1969 AMC 12 Problem 29

Attempt Problem 29 of the 1969 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 1969 AMC 12 solutions, or check the answer key.

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29.

If

x=t^{\frac{1}{t-1}}

and

y=t^{\frac{t}{t-1}},

t\gt0,

t\ne1,

a relation between

x

and

y

is:

y^x=x^{\frac{1}{y}}

y^{\frac{1}{x}}=x^y

y^x=x^y

x^x=y^y

none of these

Answer: C

Concepts:exponent algebraic manipulation substitution

Difficulty rating: 2100

Small hint:

Divide

y

by

x

to express

t

in terms of

x

and

y

Big hint:

Also observe that

y=x^t

, then eliminate

t

Solution:

Dividing the definitions gives

\frac yx=t^{\frac{t-1}{t-1}}=t.

Also,

y=t^{\frac{t}{t-1}}=\left(t^{\frac{1}{t-1}}\right)^t=x^t.

Substituting

t=\frac{y}{x}

gives

y=x^{\frac{y}{x}}.

Raising both sides to the power

x

yields

y^x=x^y.

Therefore, the correct answer is C.