HMMT 二月 2005 · 冲刺赛 · 第 8 题
HMMT February 2005 — Guts Round — Problem 8
题目详情
英文原题
- [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
英文解析
- Let ABCD be a convex quadrilateral inscribed in a circle with shortest side AB . Theratio [ 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, findthe largest possible value of AB .
Solution: 5
Note that
BC · CD · sin C1
[ BCD ] BC · CD = =2
[ ABD ] DA · AB1
DA · AB · sin Asince ∠ A and ∠ C are supplementary. If AB ≥ 6, it is easy to check that no assignment 2
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.