HMMT 二月 2019 · 冲刺赛 · 第 19 题

HMMT February 2019 — Guts Round — Problem 19

[ 9 ] Complex numbers a , b , c form an equilateral triangle with side length 18 in the complex plane. If | a + b + c | = 36, find | bc + ca + ab | .

解析

[ 9 ] Complex numbers a , b , c form an equilateral triangle with side length 18 in the complex plane. If | a + b + c | = 36, find | bc + ca + ab | . Proposed by: Henrik Boecken Answer: 432 a + b + c Using basic properties of vectors, we see that the complex number d = is the center of the 3 ′ ′ ′ triangle. From the given, | a + b + c | = 36 = ⇒ | d | = 12. Then, let a = a − d, b = b − d , and c = c − d . ′ ′ ′ ′ ′ ′ ′ ′ ′ Due to symmetry, | a + b + c | = 0 and | b c + c a + a b | = 0. Finally, we compute ′ ′ ′ ′ ′ ′ | bc + ca + ab | = | ( b + d )( c + d ) + ( c + d )( a + d ) + ( a + d )( b + d ) | ′ ′ ′ ′ ′ ′ ′ ′ ′ 2 = | b c + c a + a b + 2 d ( a + b + c ) + 3 d | 2 2 = | 3 d | = 3 · 12 = 432 .