HMMT 二月 2007 · 冲刺赛 · 第 3 题

HMMT February 2007 — Guts Round — Problem 3

[ 5 ] The equation x + 2 x = i has two complex solutions. Determine the product of their real parts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . th 10 HARVARD-MIT MATHEMATICS TOURNAMENT, 24 FEBRUARY 2007 — GUTS ROUND

解析

[ 5 ] The equation x + 2 x = i has two complex solutions. Determine the product of their real parts. √ √ iπ 1 − 2 2 4 Answer: . Complete the square by adding 1 to each side. Then ( x + 1) = 1 + i = e 2, so 2 √ iπ 4 8 x + 1 = ± e 2. The desired product is then ( ( )) √ π ( ( ) ) ( ( ) ) ( ) √ √ √ √ 1 + cos π π π 1 − 2 4 4 2 4 − 1 + cos 2 − 1 − cos 2 = 1 − cos 2 = 1 − 2 = . 8 8 8 2 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . th 10 HARVARD-MIT MATHEMATICS TOURNAMENT, 24 FEBRUARY 2007 — GUTS ROUND