HMMT 二月 2006 · TEAM2 赛 · 第 8 题

HMMT February 2006 — TEAM2 Round — Problem 8

[25] A regular 12-sided polygon is inscribed in a circle of radius 1. How many chords of the circle that join two of the vertices of the 12-gon have lengths whose squares are rational? (No proof is necessary.)

解析

[25] A regular 12-sided polygon is inscribed in a circle of radius 1. How many chords of the circle that join two of the vertices of the 12-gon have lengths whose squares are rational? (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 a chord with central angle θ , we can draw radii from the two endpoints of the chord to the 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.