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Last edited by hbghlyj 2022-12-19 02:50University of Rochester Math Olympiad – 2015Olympiad
4. Let $\mathcal{C}$ be a smooth closed curve in the plane that intersects itself in only finitely many points. (The curve may pass through a single point several times, but is not tangent to itself.) Orient $\mathcal{C}$ by tracing out the entire curve in a particular direction, indicated by the arrows in the diagram. Prove there exists an enclosed region, such as the shaded one, whose boundary is consistently oriented by $\mathcal{C}$, meaning that every portion of the boundary is traversed in the same direction by $\mathcal{C}$, either clockwise or counterclockwise.
4. 设 $\mathcal{C}$ 是平面中的一条光滑闭合曲线，它仅有有限多个自交点（可能多次通过一个点，但不与自身相切）。以顺时针或逆时针描绘整个曲线来定向 $\mathcal{C}$，如图中的箭头所示。
证明存在一个封闭区域，如图中的阴影区域，其边界被 $\mathcal{C}$ 沿一致的方向描绘，即$\mathcal{C}$ 以相同的方向（顺时针或逆时针）描绘边界的每一段。
我今天看到的，觉得蛮有意思，发上来找高手解答，然后学习之 |
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Czhang271828
Posted 2022-12-19 18:08
