Question 54
Let $ABCD$ be a regular tetrahedron with side length $1$. Let $X$ be the point in triangle $BCD$ such that $[XBC] = 2[XBD] = 4[XCD]$, where $[\varpi]$ denotes the area of figure $\varpi$. Let $Y$ lie on segment $AX$ such that $2AY = YX$. Let $M$ be the midpoint of $BD$. Let $Z$ be a point on segment $AM$ such that the lines $YZ$ and $BC$ intersect at some point. Find $\frac{AZ}{ZM}$.