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Здравствуйте! У меня возникли сложности с таким вопросом: Исследовательская задача: Дан единичный квадрат. Даны еще n равных окружностей, радиусы которых нужно будет найти. Требуется разместить эти окружности внутри квадрата, так, чтобы их радиусы были максимально возможного радиуса. При этом окружности между собой могут только касаться, но не пересекаться. Еще окружности могут пересекать стороны квадрата, но их центры должны быть внутри квадрата или на стороне квадрата. Начните со случая n=3, затем 4, 5, и т.д
Неужели исследовательская задача может быть задана тому, кто не знает, как её решить? Обычно её дают студентам, претендующим на дальнейшее обучение в магистратуре. Или Вы учитесь в математической школе и так же должны показать степень своей одарённости?
Попробуйте всё-таки начать со случая n=3. Зафиксируем на плоскости три круга одинакового радиуса, касающихся друг друга вплотную. Они покроют некоторую часть этой плоскости. Какова будет площадь "внутренней" части плоскости, не покрытой этими кругами?
Здесь я интуитивно предполагаю, что максимальному радиусу кругов соответствует вполне определённая (возможно, минимальная) площадь, не покрытая этими кругами. В частном случае, по-видимому, нужно искать наиболее плотную упаковку кругов на плоскости. Вы решали подобные задачи?
И ещё: наверняка решение задачи должно сопровождаться доказательствами. Какой базой теоретических сведений мы можем располагать? Возможно, данная задача может быть решена элементарными методами, не выходя за рамки стандартной школьной программы. Но я в этом не уверен...
Интуиция подсказывает мне, что при n=3 задача сводится к нахождению лежащих на сторонах квадрата вершин равностороннего треугольника наибольшего периметра. Правда, это нужно, по-видимому, доказать.
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Facta loquuntur.
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