HMMT 十一月 2015 · 冲刺赛 · 第 9 题
HMMT November 2015 — Guts Round — Problem 9
题目详情
- [ 7 ] Find the smallest positive integer n such that there exists a complex number z , with positive realn nand imaginary part, satisfying z = ( z ) .
HMMT November 2015, November 14, 2015 — GUTS ROUND
Organization Team Team ID#
英文原题
[ 7 ] Find the smallest positive integer n such that there exists a complex number z , with positive real
and imaginary part, satisfying z n = ( z ) n .
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HMMT November 2015, November 14, 2015 — GUTS ROUND
Organization Team Team ID#
解析
英文解析
- [ 7 ] Find the smallest positive integer n such that there exists a complex number z , with positive realn nand imaginary part, satisfying z = ( z ) .
Proposed by: Alexander Katz
Since | z | = | z | we may divide by | z | and assume that | z | = 1. Then z = , so we are looking for the 1
smallest positive integer n such that there is a 2 n root of unity in the first quadrant. Clearly therethzis a sixth root of unity in the first quadrant but no fourth or second roots of unity, so n = 3 is thesmallest.