### 9.305 Find $a$, for which the inequality

$x^{2}-2^{a+2} \cdot x-2^{a+3}+12>0$ is true for any $x$.
Solution. The requirement of the problem is equivalent to the condition: the discriminant $D$ is less than 0, where $D=\left(2^{a+2}\right)^{2}-4\left(12-2^{a+3}\right)=16\left(2^{2 a}+2 \cdot 2^{a}-3\right)$. $D<0 \Leftrightarrow 2^{2 a}+2 \cdot 2^{a}-3<0 \Leftrightarrow\left(2^{a}+3\right)\left(2^{a}-1\right)<0 \Leftrightarrow 2^{a}<1 \Leftrightarrow a \in(-\infty ; 0)$. Answer: $a \in(-\infty ; 0)$.

## PROBLEMS IN PLANIMETRY

In the solutions of geometric problems, formulas and theorems from sections XII, XIV, XV of the reference materials are used.