2021 Putnam B3

Let $h(x,y)$ be a real-valued function that is twice continuously differentiable throughout $\mathbb{R}^2$, and define

$$

\rho(x,y) = yh_x - xh_y.

$$

Prove or disprove: For any positive constants $d$ and $r$ with $d>r$, there is a circle $\mathcal{S}$ of radius $r$ whose center is a distance $d$ away from the origin such that the integral of $\rho$ over the interior of $\mathcal{S}$ is zero.

设 $h(x,y)$ 为实值函数，在 $\mathbb{R}^2$ 中连续两次可微，并定义

$$

\rho(x,y) = yh_x - xh_y。

$$

证明或反证：对于任何正常数$d$和$r$且$d>r$，存在一个半径为$r$的圆$\mathcal{S}$，圆心距原点$d$，使得$\rho$在$\mathcal{S}$内部的积分为零。