2000 AMC 12 Problems/Problem 20: Difference between revisions

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Problem

If $x,y,$ and $z$ are positive numbers satisfying

$x + 1/y = 4,\qquad y + 1/z = 1, \qquad \text{and} \qquad z + 1/x = 7/3$

Then what is the value of $xyz$ ?

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

Solution

Solution 1

We can multiply all given expressions to get: $(1) xyz + x + y + z + \frac{1}{x} + \frac{1}{y} + \frac{1}{z} + \frac{1}{xyz} = \frac{28}{3}$ Adding all the given expressions gives that $(2) x + y + z + \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 4 + \frac{7}{3} + 1 = \frac{22}{3}$ We subtract $(2)$ from $(1)$ to get that $xyz + \frac{1}{xyz} = 2$ . Hence, $xyz = 1 \rightarrow B$ .

Solution 2

We have a system of three equations and three variables, so we can apply repeated substitution.

$4 = x + \frac{1}{y} = x + \frac{1}{1 - \frac{1}{z}} = x + \frac{1}{1-\frac{1}{7/3-1/x}} = x + \frac{7x-3}{4x-3}$

Multiplying out the denominator and simplification yields $4(4x-3) = x(4x-3) + 7x - 3 \Longrightarrow (2x-3)^2 = 0$ , so $x = \frac{3}{2}$ . Substituting leads to $y = \frac{2}{5}, z = \frac{5}{3}$ , and the product of these three variables is $1$ .

Also see

2000 AMC 12 (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: Unable to save thumbnail to destination

@@ Line 12: / Line 12: @@
 == Solution ==
 === Solution 1 ===
-Multiplying all three expressions together,
+We can multiply all given expressions to get:
+<math>(1) xyz + x + y + z + \frac{1}{x} + \frac{1}{y} + \frac{1}{z} + \frac{1}{xyz} = \frac{28}{3}</math>
-<cmath>\begin{align*}
+Adding all the given expressions gives that
-\left( x + \frac 1y \right) \left( y + \frac 1z \right) \left( z + \frac 1x \right) &= xyz + x + y + z + \frac 1x + \frac 1y + \frac 1z + \frac 1{xyz}\\
+<math>(2) x + y + z + \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 4 + \frac{7}{3} + 1 = \frac{22}{3}</math>
-(4)(1)\left(\frac 73\right) &= 4 + 1 + \frac 73 + xyz + \frac 1{xyz}\\
+We subtract <math>(2)</math> from <math>(1)</math> to get that <math>xyz + \frac{1}{xyz} = 2</math>. Hence, <math>xyz = 1 \rightarrow B</math>.
-&= xyz + \frac 1{xyz}\\
-&= (xyz - 1)^2
-\end{align*}</cmath>
-Thus <math>xyz = 1 \Rightarrow B</math>
 === Solution 2 ===

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