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 \(|\cos(x)|+|\cos(y+z)|+|\cos(x+y+z)|\ge1\).

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Double hint: show that \(|\cos a|+|\cos b|+|\cos(a+b)|\ge1\).

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What is the equality case?

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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 \(\cos a=\cos b=\cos c=0\).

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Ah yea, thanks.

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!

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Well sorry to interrupt in a different question but can you please give the solution to charge oscillating above a charged sheet.

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Done.

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Thanks and good solution yaar!

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Come on lucky number 7 ?

Could you post a solution toLog in to reply

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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.

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