HMMT 二月 2006 · TEAM2 赛 · 第 8 题
HMMT February 2006 — TEAM2 Round — Problem 8
题目详情
英文原题
- [25] A regular 12-sided polygon is inscribed in a circle of radius 1. How many chords of the circle thatjoin two of the vertices of the 12-gon have lengths whose squares are rational? (No proof is necessary.)
解析
英文解析
- [25] A regular 12-sided polygon is inscribed in a circle of radius 1. How many chordsof the circle that join two of the vertices of the 12-gon have lengths whose squares arerational? (No proof is necessary.)
Answer: 42
° ° ° ° °
Solution: The chords joining vertices subtend minor arcs of 30 , 60 , 90 , 120 , 150 ,
°
or 180 . There are 12 chords of each of the first five kinds and 6 diameters. For achord with central angle θ , we can draw radii from the two endpoints of the chord tothe center of the circle. By the law of cosines, the square of the length of the chord is
° ° ° °
1 + 1 − 2 cos θ, which is rational when θ is 60 , 90 , 120 , or 180 . The answer is thus
12 + 12 + 12 + 6 = 42.