Question 173
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that, for any real numbers $x, y$, the following two multisets are equal
$$
\{f(xf(y) + 1), f(yf(x) - 1)\} = \{xf(f(y)) - 1, yf(f(x)) + 1\}. \qquad (*)
$$
*Remark: $\{a, b\} = \{c, d\}$ are equal as multisets if $a = c$ and $b = d$, or $a = d$ and $b = c$.*
$$
\{f(xf(y) + 1), f(yf(x) - 1)\} = \{xf(f(y)) - 1, yf(f(x)) + 1\}. \qquad (*)
$$
*Remark: $\{a, b\} = \{c, d\}$ are equal as multisets if $a = c$ and $b = d$, or $a = d$ and $b = c$.*