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Cosines everywhere (Posted on 2005-03-21) Difficulty: 4 of 5
Prove that in any triangle ABC, 8.cos(A).cos(B).cos(C) < 1.

See The Solution Submitted by Federico Kereki    
Rating: 4.5000 (2 votes)

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Solution Solution | Comment 1 of 6

If ABC is obtuse or right, the left hand side is nonpositive so the inequality holds.

Otherwise, note that the function f(x)= ln(cos(x)) has second derivative f''(x)=-sec©÷(x)<0 on the interval (0,pi/2).  By Jensen's Inequality, then,

f(A)+f(B)+f(C) ¡Â 3*f((A+B+C)/3) = 3*f(60).

Exponentiating yields the desired result.  (The problem statement should have ¡Â rather than <)


  Posted by David Shin on 2005-03-21 19:13:57
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