Show that the system of equations $$\begin{cases} x^4+y^4+\phantom{2}z^4=1\\ x^2+y^2+2z^2=\sqrt 7 \\ \end{cases}$$ has no real solutions.
At a first glance I thought about Newton sums, but the second equation is not that great, maybe if we had $z^2$ instead of $2z^2$. Then I though about somehow using some clever inequalities, but I wasn't successful.
Write $a=x^2,b=y^2,c=z^2$ so we have $$a^2+b^2+c^2=1\qquad a+b+2c=\sqrt7$$ The first equation is the unit sphere and the second is a plane; a real solution corresponds to an intersection, where the origin-plane distance is $1$. But the shortest distance from $a+b+2c=\sqrt7$ to the origin is $\sqrt{\frac76}>1$. Thus the plane does not intersect the sphere and the system in $a,b,c$ has no real solution, implying that the original system also has no real solutions.