Problem 310

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Problem 310: (2021 AMC 12b, 14)

Let $ABCD$ be a rectangle and let $\overline{DM}$ be a segment perpendicular to the plane of $ABCD$. Suppose that $\overline{DM}$ has integer length, and the lengths of $\overline{MA},\overline{MC},$ and $\overline{MB}$ are consecutive odd positive integers (in this order). What is the volume of pyramid $MABCD?$

$\textbf{(A) }24\sqrt5 \qquad$ $ \textbf{(B) }60 \qquad$ $ \textbf{(C) }28\sqrt5\qquad$ $ \textbf{(D) }66 \qquad$ $ \textbf{(E) }8\sqrt{70}$

Solution

Exercise 310: Solution

Let $ABCD$ be a rectangle and let $\overline{DM}$ be a segment perpendicular to the plane of $ABCD$. Suppose $DM$ has integer length, and the lengths $MA$, $MC$, $MB$ are positive integers that increase by $3$ at each step (in this order). What is the volume of pyramid $MABCD$?

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