ГДЗ по алгебре 8 класс Макарычев ФГОС Задание 1007

\[\boxed{\text{1007.\ }\text{еуроки}\text{-}\text{ответы}\text{\ }\text{на}\text{\ }\text{пятёрку}}\]

\[a > 0,\ \ b > 0,\ \ c > 0\]

\[\textbf{а)}\ \frac{a + b}{c} + \frac{b + c}{a} + \frac{a + c}{b} \geq 6\]

\[a^{2}b + ab^{2} + cb^{2} + c^{2}b + a^{2}c +\]

\[+ ac^{2} \geq 6abc\]

\[\left( a^{2}b + bc^{2} \right) + \left( ab^{2} + ac^{2} \right) +\]

\[+ \left( b^{2}c + a^{2}c \right) \geq 6abc\]

\[b\left( a^{2} + c^{2} \right) + a\left( b^{2} + c^{2} \right) +\]

\[+ c\left( b^{2} + a^{2} \right) \geq 6abc\]

\[\left. \ \begin{matrix} b\left( a^{2} + c^{2} \right) \geq 2\sqrt{a^{2}b^{2}c^{2}} = 2abc \\ a\left( b^{2} + c^{2} \right) \geq 2\sqrt{a^{2}b^{2}c^{2}} = 2abc \\ c\left( b^{2} + a^{2} \right) \geq 2\sqrt{a^{2}b^{2}c^{2}} = abc \\ \end{matrix} \right\} \Longrightarrow\]

\[\left( a^{2}b + bc^{2} \right) + \left( ab^{2} + ac^{2} \right) +\]

\[+ \left( b^{2}c + a^{2}c \right) = 3 \cdot (2abc)\]

\[\frac{a + b}{c} + \frac{b + c}{a} + \frac{a + c}{b} \geq\]

\[\geq 6 \Longrightarrow ч.т.д.\]

\[\textbf{б)}\ (1 + a)(1 + b)(1 + c) > 24,\]

\[\ \ abc = 9\]

\[2\sqrt{a} \cdot 2\sqrt{b} \cdot 2\sqrt{c} = 8\sqrt{\text{abc}}\]

\[8\sqrt{9} = 24 \Longrightarrow abc = 9 \Longrightarrow ч.т.д.\]