2019 APMO 第 5 题

Determine all the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that

$$f\left(x^{2}+f(y)\right)=f(f(x))+f\left(y^{2}\right)+2 f(x y)$$

for all real number $x$ and $y$.
Answer: The possible functions are $f(x)=0$ for all $x$ and $f(x)=x^{2}$ for all $x$.
Determine all the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that

$$f\left(x^{2}+f(y)\right)=f(f(x))+f\left(y^{2}\right)+2 f(x y)$$

对于所有实数 $x$ 和 $y$。

Answer: The possible functions are $f(x)=0$ for all $x$ and $f(x)=x^{2}$ for all $x$.