HMMT 十一月 2015 · 冲刺赛 · 第 9 题

HMMT November 2015 — Guts Round — Problem 9

[ 7 ] Find the smallest positive integer n such that there exists a complex number z , with positive real n n and imaginary part, satisfying z = ( z ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2015, November 14, 2015 — GUTS ROUND Organization Team Team ID#

解析

[ 7 ] Find the smallest positive integer n such that there exists a complex number z , with positive real n n and imaginary part, satisfying z = ( z ) . Proposed by: Alexander Katz 1 Since | z | = | z | we may divide by | z | and assume that | z | = 1. Then z = , so we are looking for the z th smallest positive integer n such that there is a 2 n root of unity in the first quadrant. Clearly there is a sixth root of unity in the first quadrant but no fourth or second roots of unity, so n = 3 is the smallest.