Prove that in any triangle ABC, 8.cos(A).cos(B).cos(C) < 1.
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 <)
Solution
