Problem #873

873.

If $x,y,$ and $z$ are positive numbers satisfying $x + \frac{1}{y} = 4, y + \frac{1}{z} = 1,$ and $z + \frac{1}{x} = \frac73,$ then what is the value of $xyz$ ?

$\textbf {(A)}\ 2/3 \qquad \textbf {(B)}\ 1 \qquad \textbf {(C)}\ 4/3 \qquad \textbf {(D)}\ 2 \qquad \textbf {(E)}\ 7/3$

This problem is copyrighted by the American Mathematics Competitions.

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Instructions for entering answers:

  • Reduce fractions to lowest terms and enter in the form 7/9.
  • Numbers involving pi should be written as 7pi or 7pi/3 as appropriate.
  • Square roots should be written as sqrt(3), 5sqrt(5), sqrt(3)/2, or 7sqrt(2)/3 as appropriate.
  • Exponents should be entered in the form 10^10.
  • If the problem is multiple choice, enter the appropriate (capital) letter.
  • Enter points with parentheses, like so: (4,5)
  • Complex numbers should be entered in rectangular form unless otherwise specified, like so: 3+4i. If there is no real component, enter only the imaginary component (i.e. 2i, NOT 0+2i).

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