8. Proportion. The numbers $A, B, C, p, q, r$ are related by the following ratios:

$$
A: B=p, \quad B: C=q, \quad C: A=r
$$

Write the proportion

$$
A: B: C=\square: \square: \square
$$

such that the expressions in the blanks consist of $p, q, r$, and these expressions can be transformed into one another by cyclic permutations of the letters $p, q, r$. (We understand this as follows: if we replace $p$ with $q$, $q$ with $r$, and $r$ with $p$, then the first expression will transform into the second, the second into the third, and the third into the first).

8a. Symmetric Expressions. Expressions such as $x+y+z$ or $x y z$ are symmetric. By this, we mean that their value does not change when the variables $x, y, z$ are permuted in any way. The symmetry of the examples given above is obvious; however, there are symmetric expressions whose symmetry is not obvious, for example:

$$
|| x-y|+x+y-2 z|+|x-y|+x+y+2 z
$$

Prove the symmetry of this expression and determine its value in such a way that the symmetry becomes obvious.
8. We can accept

$$
A: B: C=\sqrt[3]{\frac{\bar{p}}{r}}: \sqrt[3]{\frac{q}{p}}: \sqrt[3]{\frac{r}{q}}
$$

Indeed, due to the easily provable equality

$$
p q r=1
$$

we have:

$$
A: B=\sqrt[3]{\frac{p^{2}}{q r}}=p, \quad B: C=\sqrt[3]{\frac{q^{2}}{r p}}=q, \quad C: A=\sqrt[3]{\frac{r^{2}}{p q}}=r
$$

Using the equality \( p q r=1 \) again, we can derive another proportion with the required properties from the above proportion:

$$
A: B: C=\sqrt[3]{p^{2} q}: \sqrt[3]{q^{2} r}: \sqrt[3]{r^{2} p}
$$

8a. Let the expression whose symmetry we need to prove be denoted by \( w \). It is easy to notice that

![](https://cdn.mathpix.com/cropped/2024_05_21_5d77f4829f88503a49eag-063.jpg?height=134&width=1254&top_left_y=1181&top_left_x=398)

Considering the cases \( x-y \geqslant 0 \) and \( x-y \leqslant 0 \), we conclude that

![](https://cdn.mathpix.com/cropped/2024_05_21_5d77f4829f88503a49eag-063.jpg?height=199&width=865&top_left_y=1448&top_left_x=601)

i.e., \( w=4 \max \{x, y, z\} \), from which the symmetry of this expression follows directly.