Maths Olympiad Prep

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HMMO, 2020

Algebra Difficulty 3.9 AMC 10/12 Find the answer United States

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

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

Solution

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

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.