HMMT February

Algebra Difficulty 5.0 AIME, harder Find the answer

If $x, y, k$ are positive reals such that $3=k^{2}\left(\frac{x^{2}}{y^{2}}+\frac{y^{2}}{x^{2}}\right)+k\left(\frac{x}{y}+\frac{y}{x}\right)$ find the maximum possible value of $k$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

We have $3=k^{2}(x^{2} / y^{2}+y^{2} / x^{2})+k(x / y+y / x) \geq 2 k^{2}+2 k$ , 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 $x=y=1$ .

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