Three variable equations

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Middle School Math Grades 5-8, Ages 10-13, MATHCOUNTS, AMC 8

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Middle School Math Grades 5-8, Ages 10-13, MATHCOUNTS, AMC 8

Grades 5-8, Ages 10-13, MATHCOUNTS, AMC 8

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Three variable equations

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#1 h Mar 21, 2025, 1:55 PM

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1. For integers $x$ , $y$ , and $z$ such that $\frac{\sqrt{x}}{\sqrt{y}} = z$ , find $\frac{\sqrt{z}}{\sqrt{x}}$ in terms of $x$ , $y$ , and $z$ .

2. For integers $x$ , $y$ , and $z$ such that $x = y - 1$ and $z = y + 1$ , prove that $y^3 = xyz + y$ .

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#2 h Mar 21, 2025, 2:45 PM

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1: we have x/y=z^2, so x/z=yz, so sqrt(x)/sqrt(z)=sqrt(yz), and thus the answer is sqrt(yz)/yz.

2: we have that xyz+y=(y-1)y(y+1)+y=y(y^2-1)+y=y^3, as desired.

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