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求证三个类似的不等式：
(1）若$n$为正整数，且$x_1,x_2,\cdots ,x_n>1,x_1x_2\cdots x_n=n+1$，证明：$(\dfrac{1}{1^2(x_1-1)}+1)(\dfrac{1}{2^2(x_2-1)}+1)\cdots (\dfrac{1}{n^2(x_n-1)}+1)\geqslant n+1.$
(2)若$n$为正整数，且$x_1,x_2,\cdots ,x_n>1,x_1x_2\cdots x_n=n+1$，证明：$(\dfrac{1}{1^2x_1(x_1-1)}+1)(\dfrac{1}{2^2x_2(x_2-1)}+1)\cdots (\dfrac{1}{n^2x_n(x_n-1)}+1)\geqslant \dfrac{n+2}{2}.$
(3)若$n$为正整数，且$x_1,x_2,\cdots ,x_n>1,x_1x_2\cdots x_n=n+1$，证明：$(\dfrac{1}{1^2(x_1-1)}+x_1)(\dfrac{1}{2^2(x_2-1)}+x_2)\cdots (\dfrac{1}{n^2(x_n-1)}+x_n)\geqslant \dfrac{(n+1)(n+2)}{2}.$
此外，还有其它的类似不等式吗？ |
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