## Task $1 / 85$

Find all solutions of the equation $3^{x}+4^{x}=5^{x}$ in real numbers $x$.
1. Observation: $x_{1}=2$ is a solution.
2. Observation: The given equation is equivalent to the equation

$$
\left(\frac{3}{5}\right)^{x}+\left(\frac{4}{5}\right)^{x}-1=0
$$

3. Observation: The function $y=a^{x}$ with $0<a<1$ is strictly decreasing. Consequently, the function

$$
y=\left(\frac{3}{5}\right)^{x}+\left(\frac{4}{5}\right)^{x}-1
$$

is also a strictly decreasing function.

4. Observation: A strictly monotonic function takes each value at most once. Therefore, it has at most one root. Thus, $x_{1}=2$ is the only solution.