Maths Olympiad Prep

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Problem 878

AMC 12 late, AIME early
Algebra Difficulty 4.6 Find the answer HMMT November

If x,y,zx, y, z are real numbers such that xy=6,x−z=2xy=6, x-z=2, and x+y+z=9x+y+z=9, compute xy−zx−z2xy\frac{x}{y}-\frac{z}{x}-\frac{z^{2}}{xy}.

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

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Official solution

Let k=xy−zx−z2xy=x2−yz−z2xyk=\frac{x}{y}-\frac{z}{x}-\frac{z^{2}}{xy}=\frac{x^{2}-yz-z^{2}}{xy}. We have k+1=x2+xy−yz−z2xy=x2−xz+xy−yz+zx−z2xy=(x+y+z)(x−z)xy=9⋅26=3k+1=\frac{x^{2}+xy-yz-z^{2}}{xy}=\frac{x^{2}-xz+xy-yz+zx-z^{2}}{xy}=\frac{(x+y+z)(x-z)}{xy}=\frac{9 \cdot 2}{6}=3 so k=2k=2.

Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.