1992 AMC 12 Problem 16

Attempt Problem 16 of the 1992 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1992 AMC 12 solutions, or check the answer key.

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16.

\frac{y}{x-z}=\frac{x+y}{z}=\frac{x}{y}

for three positive numbers

x,

y,

and

z,

all different, then

\frac{x}{y}=

\frac12

\frac35

\frac23

\frac53

2

Let

r=\frac{x}{y}

and express

\frac{z}{y}

using the middle ratio

Substitute

\frac{z}{y}=\frac{r+1}{r}

into the first ratio

Let

r=\frac{x}{y}\gt0.

From

\frac{x+y}{z}=r,

we get

\frac{z}{y}=\frac{r+1}{r}.

Then

\frac{y}{x-z} =\frac{1}{r-\frac{r+1}{r}} =r.

Thus

r^2-r-1=1,

r^2-r-2=0.

Hence

r=2

-1,

and positivity gives

r=2.

Thus the correct answer is E.

Problem 16 in other years