5. A cube with a side of 5 is made up of 125 smaller cubes with a side of 1. How many small cubes does a plane perpendicular to one of the cube's diagonals and passing through its midpoint intersect?
Answer: 55.

Let's introduce a coordinate system such that the cube is located in the first octant (the set of points with non-negative coordinates) and the mentioned diagonal extends from the origin $O$. The midpoint of the cube's diagonal has coordinates $(5 / 2, 5 / 2, 5 / 2)$, so the specified plane is given by the equation $x + y + z = 15 / 2$. Consider one of the 125 small cubes. Let the vertex closest to point $O$ have coordinates $(k, m, n)$, where the integers $k, m, n$ satisfy the inequalities $0 \leqslant k, m, n \leqslant 4$. The vertex farthest from point $O$ has coordinates $(k+1, m+1, n+1)$. Thus, the small cube intersects the plane if and only if $k + m + n \leqslant 15 / 2 \leqslant k + 1 + m + 1 + n + 1$, which is equivalent to the condition $k + m + n \in (9 / 2 ; 15 / 2)$. Considering the integer nature of the sum $k + m + n$, we get only three cases: $k + m + n = 5$, $k + m + n = 6$, or $k + m + n = 7$. Let's calculate the number of small cubes for each of the three cases.

(i) $k + m + n = 5$. List the ways to split 5 into three summands and indicate the number of different permutations of these summands: $0 + 1 + 4$ (6 solutions), $0 + 2 + 3$ (6 solutions), $1 + 1 + 3$ (3 solutions), $1 + 2 + 2$ (3 solutions) - a total of 18.

(ii) $k + m + n = 6. 0 + 2 + 4$ (6 solutions), $0 + 3 + 3$ (3 solutions), $1 + 1 + 4$ (3 solutions), $1 + 2 + 3$ (6 solutions), $2 + 2 + 2$ (1 solution) - a total of 19.

(iii) $k + m + n = 7$. All such small cubes are symmetric to those with the condition $k + m + n = 5$ relative to the center of the cube - a total of 18.

In total: $18 + 19 + 18 = 55$.
t The intersected small cubes are divided into three layers, and the number of small cubes in at least two layers is correctly calculated with justification. 6 points

± The intersected small cubes are divided into three layers, and the number of small cubes in at least one layer is correctly calculated with justification. 5 points

± The intersected small cubes are divided into three layers, and the number of small cubes in at least one layer is correctly calculated, but the justification is not strict or is missing. 4 points

Ғ The intersected small cubes are divided into three layers, but there is no further progress. 3 points

- The correct answer is given without justification. 2 points
- The equation of the plane is written. 1 point