11.2. In the city of Perpendicularinsk, it was decided to build new multi-story houses (some of them may be single-story), but in such a way that the total number of floors would be 30. The city architect, Parallelnikov, proposed a project according to which, if after construction one climbs to the roof of each new house, counts the number of lower new houses, and adds up all these numbers, the resulting sum would be the largest possible. What is the indicated sum? How many houses, and of what height, are proposed to be built?
Solution: 1) We will show that the project does not involve building houses with more than two floors. Assume the opposite, that such houses are planned. Take the lowest of them and reduce it by one floor, building an additional one-story house as a result. The sum of the numbers in question will decrease by the number of new two-story houses (and only in the case where we reduced a three-story house) and increase by the number of new houses with 2 or more floors (due to the built one-story house). Therefore, the sum of the new project is greater. Contradiction. Thus, Parallelnikov proposes to build only one- and two-story houses.

2) Let the project involve building $x$ one-story houses and $y$ two-story houses. Then $x+2y=30$, and the sum will be equal to the product $xy$. We need to maximize the expression $y(30-2y)$ on the interval $[0; 15]$. The maximum is achieved at the vertex of the parabola, at the point $y=7.5$. Considering that $y$ is an integer, we get that the maximum value is achieved at $y=7$ or $y=8$ and is equal to 112. Accordingly, $x$ is either 16 or 14.

Answer: The sum is 112. The project involves building either 14 one-story and 8 two-story houses, or 16 one-story and 7 two-story houses.

| in progress | points |
| :--- | :--- |
| Correct justified answer | 7 points |
| One of the two sets of houses that make up the answer is "lost" in the correct solution process and/or the sum is determined incorrectly due to arithmetic errors | 6 points |
| It is proven that the project does not involve houses higher than three floors, but further reasoning is incorrect | 4 points |
| The case where all houses are one- or two-story is correctly analyzed, but it is not proven that the maximum is achieved precisely on such sets of houses | 3 points |
| Some statements about the properties of the optimal set of houses are proven, but they do not lead to a solution (for example, that there must be one-story houses, or that the heights of the houses go without gaps) | no more than 2 points |
| Both possible sets of houses are provided without any justification, and the required sum is correctly calculated | 2 points |
| The required sum is correctly indicated without justification, or one of the two examples on which this sum is achieved is provided | 1 point |
| Incorrect answer | 0 points |