Question 39
Determine the smallest positive real number $k$ with the following property.
Let $ABCD$ be a convex quadrilateral, and let points $A_{1}, B_{1}, C_{1}$ and $D_{1}$ lie on sides $AB, BC$, $CD$ and $DA$, respectively. Consider the areas of triangles $AA_{1}D_{1}, BB_{1}A_{1}, CC_{1}B_{1}$, and $DD_{1}C_{1}$; 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 $k S_{1} \geq S$.
Let $ABCD$ be a convex quadrilateral, and let points $A_{1}, B_{1}, C_{1}$ and $D_{1}$ lie on sides $AB, BC$, $CD$ and $DA$, respectively. Consider the areas of triangles $AA_{1}D_{1}, BB_{1}A_{1}, CC_{1}B_{1}$, and $DD_{1}C_{1}$; 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 $k S_{1} \geq S$.