Problem 2. Determine the integers $x, y$, for which

$$
5^{x}-\log _{2}(y+3)=3^{y} \text { and } 5^{y}-\log _{2}(x+3)=3^{x}
$$
Solution. Subtracting the equalities, we obtain

$$
5^{x}+3^{x}+\log _{2}(x+3)=5^{y}+3^{y}+\log _{2}(y+3)
$$

Since the function $f(t)=5^{t}+3^{t}+\log _{2}(t+3)$ is strictly increasing, it follows that $x=y$.

For solving the equation in $Z$

$$
5^{x}=3^{x}+\log _{2}(x+3)
$$

it is observed that $x \in\{-2,-1,0\}$ do not satisfy, while $x=1$ is a solution.

For $x \geq 2$, it is shown that

$$
5^{x} \geq 3^{x}+4^{x}
$$

using the monotonicity of the function $g(t)=\left(\frac{3}{5}\right)^{x}+\left(\frac{4}{5}\right)^{x}$

and then the inequality

$$
4^{x}>\log _{2}(x+3)
$$

is demonstrated by mathematical induction.