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 \(x=t s, y=\frac{t}{s}, z=\frac{s}{t}\) in (i) gives 
\[ f(t) f(s)=f(t s)+f(t / s) \]
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 \(g\left(t^{q}\right)=g(t)^{q}\) for every rational \(q\). Consequently, if \(t>1\) is fixed, we have \(f\left(t^{q}\right)=a^{q}+a^{-q}\), where \(a=g(t)\). But since the set of \(a^{q} (q \in \mathbb{Q})\) 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 \(t^{k}=a\), we have 
\[ f(x)=x^{k}+x^{-k} \quad \text{for every } x \in \mathbb{R} \]