1. Find all values of $a$, for each of which the maximum of the two numbers $x^{3}+3 x+a-9$ and $a+2^{5-x}-3^{x-1}$ is positive for any $x$. Answer: $a>-5$.
Solution. The first function (of $x$) is strictly increasing, while the second is decreasing. Therefore, the requirement for the parameter $a$ specified in the problem means that the intersection point of the graphs of these functions lies above the x-axis, i.e., their value at the root (guessed) of the equation

$$
x^{3}+3 x+a-9=a+2^{5-x}-3^{x-1} \Leftrightarrow x=2
$$

is positive, i.e.,

$$
a+2^{3}-3^{1}>0 \Leftrightarrow a>-5
$$