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[luongminh1964]

Các bạn giải giúp!

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[Unrruly Kid]

3 thay xy+yz+xz=1 vào, quá dễ

4) c+ab=c(a+b+c)+ab=(c+a)(c+b), rồi dùng AM-GM

[Tea Coffee]

1) $\left\{\begin{matrix}\frac{y^{2}-x^{2}}{\sqrt{x-1}}=2y-\sqrt{x-1} \\ x^{2}+y^{2}=3x-1 \end{matrix}\right. <=> \left\{\begin{matrix}y^{2}-x^{2}=2y\sqrt{x-1}-x+1 \\ x^{2}+y^{2}=3x-1 \end{matrix}\right. <=> \left\{\begin{matrix}(y-\sqrt{x-1})^{2}=x^{2} \\ x^{2}+y^{2}=3x-1 \end{matrix}\right.$

+)T/H1: $\left\{\begin{matrix}y-\sqrt{x-1}=x \\ x^{2}+y^{2}=3x-1 \end{matrix}\right. <=>\left\{\begin{matrix}y^{2}=x^{2}+x-1+2x\sqrt{x-1} \\ y^{2}=3x-1-x^{2} \end{matrix}\right. =>2x^{2}+2x\sqrt{x-1}-2x=0<=>x(x-1+\sqrt{x-1})=0...$

+)T/H2: $\left\{\begin{matrix}y-\sqrt{x-1}=-x \\ x^{2}+y^{2}=3x-1 \end{matrix}\right. <=> \left\{\begin{matrix}y=\sqrt{x-1}-x \\ y^{2}=3x-1-x^{2} \end{matrix}\right. <=>\left\{\begin{matrix}y^{2}=x^{2}+x-1-2x\sqrt{x-1} \\ y^{2}=3x-1-x^{2} \end{matrix}\right. => 2x^{2}-2x\sqrt{x-1}-2x=0<=>x(x-1-\sqrt{x-1})=0...$

4) $P=\sqrt{\frac{ab}{c(a+b+c)+ab}}+\sqrt{\frac{bc}{a(a+b+c)+bc}}+\sqrt{\frac{ca}{b(a+b+c)+ac}} = \sqrt{\frac{ab}{(a+c)(b+c)}}+\sqrt{\frac{ac}{(a+b)(b+c)}}+\sqrt{\frac{bc}{(a+b)(a+c)}}\leq \frac{1}{2}(\frac{a}{a+c}+\frac{b}{b+c}+\frac{a}{a+b}+\frac{c}{b+c}+\frac{b}{a+b}+\frac{c}{a+c})=\frac{3}{2}<=>a=b=c=\frac{1}{3}$

5) $x^{2}+2y^{2}+3xy+3x+5y-15=0<=>x^{2}+x(3y+3)+(2y^{2}+5y-15)=0=>\Delta x...$

6) $P=x^{3}y^{3}(x^{3}+y^{3})=(x+y)x^{3}y^{3}(x^{2}-xy+y^{2})=2x^{3}y^{3}(x^{2}-xy+y^{2})\leq 2x^{2}y^{2}(\frac{x^{2}-xy+y^{2}+xy}{2})^{2}=\frac{1}{2}x^{2}y^{2}(x^{2}+y^{2})^{2}=\frac{1}{8}\left [ 2xy(x^{2}+y^{2}) \right ]^{2}\leq \frac{1}{8}\frac{(x+y)^{16}}{16} =\frac{2^{16}}{2^{7}}=2^{9}$

Edited by Tea Coffee, 04-04-2018 - 20:51.

[luongminh1964]

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