2007 IMO Shortlist G6

Determine the smallest positive real number $k$ with the following property. Let $A B C D$ be a convex quadrilateral, and let points $A_{1}, B_{1}, C_{1}$ and $D_{1}$ lie on sides $A B, B C$, $C D$ and $D A$, respectively. Consider the areas of triangles $A A_{1} D_{1}, B B_{1} A_{1}, C C_{1} B_{1}$, and $D D_{1} C_{1}$; let $S$ be the sum of the two smallest ones, and let $S_{1}$ be the area of quadrilateral $A_{1} B_{1} C_{1} D_{1}$. Then we always have $k S_{1} \geq S$. (U.S.A.) Answer. $k=1$.

使用以下属性确定最小正实数 $k$。设$A B C D$为凸四边形，并设点$A_{1}、B_{1}、C_{1}$和$D_{1}$分别位于边$A B、B C$、$C D$和$D A$上。考虑三角形 $A A_{1} D_{1}、B B_{1} A_{1}、C C_{1} B_{1}$ 和 $D D_{1} C_{1}$ 的面积；设 $S$ 为两个最小数之和，$S_{1}$ 为四边形 $A_{1} B_{1} C_{1} D_{1}$ 的面积。那么我们总是有 $k S_{1} \geq S$。 (美国)回答。 $k=1$。