2004 IMO Shortlist S01

Let $P$ be a convex polygon. Prove that there exists a convex hexagon that is contained in $P$ and whose area is at least $\frac34$ of the area of the polygon $P$ .

*Alternative version.* Let $P$ be a convex polygon with $n\geq 6$ vertices. Prove that there exists a convex hexagon with **a)** vertices on the sides of the polygon (or)**b)** vertices among the vertices of the polygon

such that the area of the hexagon is at least $\frac{3}{4}$ of the area of the polygon.

*Proposed by Ben Green and Edward Crane, United Kingdom*

令 $P$ 为凸多边形。证明存在一个包含在 $P$ 中的凸六边形，其面积至少为多边形 $P$ 面积的 $\frac34$ 。

*替代版本。* 令 $P$ 为具有 $n\geq 6$ 个顶点的凸多边形。证明存在一个凸六边形，其多边形各边上有**a)**顶点(或多边形顶点中有**b)**顶点

使得六边形的面积至少为多边形面积的 $\frac{3}{4}$ 。

*由英国 Ben Green 和 Edward Crane 提议*