1996 Putnam B6

Let $(a_1, b_1), (a_2, b_2), \ldots, (a_n, b_n)$ be the vertices of a

convex polygon which contains the origin in its interior. Prove that

there exist positive real numbers $x$ and $y$ such that

\begin{gather*}

(a_1, b_1)x^{a_1} y^{b_1} + (a_2, b_2)x^{a_2}y^{b_2} + \cdots \\

+ (a_n, b_n)x^{a_n}y^{b_n} = (0,0).

\end{gather*}

设 $(a_1, b_1), (a_2, b_2), \ldots, (a_n, b_n)$ 为 a 的顶点

内部包含原点的凸多边形。证明

存在正实数 $x$ 和 $y$ 使得

\开始{收集*}

(a_1, b_1)x^{a_1} y^{b_1} + (a_2, b_2)x^{a_2}y^{b_2} + \cdots \\

+ (a_n, b_n)x^{a_n}y^{b_n} = (0,0)。

\结束{聚集*}