2000 AMC 12 Problems/Problem 20: Difference between revisions
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== Solution == | == Solution == | ||
=== Solution 1 === | === Solution 1 === | ||
We | We multiply all given expressions to get: | ||
<cmath>(1) xyz + x + y + z + \frac{1}{x} + \frac{1}{y} + \frac{1}{z} + \frac{1}{xyz} = \frac{28}{3}</cmath> | <cmath>(1) xyz + x + y + z + \frac{1}{x} + \frac{1}{y} + \frac{1}{z} + \frac{1}{xyz} = \frac{28}{3}</cmath> | ||
Adding all the given expressions gives that | Adding all the given expressions gives that | ||
<cmath>(2) x + y + z + \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 4 + \frac{7}{3} + 1 = \frac{22}{3}</cmath> | <cmath>(2) x + y + z + \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 4 + \frac{7}{3} + 1 = \frac{22}{3}</cmath> | ||
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>. | We subtract <math>(2)</math> from <math>(1)</math> to get that <math>xyz + \frac{1}{xyz} = 2</math>. Hence, by inspection, <math>\boxed{xyz = 1 \rightarrow B}</math>. | ||
=== Solution 2 === | === Solution 2 === | ||
Revision as of 11:01, 6 July 2019
Problem
If 
and 
are positive numbers satisfying
![\[x + 1/y = 4,\qquad y + 1/z = 1, \qquad \text{and} \qquad z + 1/x = 7/3\]](http://latex.artofproblemsolving.com/8/0/9/809268c34d341e924e9b6500bbdc5de487d5445d.png)
Then what is the value of 
?
Solution
Solution 1
We multiply all given expressions to get:
![\[(1) xyz + x + y + z + \frac{1}{x} + \frac{1}{y} + \frac{1}{z} + \frac{1}{xyz} = \frac{28}{3}\]](http://latex.artofproblemsolving.com/8/5/2/852e2a372eafd5e5dce43268c7d3b1531c1eb743.png)
Adding all the given expressions gives that
![\[(2) x + y + z + \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 4 + \frac{7}{3} + 1 = \frac{22}{3}\]](http://latex.artofproblemsolving.com/b/8/b/b8b3ee76ca8fe7a060cceabcc53fac838de13cc6.png)
We subtract 
from 
to get that 
. Hence, by inspection, 
.
Solution 2
We have a system of three equations and three variables, so we can apply repeated substitution.
![\[4 = x + \frac{1}{y} = x + \frac{1}{1 - \frac{1}{z}} = x + \frac{1}{1-\frac{1}{7/3-1/x}} = x + \frac{7x-3}{4x-3}\]](http://latex.artofproblemsolving.com/9/f/0/9f0d43b0925f112e173739d093abd90c9a6ecf41.png)
Multiplying out the denominator and simplification yields 
, so 
. Substituting leads to 
, and the product of these three variables is 
.
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 | |
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