## Task A-3.4.

How many ordered pairs of natural numbers $(a, b)$ satisfy

$$
\log _{2023-2(a+b)} b=\frac{1}{3 \log _{b} a} ?
$$
## Solution.

For the equality to be well-defined, it is necessary that

$$
\begin{aligned}
2023-2(a+b) & \in (0,1) \cup (1,+\infty) \\
b & \in (0,1) \cup (1,+\infty) \\
\log _{b} a & \neq 0
\end{aligned}
$$

Since $a$ and $b$ are natural numbers, the first two conditions become $2023-2(a+b) \geqslant 2$ and $b \geqslant 2$. The last condition is equivalent to $a \geqslant 2$.

We can rearrange the initial equation as follows:

$$
\begin{aligned}
\frac{1}{\log _{b}(2023-2(a+b))} & =\frac{1}{\log _{b} a^{3}} \\
\log _{b}(2023-2(a+b)) & =\log _{b} a^{3} \\
2023-2(a+b)) & =a^{3} \\
a^{3}+2(a+b) & =2023 \\
a^{3}+2 a+2 b & =2023 .
\end{aligned}
$$

Notice that the condition $a^{3}=2023-2(a+b) \geqslant 2$ is automatically satisfied as long as $a \geqslant 2$. Therefore, we are looking for the number of natural solutions $(a, b)$ of the obtained equation that satisfy $a, b \geqslant 2$.

The number of solutions to this equation is equal to the number of odd natural numbers $a$ for which

$$
a^{3}+2 a \leqslant 2019
$$

Indeed, for the left and right sides of the equation to have the same parity, $a^{3}$ must be odd, i.e., $a$ must be odd. On the other hand, for any odd $a$ that satisfies the above inequality, the number $b=\frac{2023-a^{3}-2 a}{2}$ is a unique natural number that satisfies the desired equation.

For $a \leqslant 11$, we have $a^{3}+2 a \leqslant 11^{3}+2 \cdot 11=1353<2019$, while for $a \geqslant 13$, we have $a^{3}+2 a \geqslant 13^{3}+2 \cdot 13=2223$. Since there are 5 odd natural numbers less than or equal to 11 and greater than 2, we conclude that there are 5 ordered pairs that are solutions to the equation in the problem.

Note: The second point in the scoring scheme is achieved if an expression is obtained in which the number $(2023-2(a+b))$ does not appear in the base of the logarithm. The third point is achieved if an equation is obtained that does not include logarithms.