Question 60, Given real numbers $a$, $b$, and $c$ satisfy: $f(x)=a \cos x+b \cos 2 x+c \cos 3 x \geq-1$ for any real number $x$, try to find the maximum value of $a+b+c$.

 untranslated part:
 已知实数 $a 、 b$ 、 c满足: $f(x)=a \cos x+b \cos 2 x+c \cos 3 x \geq-1$ 对任意实数 $\mathrm{x}$ 均成立, 试求 $\mathrm{a}+\mathrm{b}+\mathrm{c}$ 的最大值。

translated part:
Given real numbers $a$, $b$, and $c$ satisfy: $f(x)=a \cos x+b \cos 2 x+c \cos 3 x \geq-1$ for any real number $x$, try to find the maximum value of $a+b+c$.
Question 60, Answer: Frog Intention
$$
\begin{array}{l}
\mathrm{f}(0)+\mathrm{f}\left(\frac{\pi}{2}\right)+\mathrm{f}(\pi)+\mathrm{f}\left(\frac{3 \pi}{2}\right)=0 \\
\Rightarrow \mathrm{a}+\mathrm{b}+\mathrm{c}=\mathrm{f}(0)=-\left(\mathrm{f}\left(\frac{\pi}{2}\right)+\mathrm{f}(\pi)+\mathrm{f}\left(\frac{3 \pi}{2}\right)\right) \leq 3
\end{array}
$$

On the other hand, when $a=\frac{3}{2}, b=1, c=\frac{1}{2}$, $a+b+c=3$, and for any real number $x$, we have
$$
f(x)=\frac{3}{2} \cos x+\cos 2 x+\frac{1}{2} \cos 3 x=2 \cos ^{2} x(1+\cos x)-1 \geq-1
$$

In summary, the maximum value of $a+b+c$ is 3.