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Консультация № 182552
18.03.2011, 21:21
45.92 руб.
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Уважаемые эксперты! Пожалуйста, ответьте на вопрос:Найдите число натуральных корней уравнения:целая часть(х/2010)=целая часть(x/2011)+1.Заранее благодарен.
Обсуждение
Неизвестный
18.03.2011, 22:41
общий
это ответ
Здравствуйте, Тимофеев Алексей Валентинович!
При натуральном x (положительным) легко видеть сразу несколько корней:x [$8712$]{2010, 2*2010, 3*2010, ..., k*2010, ...}. Найдем все натуральные k, для которых решения имеют вид: x=k*2010. Для удобства уравнение запишем в виде: [x/2010]-[x/2011]=1.
Имеем [k*2010/2010]=k, [k*2010/2011]<k. Если к тому же будет k*2010/2011>=k-1, то тогда будем иметь [k*2010/2011]=k-1 и уравнение будет удовлетворяться. Находим из этого условия k: k*2010/2011-k>=-1 -> k*((2010/2011)-1)>=-1 -> k*(-1/2011)>=-1 -> k<=2011.
Итак, все решения уравнения имеют вид: x=k*2010, где k=1, 2, ... ,2011. Таких решений 2011 штук.
P.S. Легко показать подстановкой в уравнение, что любое натуральное число из промежутка ((k-1)*2010, k*2010) не будет решением уравнения.
Если что непонятно, то обращайтесь в мини-форум.
С уважением
При натуральном x (положительным) легко видеть сразу несколько корней:x [$8712$]{2010, 2*2010, 3*2010, ..., k*2010, ...}. Найдем все натуральные k, для которых решения имеют вид: x=k*2010. Для удобства уравнение запишем в виде: [x/2010]-[x/2011]=1.
Имеем [k*2010/2010]=k, [k*2010/2011]<k. Если к тому же будет k*2010/2011>=k-1, то тогда будем иметь [k*2010/2011]=k-1 и уравнение будет удовлетворяться. Находим из этого условия k: k*2010/2011-k>=-1 -> k*((2010/2011)-1)>=-1 -> k*(-1/2011)>=-1 -> k<=2011.
Итак, все решения уравнения имеют вид: x=k*2010, где k=1, 2, ... ,2011. Таких решений 2011 штук.
P.S. Легко показать подстановкой в уравнение, что любое натуральное число из промежутка ((k-1)*2010, k*2010) не будет решением уравнения.
Если что непонятно, то обращайтесь в мини-форум.
С уважением
3
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