HMMT 二月 2022 · 冲刺赛 · 第 14 题

HMMT February 2022 — Guts Round — Problem 14

[9] The area of the largest regular hexagon that can fit inside of a rectangle with side lengths 20 and 22 √ can be expressed as a b − c , for positive integers a , b , and c , where b is squarefree. Compute 100 a +10 b + c .

解析

[9] The area of the largest regular hexagon that can fit inside of a rectangle with side lengths 20 and √ 22 can be expressed as a b − c , for positive integers a , b , and c , where b is squarefree. Compute 100 a + 10 b + c . Proposed by: Akash Das Answer: 134610 Solution: Let s be the sidelength of the hexagon. We can view this problem as finding the maximal √ rectangle of with sides s and s 3 that can fit inside this rectangle. Let ABCD be a rectangle with AB = 20 and BC = 22 and let XY ZW be an inscribed rectangle with X on AB and Y on BC with √ √ XY = s and Y Z = s 3. Let BX = a and BY = b . Then, by similar triangles, we have AX = b 3 √ √ √ √ and CY = a 3. Thus, we have a + b 3 = 20 and a 3 + b = 22 . Solving gives us a = 11 3 − 10 and √ √ √ √ 2 s · 3 3 2 2 2 b = 10 3 − 11 , so s = a + b = 884 − 440 3. Thus, the area of the hexagon is = 1326 3 − 1980. 2