MathDB

Complex numbers

Source: 2012 China TST Quiz 1 Day 1 P1

March 14, 2012

inequalitiescomplex numberstriangle inequalityalgebra proposedalgebra

Problem Statement

Complex numbers

{x_i},{y_i}

satisfy

\left| {{x_i}} \right| = \left| {{y_i}} \right| = 1

for

i=1,2,\ldots ,n

. Let

x=\frac{1}{n}\sum\limits_{i=1}^n{{x_i}}

y=\frac{1}{n}\sum\limits_{i=1}^n{{y_i}}

and

z_i=x{y_i}+y{x_i}-{x_i}{y_i}

. Prove that

\sum\limits_{i=1}^n{\left| {{z_i}}\right|}\leqslant n