HMMT November

Observe that $[X B D]=[B A D]+[B A X]+[D A X]$. We will find the area of each of these triangles individually. - We have $[A B D]=\frac{1}{2}[A B C D]$. - Because $A M=A X,[B A X]=[B A M]$ as the triangles have the same base and height. Thus, as $[B A M]$ have the same base and height as $A B C D,[B A X]=[B A M]=\frac{1}{2}[A B C D]$. - From similar reasoning, we know that $[D A X]=[D A M]$. We have that $D A M$ has the same base and half the height of the rectangle. Thus, $[D A X]=[D A M]=\frac{1}{4}[A B C D]$. Hence, we have $$\begin{aligned} {[X B D] } & =[B A D]+[B A X]+[D A X] \\ & =\frac{1}{2}[A B C D]+\frac{1}{2}[A B C D]+\frac{1}{4}[A B C D] \\ & =\frac{5}{4}[A B C D] \end{aligned}$$ Thus, our answer is $\frac{5}{4}[A B C D]=\frac{5}{4}(20 \cdot 23)=575$.