1960 AMC 12 Problem 18

Attempt Problem 18 of the 1960 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 1960 AMC 12 solutions, or check the answer key.

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18.

The pair of equations

3^{x+y}=81

and

81^{x-y}=3

has:

no common solution

the solution

x=2,

y=2

the solution

x=2\dfrac12,

y=1\dfrac12

a common solution in positive and negative integers

none of these

Rewrite

81

3^4

in both equations

Solve the resulting system for

x+y

and

x-y

The first equation gives

x+y=4.

The second gives

3^{4(x-y)}=3,

x-y=\frac{1}{4}.

Therefore

x=\frac{17}{8}

and

y=\frac{15}{8}.

This pair is not listed among the first four choices.

Thus, the correct answer is E.

Problem 18 in other years