Hi, this one came in proofathon contest and had an average score of \(0\).

**Problem**

Prove that

\(|\cos (x)| + |\cos (y)| + |\cos (z)| + |\cos(y+z)| + |\cos(z+x)| + |\cos(x+y)| + 3|\cos(x+y+z)| \geq 3\)

for all real \(x, y,\) and \(z\).

Hi, this one came in proofathon contest and had an average score of \(0\).

**Problem**

Prove that

\(|\cos (x)| + |\cos (y)| + |\cos (z)| + |\cos(y+z)| + |\cos(z+x)| + |\cos(x+y)| + 3|\cos(x+y+z)| \geq 3\)

for all real \(x, y,\) and \(z\).

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## Comments

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TopNewestHint: show that \(|\cos(x)|+|\cos(y+z)|+|\cos(x+y+z)|\ge1\). – Cody Johnson · 3 years, 2 months ago

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– Cody Johnson · 3 years, 1 month ago

Double hint: show that \(|\cos a|+|\cos b|+|\cos(a+b)|\ge1\).Log in to reply

What is the equality case? – Daniel Liu · 3 years, 2 months ago

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– Cody Johnson · 3 years, 1 month ago

Well there is a lemma necessary in solving this problem: \(|\cos a|+|\cos b|+|\cos(a+b)|\ge1\) which has an equality case where \(\cos a=\cos b=0\). This leads us to the equality case that \(\cos a=\cos b=\cos c=0\).Log in to reply

I figured that out later, after thinking the problem over again. Did you make this problem? If you did, it's a really really good problem! – Daniel Liu · 3 years, 1 month ago

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– Cody Johnson · 3 years, 1 month ago

Proofathon makes all of its problems.Log in to reply

Well sorry to interrupt in a different question but can you please give the solution to charge oscillating above a charged sheet. – Milun Moghe · 3 years, 2 months ago

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– Jatin Yadav · 3 years, 2 months ago

Done.Log in to reply

– Milun Moghe · 3 years, 2 months ago

Thanks and good solution yaar!Log in to reply

Come on lucky number 7 ? – Jatin Yadav · 3 years, 1 month ago

Could you post a solution toLog in to reply

– Milun Moghe · 3 years, 1 month ago

Can you post a Sol to the flight of a housefly if solved please...Log in to reply

– Milun Moghe · 3 years, 1 month ago

Done , but don't know if it's correct. I am very eager to get clarified.Log in to reply

– Milun Moghe · 3 years, 1 month ago

I had posted the time before they met 1 and 2, it didnt get selected but roger kepstiens problem's what did the proton say to the electron got selected thats not fare , i posted it first. and a better question related to electrostatic imaging tooLog in to reply

We had greatly cleaned up Cricical Angle Of Precession Of A Re-assembled Top to give you an example of how smoother phrasing, better presentation and a clearer picture can greatly improve the quality of your problem.

The easier a problem is to understand, the more others will like and share it, which increases the likelihood that it would be selected. You can read my note for further guidelines to improve your problem. – Calvin Lin Staff · 3 years, 1 month ago

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– Milun Moghe · 3 years, 1 month ago

Well thanks for the advice, I'll try to make my problems more precise and to the pointLog in to reply

– Calvin Lin Staff · 3 years, 1 month ago

To check, did you get the email that I sent? We've sent some clarification requests about your other problems too. Responding to those would help you understand what aspects of your problems are confusing, and how to correct them.Log in to reply

– Milun Moghe · 3 years, 1 month ago

Yes sir I have received all emails and have made corrections Accordingly.Log in to reply

– Raja Ram epsilon Roy · 3 years, 1 month ago

Same in need over here. I had posted my solution in one of the comments which got deletedLog in to reply