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 $z$ .

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Comments

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

Cody Johnson - 7 years, 2 months ago

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

Cody Johnson - 7 years, 2 months ago

What is the equality case?

Daniel Liu - 7 years, 2 months ago

Cody Johnson - 7 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!

Daniel Liu - 7 years, 2 months ago

@Daniel Liu – Proofathon makes all of its problems.

Cody Johnson - 7 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.