# A nice trig. problem

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 $$z$$.

4 years, 2 months ago

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Hint: show that $$|\cos(x)|+|\cos(y+z)|+|\cos(x+y+z)|\ge1$$.

- 4 years, 2 months ago

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

- 4 years, 2 months ago

What is the equality case?

- 4 years, 2 months 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 $$\cos a=\cos b=\cos c=0$$.

- 4 years, 2 months ago

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!

- 4 years, 2 months ago

Proofathon makes all of its problems.

- 4 years, 2 months ago

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

- 4 years, 2 months ago

Done.

- 4 years, 2 months ago

Thanks and good solution yaar!

- 4 years, 2 months ago

Could you post a solution to Come on lucky number 7 ?

- 4 years, 2 months ago

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

- 4 years, 2 months ago

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

- 4 years, 2 months 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 too

- 4 years, 2 months ago

As I have explained to you (in email), several of your problems have unnecessarily complicated scenarios and convoluted phrasing, which make it hard to understand what you are asking.

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.

Staff - 4 years, 2 months ago

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

- 4 years, 2 months 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.

Staff - 4 years, 2 months ago

Yes sir I have received all emails and have made corrections Accordingly.

- 4 years, 2 months ago

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

- 4 years, 2 months ago