HMMT 十一月 2024 · 团队赛 · 第 5 题

HMMT November 2024 — Team Round — Problem 5

[40] Let ABCD be a convex quadrilateral with area 202 , AB = 4 , and ∠ A = ∠ B = 90 such that ◦ there is exactly one point E on line CD satisfying ∠ AEB = 90 . Compute the perimeter of ABCD .

解析

[40] Let ABCD be a convex quadrilateral with area 202, AB = 4, and ∠ A = ∠ B = 90 such that ◦ there is exactly one point E on line CD satisfying ∠ AEB = 90 . Compute the perimeter of ABCD . Proposed by: Benjamin Shimabukuro Answer: 206 Solution: A D E B C ◦ The locus of point E such that ∠ AEB = 90 is the circle ω with diameter AB . Thus, if there exists unique point E , the circle ω must intersect line CD at exactly one point and hence line CD must be tangent to ω . ◦ Now, since ∠ DAB = 90 , we get that AD is tangent to ω , so DA = DE by equal tangents property. Similarly, CB = CE . Thus, CD = CE + DE = AD + BC. However, equating the given area of the quadrilateral gives 1 ( AD + BC ) · AB = 202 = ⇒ AD + BC = 101 . 2 Hence, the final answer is CD + ( AD + BC ) + AB = 101 + 101 + 4 = 206 .