Question

In rectangle $PQRS,$ $PQ = 8$ and $QR = 6.$ Points $A$ and $B$ lie on $PQ,$ points $C$ and $D$ lie on $QR,$ points $E$ and $F$ lie on $RS,$ and points $G$ and $H$ lie on $SP$ 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 $k + m + n?$

$1$

$7$

$21$

$92$

$106$

Solution:

By symmetry the four cut corners are congruent right triangles, with legs $x$ along the sides of length $8$ and $y$ along the sides of length $6.$ The octagon's sides come in three types, $8 - 2x,$ $6 - 2y,$ and $\sqrt{x^2 + y^2},$ and they're all equal. From $8 - 2x = 6 - 2y$ we get $y = x - 1.$ Substitute into $8 - 2x = \sqrt{x^2 + (x-1)^2}$ and square: $2x^2 - 30x + 63 = 0,$ so $x = \tfrac{15 - 3\sqrt{11}}{2}$ (taking the root with $x < 4$ ). The side length is $8 - 2x = -7 + 3\sqrt{11},$ so $k + m + n = -7 + 3 + 11 = 7.$ Thus, B is the correct answer.

2018 AMC 10B Problem 17

Solution:

Problem 17 in Other Years