<s>[INST] for any acute triangle $ABC:\cos^3 (A)+ \cos^3 (B)+\cos^3 (C)+\cos(A)\cdot\cos(B)\cdot\cos(C)\ge\frac{1}{2} $ What is the proof that for any acute triangle $ABC$,then :
$$\cos^3 (A)+ \cos^3 (B)+\cos^3 (C)+\cos(A)\cdot\cos(B)\cdot\cos(C)\ge\frac{1}{2} $$
 [/INST] Hints: Note that for acute triangle, cosines of all the angles are positive. So, by Arithmetic–Geometric means inequality $$\cos^3 x+\frac{\cos x}{4}\ge \cos^2 x\Leftrightarrow\cos^3 x\ge \cos^2 x-\frac{\cos x}{4},\quad\forall x=A,B,C$$
Hence we are left with the following quantity, $$\sum \cos^2x -\frac{1}{4}\sum \cos x +\Pi \cos x$$
Now there are many useful things, you have to use some of them: 


*

*For any triangle, $\sum \cos^2x +2\Pi \cos x=1$ 

*$\sum\cos x\le \frac{3}{2}$ 

*$\sum\cos^2 x\ge \frac{3}{4}$

*$\Pi\cos x\le \frac{1}{8}$


A Remark: Equality holds iff $A=B=C$
</s>