4. Determine for which values of the real parameter $k$ the system of equations

$$
\begin{aligned}
|x+6|+2|y| & =24 \\
|x+y|+|x-y| & =2 k
\end{aligned}
$$

has an odd number of solutions in the set of real numbers.

(Pavel Calábek)
SOLUTION. Notice that if the pair $(x, y)$ is a solution to the system, so is the pair $(x, -y)$. Solutions, therefore, typically "come in pairs." For example, if the pair $(x, y) = (2, 8)$ is a solution, then the pair $(2, -8)$ is also a solution. Only in the case where $y = -y$, i.e., $y = 0$, are the two solutions $(x, y)$ and $(x, -y)$ not distinct.

The given system can have an odd number of solutions only in the case where for $y = 0$ there exists an $x$ that satisfies both equations. In such a case, the first equation simplifies to $|x + 6| = 24$, so it must be that either $x = 18$ or $x = -30$. From the second equation, it follows that the pair $(18, 0)$ is a solution for $k = 18$ and the pair $(-30, 0)$ is a solution for $k = 30$. In both cases, there is no other solution $(x, y)$ for which $y = 0$, so the system has an odd number of solutions for both $k = 18$ and $k = 30$, provided that for these two $k$ there are not infinitely many solutions. The fact that the given system has only a finite number of solutions (for any $k$) follows from the fact that each of them is a solution to one of the 16 systems we get when in the "indeterminate" system

\[
\begin{aligned}
\pm(x + 6) \pm 2 y & = 24 \\
\pm(x + y) \pm (x - y) & = 2 k
\end{aligned}
\]

we choose the signs arbitrarily. We will now explain that each of these 16 systems has exactly one solution. The left side of the second equation is equal to one of the four expressions $\pm 2x$ or $\pm 2y$, so for a given $k$, the second equation always leads to a unique determination of one of the unknowns $x$ or $y$, while the other unknown can take any value - which we then uniquely determine from the first equation by substituting the determined value of the first unknown. Therefore, the system of equations from the problem statement always has at most 16 solutions.

Conclusion. The sought values of the parameter $k$ are exactly two, namely the numbers 18 and 30.