Gerver M.L.

a) Each side of an equilateral triangle is divided into $m$ equal parts, and lines parallel to the sides are drawn through the division points, cutting the triangle into $m^{2}$ small triangles. Among the vertices of the resulting triangles, $N$ vertices need to be marked such that for no two marked vertices $A$ and $B$ is the segment $A B$ parallel to any of the sides. What is the largest possible value of $N$ (for a given $m$)?

b) Divide each edge of a tetrahedron into $m$ equal parts and draw planes through the division points parallel to the faces. Among the vertices of the resulting polyhedra, $N$ vertices need to be marked such that no two marked vertices lie on a line parallel to one of the faces. What is the largest possible value of $N$?

c) Among the solutions of the equation $x_{1}+x_{2}+\ldots+x_{k}=m$ in non-negative integers, $N$ solutions need to be chosen such that in no two of the chosen solutions does any variable $x_{i}$ take the same value. What is the largest possible value of $N$?
Let's show that problem a) is equivalent to problem v) when $k=3$. Let's take the distance between adjacent parallel lines cutting the triangle as the unit of distance. To each point of the triangle, we assign three numbers: $x_{1}, x_{2}$, and $x_{3}$, representing the distances from this point to the sides of the triangle. It is easy to see that $x_{1}+x_{2}+x_{3}=m$ for all points of the triangle.

Note that the vertices of the small triangles are assigned non-negative integers. In problem a), it is required that for no two marked vertices $A$ and $B$, the segment $AB$ is parallel to any side of the original triangle. In problem v), in none of the selected solutions should any unknown $x_{i}$ take the same value. Both requirements mean the same thing.

Similarly, it can be verified that problem b) is equivalent to problem v) when $k=4$.

Let's restate problem v).

Suppose in a rectangular table with $k$ columns and $n$ rows, non-negative integers can be arranged such that the sum of the numbers in each row is $m$ and no numbers in any column repeat. $k$ and $m$ are fixed.

What is the largest possible value of $n$?

In each column, there are $n$ numbers, and they do not repeat. Therefore, their sum is at least $0+1+\ldots+(n-1)=1 / 2 n(n-1)$, and the sum of the numbers in the entire table is at least $1 / 2 k n(n-1)$.

On the other hand, the sum of the numbers in the entire table is $m n$. Thus, $k n(k n-1) \leq 2 m n$, which means $n-1 \leq 2 m / k$. Therefore, $n \leq [2 m / k] + 1$.

Let's construct for any integers $k \geq 2$ and $m \geq 0$ tables $k \times N$ that satisfy the conditions of the problem, where $N = \left[2 m / k\right] + 1$.

1) Let $k=2 r$ be even. If $m=r l+q (l \geq 0, 0 \leq q < r)$, then:

1.1) If $q=0$, we construct a table with $N = l + 1$ rows and $k$ columns as follows:
- The first $l$ rows contain the numbers $0, 1, \ldots, l-1$ in each column.
- The last row contains the numbers $0, 1, \ldots, r-1$ in the first $r$ columns and the numbers $l, l+1, \ldots, l+r-1$ in the last $r$ columns.

1.2) If $q > 0$, we construct a table with $N = l + 1$ rows and $k$ columns as follows:
- The first $l$ rows contain the numbers $0, 1, \ldots, l-1$ in each column.
- The last row contains the numbers $0, 1, \ldots, q-1$ in the first $q$ columns, the numbers $l, l+1, \ldots, l+q-1$ in the next $q$ columns, and the numbers $0, 1, \ldots, r-q-1$ in the last $r-q$ columns.

2) Let $k=2 r + 1$ be odd. If $m = r l + q (l \geq 0, 0 \leq q < r + 1)$, then:

2.1) If $q=0$, we construct a table with $N = l + 1$ rows and $k$ columns as follows:
- The first $l$ rows contain the numbers $0, 1, \ldots, l-1$ in each column.
- The last row contains the numbers $0, 1, \ldots, r$ in the first $r+1$ columns and the numbers $l, l+1, \ldots, l+r-1$ in the last $r$ columns.

2.2) If $q > 0$, we construct a table with $N = l + 1$ rows and $k$ columns as follows:
- The first $l$ rows contain the numbers $0, 1, \ldots, l-1$ in each column.
- The last row contains the numbers $0, 1, \ldots, q-1$ in the first $q$ columns, the numbers $l, l+1, \ldots, l+q-1$ in the next $q$ columns, and the numbers $0, 1, \ldots, r-q$ in the last $r+1-q$ columns.