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Positive numbers $A, B, C$, and $D$ are such that the system of equations

$x^{2}+y^{2}=A$,

$|x|+|y|=B$

has $m$ solutions, and the system of equations

$x^{2}+y^{2}+z^{2}=C$,

$|x|+|y|+|z|=D$

has $n$ solutions. It is known that $m>n>1$. Find $m$ and $n$.
The first equation is the equation of a circle, and the second is satisfied by the points of a square centered at the origin with diagonals lying on the coordinate axes. Therefore, the first system, depending on $A$ and $B$, either has no solutions, or four solutions, or eight. Thus, $m$ can be either 0, or 4, or 8.

The first equation of the second system is the equation of a sphere. The second is satisfied by the points of an octahedron centered at the origin with vertices lying on the coordinate axes at equal distances from the center. This system, depending on $C$ and $D$, either has no solutions, or 6 solutions (the vertices of the octahedron lie on the sphere), or 8 solutions (the sphere touches the faces of the octahedron), or an infinite number of solutions (the sphere intersects the faces of the octahedron along circles or several arcs of circles). Thus, $n$ can be either 0, or 6, or 8. The condition $m>n>1$ is satisfied only by the variant $m=8, n=6$.

## Answer

$m=8, n=6$.