7. If $x, y, k$ are positive reals such that
$$
3=k^{2}\left(\frac{x^{2}}{y^{2}}+\frac{y^{2}}{x^{2}}\right)+k\left(\frac{x}{y}+\frac{y}{x}\right) \text {, }
$$
find the maximum possible value of $k$.
Solution: $(-1+\sqrt{7}) / 2$
We have $3=k^{2}\left(x^{2} / y^{2}+y^{2} / x^{2}\right)+k(x / y+y / x) \geq 2 k^{2}+2 k$, hence $7 \geq 4 k^{2}+4 k+1=$ $(2 k+1)^{2}$, hence $k \leq(\sqrt{7}-1) / 2$. Obviously $k$ can assume this value, if we let $x=y=1$.