbers $x, y, z$ be non-zero. Find all values that the expression

$$
\left(\frac{x}{|y|}-\frac{|x|}{y}\right) \cdot\left(\frac{y}{|z|}-\frac{|y|}{z}\right) \cdot\left(\frac{z}{|x|}-\frac{|z|}{x}\right)
$$

can take.
Solution: By the Pigeonhole Principle, among the numbers $x, y$, and $z$, there will be two numbers of the same sign. Then the corresponding bracket will be equal to 0, and the entire product will also be equal to 0.

Answer: Only the number 0.

Recommendations for checking:

| is in the work | points |
| :--- | :--- |
| correct and justified answer | 7 points |
| some (not all) cases of modulus expansion are correctly analyzed | 3 points |
| an example is provided showing that 0 can be obtained | 1 point |
| correct answer without justification (or with incorrect justification) | 0 points |