Problem 5. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$such that
$$
(z+1) f(x+y)=f(x f(z)+y)+f(y f(z)+x),
$$
for all positive real numbers $x, y, z$.
Answer: The only solution is $f(x)=x$ for all positive real numbers $x$.
Solution. The identity function $f(x)=x$ clearly satisfies the functional equation. Now, let $f$ be a function satisfying the functional equation. Plugging $x=y=1$ into (3) we get $2 f(f(z)+1)=(z+1)(f(2))$ for all $z \in \mathbb{R}^{+}$. Hence, $f$ is not bounded above.

Lemma. Let $a, b, c$ be positive real numbers. If $c$ is greater than $1, a / b$ and $b / a$, then the system of linear equations
$$
c u+v=a \quad u+c v=b
$$
has a positive real solution $u, v$.
Proof. The solution is
$$
u=\frac{c a-b}{c^{2}-1} \quad v=\frac{c b-a}{c^{2}-1} .
$$

The numbers $u$ and $v$ are positive if the conditions on $c$ above are satisfied.
We will now prove that
$$
f(a)+f(b)=f(c)+f(d) \quad \text { for all } a, b, c, d \in \mathbb{R}^{+} \text {with } a+b=c+d .
$$

Consider $a, b, c, d \in \mathbb{R}^{+}$such that $a+b=c+d$. Since $f$ is not bounded above, we can choose a positive number $e$ such that $f(e)$ is greater than $1, a / b, b / a, c / d$ and $d / c$. Using the above lemma, we can find $u, v, w, t \in \mathbb{R}^{+}$satisfying
$$
\begin{array}{ll}
f(e) u+v=a, & u+f(e) v=b \\
f(e) w+t=c, & w+f(e) t=d .
\end{array}
$$

Note that $u+v=w+t$ since $(u+v)(f(e)+1)=a+b$ and $(w+t)(f(e)+1)=c+d$. Plugging $x=u, y=v$ and $z=e$ into (3) yields $f(a)+f(b)=(e+1) f(u+v)$. Similarly, we have $f(c)+f(d)=(e+1) f(w+t)$. The claim follows immediately.
We then have
$$
y f(x)=f(x f(y)) \quad \text { for all } x, y \in \mathbb{R}^{+}
$$
since by (3) and (4),
$$
(y+1) f(x)=f\left(\frac{x}{2} f(y)+\frac{x}{2}\right)+f\left(\frac{x}{2} f(y)+\frac{x}{2}\right)=f(x f(y))+f(x) .
$$

Now, let $a=f(1 / f(1))$. Plugging $x=1$ and $y=1 / f(1)$ into (5) yields $f(a)=1$. Hence $a=a f(a)$ and $f(a f(a))=f(a)=1$. Since $a f(a)=f(a f(a))$ by $(5)$, we have $f(1)=a=1$. It follows from (5) that
$$
f(f(y))=y \quad \text { for all } y \in \mathbb{R}^{+} .
$$

Using (4) we have for all $x, y \in \mathbb{R}^{+}$that
$$
\begin{aligned}
f(x+y)+f(1) & =f(x)+f(y+1), \quad \text { and } \\
f(y+1)+f(1) & =f(y)+f(2)
\end{aligned}
$$

Therefore
$$
f(x+y)=f(x)+f(y)+b \quad \text { for all } x, y \in \mathbb{R}^{+},
$$
where $b=f(2)-2 f(1)=f(2)-2$. Using (5), (7) and (6), we get
$$
4+2 b=2 f(2)=f(2 f(2))=f(f(2)+f(2))=f(f(2))+f(f(2))+b=4+b .
$$

This shows that $b=0$ and thus
$$
f(x+y)=f(x)+f(y) \text { for all } x, y \in \mathbb{R}^{+} .
$$

In particular, $f$ is strictly increasing.
We conclude as follows. Take any positive real number $x$. If $f(x)>x$, then $f(f(x))>$ $f(x)>x=f(f(x))$, a contradiction. Similarly, it is not possible that $f(x)<x$. This shows that $f(x)=x$ for all positive real numbers $x$.