1、\(\left(1-\dfrac 12+\dfrac 13-\dfrac 14+\cdots+\dfrac{1}{2011}-\dfrac{1}{2012}\right)\div\left(\dfrac{1}{1007}+\dfrac{1}{1008}+\cdots+\dfrac{1}{2012}\right)=\)_______．
2、\(\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}}=\)_______．
3、\(\dfrac{\left(3^4+4\right)\left(7^4+4\right)\left(11^4+4\right)\cdots\left(39^4+4\right)}{\left(5^4+4\right)\left(9^4+4\right)\left(13^4+4\right)\cdots\left(41^4+4\right)}=\)_______．
4、对于任意的实数\(x,y,z\)，定义运算\(\otimes\)为：\[x\otimes y=\dfrac{3x^3y+3x^2y^2+xy^3+45}{(x+1)^3+(y+1)^3-60},\]且\(x\otimes y\otimes z=\left(x\otimes y\right)\otimes z\)，则\(2013\otimes 2012\otimes 2011\otimes\cdots \otimes 3\otimes 2=\)_______．
5、\(\sqrt[3]{\dfrac{\sqrt 5-1}2-\left(\dfrac{\sqrt 5-1}{2}\right)^2}=\)_______．
6、\(1+\dfrac{1}{\sqrt 2}+\dfrac{1}{\sqrt 3}+\dfrac{1}{\sqrt 4}+\cdots+\dfrac{1}{\sqrt{100}}\)的整数部分为_______．
7、已知\(x=\sqrt[3] 9+\sqrt [3] 3+1\)，则\(\left(\dfrac{2}{x}+1\right)^3=\)_______．
8、已知\(f(x)=\dfrac{1}{\sqrt[3]{(x+1)^2}+\sqrt[3]{x^2+x}+\sqrt[3]{x^2}}\)，则\(f(1)+f(2)+f(3)+\cdots+f(511)=\)_______．
9、已知\(x-y=6\)，\(\sqrt{x^2-xy}+\sqrt{xy-y^2}=9\)，则\(\sqrt{x^2-xy}-\sqrt{xy-y^2}=\)_______．
10、\(\dfrac{x}{y+z}+\dfrac{y}{z+x}+\dfrac{z}{x+y}=1\)，则\(\dfrac{x^2}{y+z}+\dfrac{y^2}{z+x}+\dfrac{z^2}{x+y}=\)_______．
11、已知\(abc=1\)，且\[\begin{cases}\dfrac{by}{z}+\dfrac{cz}y=a,\\\dfrac{cz}{x}+\dfrac{ax}z=b,\\\dfrac{ax}y+\dfrac{by}x=c,\end{cases}\]则\(a^3+b^3+c^3=\)_______．
12、\(abc=1\)，则\(\dfrac{a}{ab+a+1}+\dfrac{b}{bc+b+1}+\dfrac{c}{ca+c+1}=\)_______．
13、已知\(a,b,c,d\)均不为\(0\)，且互不相等，若\[a+\dfrac 1b=b+\dfrac 1c=c+\dfrac 1d=d+\dfrac 1a,\]则\(a^2b^2c^2d^2=\)_______．
参考答案
1、\(1\) 提示 将被除数中的\(-\dfrac{1}{2k}\)拆为\(\dfrac 1{2k}-\dfrac 1k\)即得．
2、\(2\) 提示 令原式值为\(x\)，则解方程\(\sqrt{2+x}=x\)即得．
3、\(\dfrac{1}{353}\) 提示 \(x^4+4=\left(x^2+2\right)^2-(2x)^2=\left[(x-1)^2+1\right]\cdot\left[(x+1)^2+1\right]\)．
4、\(\dfrac{5463}{967}\) 提示 \(x\otimes 3=9\)．
5、\(\dfrac{\sqrt 5-1}{2}\) 提示 \(\dfrac{\sqrt 5-1}2\)是方程\(x^2+x-1=0\)的根，于是\(x^3+x^2-x=0\)．
6、\(18\) 提示 \(\sqrt{n+1}-\sqrt{n}<\dfrac{1}{2\sqrt n}<\sqrt n-\sqrt{n-1}\)．
7、\(3\) 提示 利用立方差公式．
8、\(7\) 提示 \(f(x)=\sqrt[3]{x+1}-\sqrt[3]{x}\)．
9、\(4\) 提示 利用平方差公式．
10、\(0\) 提示 原式即\[\left(\dfrac{x}{y+z}+\dfrac{y}{z+x}+\dfrac{z}{x+y}\right)(x+y+z)-\cdots,\]而多出来的部分恰好为\(x+y+z\)．
11、\(5\) 提示 三式相乘可整理得\[abc=\sum_{cyc}{a\left(\dfrac{by}z+\dfrac{cz}y\right)^2}-4abc.\]
12、\(1\) 提示 消元即可．
13、\(1\) 提示 \(a-b=\dfrac{c-b}{bc}\)，轮换相乘即得．