HMMT 十一月 2024 · 团队赛 · 第 3 题

HMMT November 2024 — Team Round — Problem 3

[30] Rectangle R with area 20 and diagonal of length 7 is translated 2 units in some direction to form ′ ′ a new rectangle R . The vertices of R and R that are not contained in the other rectangle form a convex hexagon. Compute the maximum possible area of this hexagon.

解析

[30] Rectangle R with area 20 and diagonal of length 7 is translated 2 units in some direction to form ′ ′ a new rectangle R . The vertices of R and R that are not contained in the other rectangle form a convex hexagon. Compute the maximum possible area of this hexagon. Proposed by: Ethan Liu Answer: 34 Solution: ′ ′ ′ ′ A A D D 2 ′ R D A A ′ ′ C C ′ B 7 R B B C C Dissect the hexagon as shown above, so that it consists of a parallelogram and two triangles which are each half the original rectangle. The parallelogram has side lengths 7 and 2, so its maximum possible area is 14. As the two triangles combined always have the same area as the original rectangle, 20, the answer is 14 + 20 = 34 .