HMMT 十一月 2023 · 冲刺赛 · 第 29 题

HMMT November 2023 — Guts Round — Problem 29

[15] Let A A . . . A be a regular hexagon with side length 11 3, and let B B . . . B be another regular 1 2 6 1 2 6 hexagon completely inside A A . . . A such that for all i ∈ { 1 , 2 , . . . , 5 } , A A is parallel to B B . 1 2 6 i i +1 i i +1 Suppose that the distance between lines A A and B B is 7, the distance between lines A A and B B 1 2 1 2 2 3 2 3 is 3, and the distance between lines A A and B B is 8. Compute the side length of B B . . . B . 3 4 3 4 1 2 6

解析

[15] Let A A . . . A be a regular hexagon with side length 11 3, and let B B . . . B be another 1 2 6 1 2 6 regular hexagon completely inside A A . . . A such that for all i ∈ { 1 , 2 , . . . , 5 } , A A is parallel 1 2 6 i i +1 to B B . Suppose that the distance between lines A A and B B is 7, the distance between lines i i +1 1 2 1 2 A A and B B is 3, and the distance between lines A A and B B is 8. Compute the side length of 2 3 2 3 3 4 3 4 B B . . . B . 1 2 6 Proposed by: Pitchayut Saengrungkongka √ Answer: 3 3 Solution: A A 2 1 X 7 B B 2 1 3 O B B 3 6 8 A A 3 B B 6 4 5 A A 4 5 Let X = A A ∩ A A , and let O be the center of B B . . . B . Let p be the apothem of hexagon B . 1 2 3 4 1 2 6 Since OA XA is a convex quadrilateral, we have 2 3 [ A A X ] = [ A XO ] + [ A XO ] − [ A A O ] 2 3 2 3 2 3 √ √ √ 11 3(7 + p ) 11 3(8 + p ) 11 3(3 + p ) = + − 2 2 2 √ 11 3(12 + p ) = . 2 √ √ 2 3 Since [ A A X ] = (11 3) , we get that 2 3 4 √ √ 12 + p 3 33 9 = (11 3) = = ⇒ p = . 2 4 4 2 √ 2 √ Thus, the side length of hexagon B is p · = 3 3. 3