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Консультация № 110013
18.11.2007, 17:11
0.00 руб.
0
1
1
Здравствуйте эксперты, помогите с задачкой:
Указать все решения в положительных числах системы уравнений:
x^y=z
y^z=x
z^x=y
Указать все решения в положительных числах системы уравнений:
x^y=z
y^z=x
z^x=y
Обсуждение
Неизвестный
19.11.2007, 09:38
общий
это ответ
Здравствуйте, Tribak!
Указать все решения в положительных числах системы уравнений:
x<sup>y</sup>=z
y<sup>z</sup>=x
z<sup>x</sup>=y
Пусть x>1 и y>1, тогда z = x<sup>y</sup> > x > 1.
Тогда по аналогии x = y<sup>z</sup> > y и y = z<sup>x</sup> > z.
Т.е. z > x > y > z. Противоречие, т.е. не может быть двух чисел > 1.
Пусть x<1 и y<1, тогда z = x<sup>y</sup> > x, но z < 1.
Тогда по аналогии x = y<sup>z</sup> > y и y = z<sup>x</sup> > z.
Т.е. z > x > y > z. Противоречие, т.е. не может быть двух чисел < 1.
Мы показали, что из трёх чисел никакие 2 не моугт быть одновременно >1 или одновременно <1.
Значит, как минимум одно число 1. Пусть это будет x.
Но z = x<sup>y</sup> = 1 и y = z<sup>x</sup> = 1, т.е. все числа 1.
Указать все решения в положительных числах системы уравнений:
x<sup>y</sup>=z
y<sup>z</sup>=x
z<sup>x</sup>=y
Пусть x>1 и y>1, тогда z = x<sup>y</sup> > x > 1.
Тогда по аналогии x = y<sup>z</sup> > y и y = z<sup>x</sup> > z.
Т.е. z > x > y > z. Противоречие, т.е. не может быть двух чисел > 1.
Пусть x<1 и y<1, тогда z = x<sup>y</sup> > x, но z < 1.
Тогда по аналогии x = y<sup>z</sup> > y и y = z<sup>x</sup> > z.
Т.е. z > x > y > z. Противоречие, т.е. не может быть двух чисел < 1.
Мы показали, что из трёх чисел никакие 2 не моугт быть одновременно >1 или одновременно <1.
Значит, как минимум одно число 1. Пусть это будет x.
Но z = x<sup>y</sup> = 1 и y = z<sup>x</sup> = 1, т.е. все числа 1.
Форма ответа
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