5. A5 (KOR) Let $\mathbb{R}^{+}$be the set of all positive real numbers. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$that satisfy the following conditions:
(i) $f(x y z)+f(x)+f(y)+f(z)=f(\sqrt{x y}) f(\sqrt{y z}) f(\sqrt{z x})$ for all $x, y, z \in$ $\mathbb{R}^{+}$.
(ii) $f(x)<f(y)$ for all $1 \leq x<y$.
5. Placing $x=y=z=1$ in (i) leads to $4 f(1)=f(1)^{3}$, so by the condition $f(1)>0$ we get $f(1)=2$. Also putting $x=t s, y=\frac{t}{s}, z=\frac{x}{t}$ in (i) gives
$$
f(t) f(s)=f\left(t_{s}\right)+f(t / s) .
$$

In particular, for $s=1$ the last equality yields $f(t)=f(1 / t)$; hence $f(t) \geq f(1)=2$ for each $t$. It follows that there exists $g(t) \geq 1$ such that $\bar{f}(t)=g(t)+\frac{1}{g(t)}$. Now it follows by induction from (1) that $g\left(t^{n}\right)=$ $g(t)^{n}$ for every integer $n$, and therefore $g(t \sigma)=g(t)^{\sigma}$ for every rational $q$. Consequently, if $t>1$ is fixed, we have $f\left(t^{\sigma}\right)=a^{q}+a^{-q}$, where $a=g(t)$. But since the set of $a^{0}(q \in Q)$ is dense in $\mathbb{R}^{+}$and $f$ is monotone on $(0,1]$ and $[1, \infty)$, it follows that $f\left(t^{r}\right)=a^{r}+a^{-r}$ for every real $r$. Therefore, if $k$ is such that $t^{k}=\alpha$, we have
$$
f(x)=x^{k}+x^{-k} \quad \text { for every } x \in \mathbb{R} \text {. }
$$