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![](https://rs.olm.vn/images/avt/0.png?1311)
Có \(\frac{xyz+x+z}{yz+1}=\frac{2012}{212}\) <=> \(\frac{x\left(yz+1\right)+z}{yz+1}=\frac{2012}{212}\)
<=> \(x+\frac{z}{yz+1}=\frac{503}{53}\)
<=> \(x+\frac{1}{\frac{yz+1}{z}}=\frac{503}{53}\)
<=> \(x+\frac{1}{y+\frac{1}{z}}=9+\frac{1}{2+\frac{1}{26}}\)
Vì PT trên chỉ có duy nhất và x,y,z nguyên dương
=> \(\left\{{}\begin{matrix}x=9\\y=2\\x=26\end{matrix}\right.\)
Vậy (x,y,z) \(\in\left\{\left(9,2,26\right)\right\}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(9,x-\sqrt{x}=\sqrt{x}\left(\sqrt{x}-1\right)\\ 10,a+3\sqrt{a}-10=\left(\sqrt{a}-1\right)\left(\sqrt{a}+5\right)\\ 11,x+\sqrt{x}-2=\left(\sqrt{x}-1\right)\left(\sqrt{x}+2\right)\\ 12,x-5\sqrt{x}+6=\left(\sqrt{x}-3\right)\left(\sqrt{x}-2\right)\)
9: \(x-\sqrt{x}=\sqrt{x}\left(\sqrt{x}-1\right)\)
10: \(a+3\sqrt{a}-10=\left(\sqrt{a}+5\right)\left(\sqrt{a}-2\right)\)
11: \(x+\sqrt{x}-2=\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)\)
12: \(x-5\sqrt{x}+6=\left(\sqrt{x}-2\right)\left(\sqrt{x}-3\right)\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(A=\left(sin^212^o+sin^278^o\right)+\left(sin^21^o+sin^289^o\right)+\left(sin^273^o+sin^217^o\right)\)
\(A=\left(sin^290^o\right)+\left(sin^290^o\right)+\left(sin^290^o\right)\)
\(A=1+1+1=3\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(=\left(\sin^212^0+\sin^278^0\right)+\left(\sin^270^0+\sin^220^0\right)-\left(\sin^235^0+\sin^255^0\right)+\sin^230^0\)
\(=1+1-1+\dfrac{1}{4}=1+\dfrac{1}{4}=\dfrac{5}{4}\)
212+1=213
212 + 1 = 213