HMMT 十一月 2011 · 冲刺赛 · 第 9 题

HMMT November 2011 — Guts Round — Problem 9

[ 7 ] Unit circle Ω has points X, Y, Z on its circumference so that XY Z is an equilateral triangle. Let W be a point other than X in the plane such that triangle W Y Z is also equilateral. Determine the area of the region inside triangle W Y Z that lies outside circle Ω. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TH 4 ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 12 NOVEMBER 2011 — GUTS ROUND Round 4

解析

[ 7 ] Unit circle Ω has points X, Y, Z on its circumference so that XY Z is an equilateral triangle. Let W be a point other than X in the plane such that triangle W Y Z is also equilateral. Determine the area of the region inside triangle W Y Z that lies outside circle Ω. √ 3 3 − π ◦ Answer: Let O be the center of the circle. Then, we note that since ∠ W Y Z = 60 = ∠ Y XZ , 3 that Y W is tangent to Ω. Similarly, W Z is tangent to Ω. Now, we note that the circular segment 1 corresponding to Y Z is equal to the area of Ω less the area of triangle OY Z . Hence, our total area 3 is √ √ √ 1 3 3 1 3 3 3 − π [ W Y Z ] − [Ω] + [ Y OZ ] = − π + = . 3 4 3 4 3 Ω Y 1 X O W Z