Problem 135

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Problem 135: (2018 AMC 10B, 17)

In rectangle $PQRS$, $PQ=8$ and $QR=6$. Points $A$ and $B$ lie on $\overline{PQ}$, points $C$ and $D$ lie on $\overline{QR}$, points $E$ and $F$ lie on $\overline{RS}$, and points $G$ and $H$ lie on $\overline{SP}$ so that $AP=BQ<4$ and the convex octagon $ABCDEFGH$ is equilateral. The length of a side of this octagon can be expressed in the form $k+m\sqrt{n}$, where $k$, $m$, and $n$ are integers and $n$ is not divisible by the square of any prime. What is $k+m+n$? $\textbf{(A) } 1 \qquad$ $ \textbf{(B) } 7 \qquad$ $ \textbf{(C) } 21 \qquad$ $ \textbf{(D) } 92 \qquad$ $ \textbf{(E) } 106$

Solution

Exercise 135: Solution

In rectangle $PQRS$, $PQ=10$ and $QR=6$. Points $A$ and $B$ lie on $\overline{PQ}$, points $C$ and $D$ lie on $\overline{QR}$, points $E$ and $F$ lie on $\overline{RS}$, and points $G$ and $H$ lie on $\overline{SP}$ so that $AP=BQ<5$ and $CQ=DR<3$. The convex octagon $ABCDEFGH$ is equilateral. Its side length can be written as $k+m\sqrt n$, where $k$, $m$, and $n$ are integers and $n$ is not divisible by the square of any prime. What is $k+m+n$?

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