Difference between revisions of "2000 AIME I Problems/Problem 7"
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== Solution 2 == | == Solution 2 == | ||
| − | Since <math>x+(1/z)=5, 1=z(5-x)=xyz</math>, so <math>5-x=xy</math>. Also, <math>y=29-(1/x)</math> by the second equation. Substitution gives <math>x=1/5</math>, y=24<math>, and < | + | Since <math>x+(1/z)=5, 1=z(5-x)=xyz</math>, so <math>5-x=xy</math>. Also, <math>y=29-(1/x)</math> by the second equation. Substitution gives <math>x=1/5</math>, <math>y=24</math>, and <math>z=5/24</math>, so the solution is <math>5</math>. |
== See also == | == See also == | ||
Revision as of 17:52, 6 January 2008
Contents
Problem
Suppose that 
and 
are three positive numbers that satisfy the equations 
and 
Then 
where 
and 
are relatively prime positive integers. Find 
.
Solution
Let 
.
Thus 
. So 
.
Solution 2
Since 
, so 
. Also, 
by the second equation. Substitution gives 
, 
, and 
, so the solution is 
.
See also
| 2000 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
