HMMT 二月 2005 · 冲刺赛 · 第 8 题

HMMT February 2005 — Guts Round — Problem 8

[6] Let ABCD be a convex quadrilateral inscribed in a circle with shortest side AB . The ratio [ BCD ] / [ ABD ] is an integer (where [ XY Z ] denotes the area of triangle XY Z .) If the lengths of AB , BC , CD , and DA are distinct integers no greater than 10, find the largest possible value of AB .

解析

Let ABCD be a convex quadrilateral inscribed in a circle with shortest side AB . The ratio [ BCD ] / [ ABD ] is an integer (where [ XY Z ] denotes the area of triangle XY Z .) If the lengths of AB , BC , CD , and DA are distinct integers no greater than 10, find the largest possible value of AB . Solution: 5 Note that 1 BC · CD · sin C [ BCD ] BC · CD 2 = = 1 [ ABD ] DA · AB DA · AB · sin A 2 since ∠ A and ∠ C are supplementary. If AB ≥ 6, it is easy to check that no assignment of lengths to the four sides yields an integer ratio, but if AB = 5, we can let BC = 10, CD = 9, and DA = 6 for a ratio of 3. The maximum value for AB is therefore 5.