2008 Putnam A1

Let $f: \mathbb{R}^2 \to \mathbb{R}$ be a function such that $f(x,y) + f(y,z)

+ f(z,x) = 0$for all real numbers$x$,$y$, and$z$. Prove that there exists

a function $g: \mathbb{R} \to \mathbb{R}$ such that $f(x,y) = g(x) - g(y)$

for all real numbers $x$ and $y$.

令 $f: \mathbb{R}^2 \to \mathbb{R}$ 为满足 $f(x,y) + f(y,z) 的函数

+ f(z,x) = 0$对于所有实数$x$、$y$和$z$。证明存在

函数 $g: \mathbb{R} \to \mathbb{R}$ 使得 $f(x,y) = g(x) - g(y)$

对于所有实数 $x$ 和 $y$。