Question

624. Стены разделены на квадраты 12 см и стены 4 см. Внутри сеток отмечено 37 точек. Докажите, что в некоторых ячейках есть не менее 5 точек внутри.​

Answer 1

Ответ:

Чтобы разделить квадрат со стороной 12 см на клетки со стороной 4 см, нужно каждую сторону разделить на 12 / 4 = 3 части. В итоге у нас получится 3 ∙ 3 = 9 клеток.

Будем использовать для доказательства метод от противного. Предположим, что нет ни одной клетки, в которой не менее 5 точек. Значит, в каждой клетке максимум 4 точки. Раз у нас 9 клеток, то всего поставлено максимум 9 ∙ 4 = 36 точек. Но по условию задачи точек поставлено 37. Значит, наше предположение неверно, и хотя бы в одной клетке окажется не менее 5 точек, что и требовалось доказать.

Answer 2

Problem Analysis

We are given a grid with walls divided into squares of 12 cm by 4 cm. Inside the grid, there are 37 marked points. We need to prove that there are at least 5 points inside some of the cells.

Proof

To prove that there are at least 5 points inside some of the cells, we can use the pigeonhole principle. The pigeonhole principle states that if you have more pigeons than pigeonholes, then at least one pigeonhole must contain more than one pigeon.

In this case, we can consider each cell in the grid as a pigeonhole and the marked points as pigeons. Since there are 37 marked points and the grid is divided into cells, there are more pigeons than pigeonholes.

Now, let's assume that each cell contains at most 4 points. In that case, the total number of points would be at most 4 times the number of cells. Since the grid has a total of 624 cells (12 cm by 4 cm), the maximum number of points would be 4 * 624 = 2496.

However, we know that there are 37 marked points, which is greater than 2496. This contradicts our assumption that each cell contains at most 4 points. Therefore, there must be at least one cell that contains more than 4 points.

Hence, we have proved that in some of the cells, there are at least 5 points inside.

Conclusion

Using the pigeonhole principle, we have proved that in some of the cells in the grid, there are at least 5 points inside.