, 2024

Solution:
Let $P(x, y)$ denote the functional equation. From $P(x, 0)$, we have $f(f(x))=x+a f(0)$. Thus, the tripling trick gives $f(x+a f(0))=f(f(f(x)))=f(x)+a f(0)$.
Now, here is the main idea: $P(f(x), y)$ gives
$$\begin{aligned} f(f(f(x))+a y) & =a f(y)+f(x) \\ f(x+a f(0)+a y) & =f(x)+a f(y) \\ f(x+a y) & =f(x)+a f(y)-a f(0) \end{aligned}$$
In particular, plugging in $x=0$ into this equation gives $f(a y)=a f(y)+(1-a) f(0)$, so inserting it back to the same equation gives
$$f(x+a y)=f(x)+f(a y)-f(0),$$
for all rational numbers $x, y$. In particular, the function $g(x)=f(x)-f(0)$ is additive, so $f$ is linear. Let $f(x)=b x+c$. By substituting it in, we have $P(x, y)$ iff
$$\begin{aligned} f(a y+b x+c) & =a(b y+c)+x \\ a b y+b^{2} x+b c+c & =a b y+a c+x \\ \left(b^{2}-1\right) x+(b+1-a) c & =0 \end{aligned}$$
Since $x$ is arbitrary, we can state that $b^{2}-1=0$ and $(b+1-a) c=0$, thus $b= \pm 1$. As $a \neq 0$, we know $b+1-a=0$ only if $b=1$ and $a=2$. When $a \neq 2$ or $b \neq 1$, we know the only solutions are $b= \pm 1, c=0$, while for $a=2, b=1$, the equation is automatically satisfied, so the final answer is
$$\left\{\begin{array}{l} f(x)=x \\ f(x)=-x \\ f(x)=x+c \text{ for all rational number } c \text{ iff } a=2 \end{array}\right.$$

Solution:
We will only prove that $f$ is linear. Then, proceed as in the end of Solution 1.
We know $f(f(x))=a f(0)+x$, so as $a f(0)+x$ can take any rational number when $x$ takes every rational number, the range of $f$ is $\mathbb{Q}$, and so $f$ is surjective. If $f\left(x_{1}\right)=f\left(x_{2}\right)$, we have $x_{1}=f\left(f\left(x_{1}\right)\right)-a f(0)=f\left(f\left(x_{2}\right)\right)-a f(0)=x_{2}$, so $x_{1}=x_{2}$, implying $f$ being injective. Thus, $f$ is bijective.
From $P(x, 0)$, we still get $f(f(x))=x+a f(0)$.
Thus, from $P(f^{-1}(0), y / a)$, we can get $f(y)=a f(y / a)+f^{-1}(0)$. Plugging again $P(f(x), y / a)$, we have $f(f(f(x))+y)=f(x+y+a f(0))=a f(y / a)+f(x)=f(x)+f(y)-f^{-1}(x)$.
Thus, we know $f(x+y)=f(x-a f(0))+f(y)-f^{-1}(0)=f(x)+f(y)-f(0)$. Hence, the function $g(x)=f(x)-f(0)$ is additive, so $g(x)=k x$ for some rational number $k$. Thus, $f$ is a linear function, and we can proceed as in above solution.