For certain positive integers \(a,b,c\) and \(d\) the set of complex numbers \(z\) that satisfy \(|z - 5 \sqrt{3} - 5i| = 5\) is equivalent to the set of complex numbers \(z\) that satisfy \(\left| \frac{1}{z} - \frac{1}{a \sqrt{b}} + \frac{i}{c} \right| = \frac{1}{d}\), where \(b\) is not divisible by the square of a prime. Find \(a+b+c+d\).

abcdefghijklmnopqrst Nov 26, 2021

#1

#2**+2 **

Melody I tried graphing it on desmos, if you would like to play around with it

abcdefghijklmnopqrst
Nov 27, 2021

#3**0 **

thanks for showing me what you have done.

Your graph looks interesting but I can't say more.

My memory of complex numbers is rudimentary at best.

Melody
Nov 28, 2021

#6**+9 **

I just played with the numbers until the circles overlapped, and I got a=5, b=3, c=15, and d=15

Then I actually did the problem

Do you want to see my work?

abcdefghijklmnopqrst
Nov 28, 2021

#7**0 **

**abcde...bs wrote:**

*I just played with the numbers until the circles overlapped, and I got a=5, b=3, c=15, and d=15*

**This is the correct solution set** ... But me thinks you are serving a big pile of BULLSHIT as an early Christmas present...

Your graph loads with **A = 5; B = 1; C = 3; D = 9** producing two non-intersecting circles that are tangent to different points on the x-axis when y = 0. Setting A, B, C, D to the solution numbers also produces two non-intersecting circles ... meaning your equations are worthless.

For sure, I’d like to see your solution method... using your presented equations. If it is doable, this would be analogous to traveling from New Jersey to Florida via the North Pole.

...While you’re at the North Poll, you should ask Father Christmas for a muckrake for Christmas, because you will have to do a massive amount of muckraking to present a valid solution with this bullshit.

GA

--. .-

GingerAle
Nov 28, 2021

edited by
Guest
Nov 28, 2021

#8**0 **

Here is your graph abcd...

I have not thought about it very hard but I also cannot see, offhand, how this graph shows anything.

Melody Nov 28, 2021

#9**+8 **

I'm very sorry, I made a rookie mistake and forgot to save my changes on the desmos

I've fixed it now if you want to see

abcdefghijklmnopqrst
Nov 29, 2021

#10**0 **

**A new link is created after saving changes on a Desmos graph. **

**You will need to post this new link. **

GA

--. .-

GingerAle
Nov 29, 2021

#13**0 **

GingerAle,

Do you still think abcde...bs served a big pile of B******T as an early Christmas present?

Also someone took the point off your post above.

Guest Dec 1, 2021

#14**0 **

*Do you still think abcde...bs served a big pile of BULLSHIT as an early Christmas present?*

Hell Yes! In addition, for the second round he served a well-muckraked, circular pile of steaming* *BULLSHIT! This is worthy of a Fields MeTal nomination. Category: Organic Houdini Processing Algorithms for Pseudo-Mathematical Solutions.

So, it seems in the twenty hours (19 hours and 47 minutes actually) it took for abcde...bs to post his link, he went to the North Pole, collected a muck rake from Farther Christmas, then went to the South Pole to consult with the penguins, waved to Melody while crossing Australia, then corrected his equation(s) by muck-raking the original BS into circles that will actually converge.

A fair amount of effort, but it is **NOT a solution** to this question! At best it will only confirm a known solution set. If the solution-set was unknown it could take hours to intuit the size and movement of the circle to find number set. Move the sliders while looking only at the graph and try to make it fit. It’s not obvious!

Incrementing the sliders sequentially from their lowest settings will require 74,235 iterations. At one step per second it will take 20 hours and 37 minutes. Some of it can be intuited, but it still will require hours.

**Abcde**, you are doing yourself disservice pursuing pseudo-mathematical solutions, because you’ve stopped looking for the correct solution methods. And others who will read your post may also slide down the slippery-slope into BS infused muck.

**You have the skills to correctly translate complex equations into circles** (perhaps this came from the consultation with the penguins) this is not a casual skill. **This means you know a third of what’s needed to actually solve these numerically.** The next step is to learn how to invert circle equations....

----------

*Also someone took the point off your post above.*

I noticed. It’s a prerogative of the trolled. I’ve also noticed through the years that no one is more concerned about my points than you. So, now I know for sure it’s you, JB (like I wouldn’t have, anyway). You should claim your copyright.

GA

--. .-

GingerAle
Dec 1, 2021