## Task B-3.5.

Let $f$ be a real function with the following properties:

- $\mathcal{D}_{f}=[-4, \infty\rangle$,
- $f(0) \cdot f(1)<0$,
- the function $f$ is strictly decreasing on its entire domain and has one root.

Determine all $x \in \mathbf{R}$ for which the inequality $f\left(a^{2 x}+2 a^{x}-7\right)<f\left(a^{x}-5\right)$ holds, where $a$ is the root of the given function $f$.
## Solution.

Let's introduce the substitution $a^{x}=y$. Then we have $f\left(y^{2}+2 y-7\right)y-5
$$

which is

$$
y^{2}+y-2>0
$$

Solving this, we get $y \in\langle-\infty,-2\rangle \cup\langle 1, \infty\rangle$, and due to the condition $\left(^{*}\right)$, we have $y \in\langle 1, \infty\rangle$.

Therefore, the original inequality is equivalent to the inequality $a^{x}>1$.

From the condition $f(0) \cdot f(1)1=a^{0}$, it follows that $x<0$.

The final solution to the given inequality is $x \in\langle-\infty, 0\rangle$.

## NATIONAL MATHEMATICS COMPETITION

## 4th grade - high school - B variant

## April 25, 2023.