Maximize complex numbers

Algebra Level 4

Determine the largest real number $\color{#3D99F6}{k}$ such that $\large | z_1z_2 + z_2z_3 + z_3z_1 | \geq \color{#3D99F6}{k} | z_1 + z_2 + z_3|$ for all complex numbers $z_1 , z_2 , z_3$ with unit absolute value.

6 None of these 2 1 7 5 4 3

1 solution

Tanishq Varshney
Oct 21, 2015

$\large{Given\quad \left| { z }_{ 1 } \right| =\left| { z }_{ 2 } \right| =\left| { z }_{ 3 } \right| =1\\ We\quad know,\quad z\bar { z } ={ \left| z \right| }^{ 2 }\\ { \left| { z }_{ 1 }{ z }_{ 2 }+{ z }_{ 2 }{ z }_{ 3 }+{ z }_{ 3 }{ z }_{ 1 } \right| }^{ 2 }\ge { k }^{ 2 }{ \left| { z }_{ 1 }+{ z }_{ 2 }+{ z }_{ 3 } \right| }^{ 2 }\\ 3+\bar { { z }_{ 1 } } { z }_{ 3 }+{ z }_{ 1 }\bar { { z }_{ 3 } } +\bar { { z }_{ 2 } } { z }_{ 3 }+{ z }_{ 2 }\bar { { z }_{ 3 } } +\bar { { z }_{ 1 } } { z }_{ 2 }+{ z }_{ 1 }\bar { { z }_{ 2 } } \ge { k }^{ 2 }\left( 3+\bar { { z }_{ 1 } } { z }_{ 3 }+{ z }_{ 1 }\bar { { z }_{ 3 } } +\bar { { z }_{ 2 } } { z }_{ 3 }+{ z }_{ 2 }\bar { { z }_{ 3 } } +\bar { { z }_{ 1 } } { z }_{ 2 }+{ z }_{ 1 }\bar { { z }_{ 2 } } \right) \\ { k }^{ 2 }\le 1\\ \boxed { \max { \left( k \right) =1 } } }$

Good use of property,

Akhil Bansal - 5 years, 7 months ago

I didn't understand your solution. Would you mind explaining it in more detail? This problem really interests me. For starters, how do you define $z$ ? If you take $z_1=z_2=z_3=1$ , then your equation says $z\bar{z}=9|z|^2$ . This would imply $z=0$ . Hopefully, you can see how I am confused...

James Wilson - 5 months, 1 week ago

I think I figured out that you are missing an equal sign there. However, I still only understand part of the solution. How do you arrive at $k^2\leq 1$ ?

James Wilson - 5 months, 1 week ago

Maximize complex numbers

1 solution

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