Solve the system of equations
$$\begin{align}
4x^2+2xy+y^2+x+y+6\sqrt{xy}=15 \\
x+\frac{6(x^2+y^2)}{x^2+xy+y^2}-\sqrt{2(x^2+y^2)}=3
\end{align}$$
The only solution is $x=y=1$.
But I can't solve this.
We can have this: $$3\bigg(\frac{6(x^2+y^2)}{x^2+xy+y^2}-\sqrt{2(x^2+y^2)}-3\bigg)^2+(x+y)^2+x+y+6\sqrt{xy}=15$$
It is very difficult to use inequality here
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Minh Hien
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@Moo I don't think it is equivalent to the original system. $$(x;y)=(1.9; 1)$$ is another root – Minh Hien Nov 29 '20 at 16:49
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@Moo You can check their graphs on geogebra – Minh Hien Nov 29 '20 at 17:00