System Of Complex Equations - Add A Constraint

Suppose that z1,z2,z3,z4,z5z_1, z_2, z_3, z_4, z_5 are complex numbers which satisfy

{z1=z2=z3=z4=z5=1,z1+z2+z3+z4+z5=0,z1z2+z2z3+z4z5+z5z1=0 \begin{cases} |z_1| = |z_2| = |z_3| = |z_4| = |z_5| =1, \\ z_1 + z_2 + z_3 + z_4 + z_5 = 0, \\ z_1 z_2 + z_2 z_3 + z_4 z_5 + z_5 z_1 = 0 \\ \end{cases}

What can we conclude about z1,z2,z3,z4,z5 z_1, z_2, z_3, z_4, z_5 ?


You may refer to System Of Complex Equations, in which Mursalin and Prakhar showed that if only the first 2 conditions are true, then we can't really conclude too much. Other than a regular pentagon, the 5 points could be that of an equilateral triangle along with a diameter, or even 5 'somewhat random' points on the circle (no easy description).

How does adding in the third condition help us?

Note by Calvin Lin
5 years, 5 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

Sort by:

Top Newest

Rearranging the third equation z2(z1+z3)=z5(z4+z1)z_2(z_1+z_3)=-z_5(z_4+z_1) and taking modulus on both the sides, one obtains: z1+z3=z1+z4|z_1+z_3|=|z_1+z_4|. This means that the distance of z1z_1 from z3z_3 and z4z_4 is same. Join the centre with z1z_1. Call this line OA. Now, OA divides the circle into two semicircles. z3z_3 and z4z_4 obviously cannot lie on the same semicircle. They lie symmetrically about OA on different semicircles. Further divide the semicircle in four quadrants. Here is an image to go with my explanation:

Notice that z3z_3 and z4z_4 cannot lie in 1st and 4th quadrant. If they did, z3+z4z_3+z_4 would be along z1z_1 and z1+z3+z4>1|z_1|+|z_3+z_4|>1 but since z2+z5<1|z_2+z_5|<1, it would be impossible to satisfy the second condition. So, z3z_3 and z4z_4 must lie in 3rd and 2nd quadrants.

By a similar argument, z2z_2 and z5z_5 must lie in 3rd and 2nd quadrant. They too must lie symmetrically about OA and on different sections of semicircles.

Hence, it is a pentagon but not necessarily a regular one.

I am not sure if this is correct, though.

Pranav Arora - 5 years, 5 months ago

Log in to reply

This is a great piece of analysis!

I just realized that I screwed up the 3rd equation, and missed out the z3z4 z_3 z_4 term. Let me post a new note to reflect that.

Calvin Lin Staff - 5 years, 5 months ago

Log in to reply

Ah!

Thanks Calvin! :)

Pranav Arora - 5 years, 5 months ago

Log in to reply

They may be 5 th roots of unity

Kanthi Deep - 5 years, 5 months ago

Log in to reply

They may be vertices of regular Pentagon inscribed the in a unit circle

Kanthi Deep - 5 years, 5 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...