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April 27th, 2019, 10:56 AM   #1
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Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems.
Question about complex absolute value

If $z$ is a complex number and $|z^2-1|=|4z-i|$ can I go ahead and say $$z^2-1=4z-i$$?

I'm using this to prove $|z|^2 \le 9+\sqrt{85}$. I'm not sure what steps to take other than what I mentioned.

PLEASE just give a hint. Do not post a full solution, just please tell me how to proceed.

Last edited by ProofOfALifetime; April 27th, 2019 at 11:07 AM.
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April 27th, 2019, 11:34 AM   #2
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Oh.. I was able to get it. Nevermind! Tricky but cool problem!
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April 27th, 2019, 12:36 PM   #3
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Originally Posted by ProofOfALifetime View Post
If $z$ is a complex number and $|z^2-1|=|4z-i|$ can I go ahead and say $$z^2-1=4z-i$$?
Not always .
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April 27th, 2019, 01:39 PM   #4
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I wanted to avoid using $z=a+bi$ but I don't think it's possible.
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April 27th, 2019, 06:50 PM   #5
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This problem is much harder than I thought!
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April 28th, 2019, 07:35 AM   #6
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Actually, it was hard because I messed up on a part of it so that problem no longer made sense. I finally solved it!
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April 29th, 2019, 02:12 PM   #7
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OMG if anyone wants to do this problem, I encourage it! I finally solved it and my goodness it was like a journey! I don't know about you all, but I found this problem challenging but the solution rewarding!

SORRY for spamming the thread!

NOTE:If you do solve it, DO NOT post a solution since it is from a math journal and technically isn't supposed to be posted online, but DEFINITELY do not post a the solution! Thanks! You guys rock!

Last edited by ProofOfALifetime; April 29th, 2019 at 02:26 PM.
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