Difference between revisions of "2002 AIME I Problems/Problem 3"
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== Solution == | == Solution == | ||
| − | { | + | Let Jane's age <math>n</math> years from now be <math>10a+b</math>, and let Dick's age be <math>10b+a</math>. If <math>10b+a>10a+b</math>, then <math>b>a</math>. The possible pairs of <math>a,b</math> are: |
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| + | <cmath>(1,2), (1,3), (2,3), (1,4), (2,4), (3,4), \dots , (8,9)</cmath> | ||
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| + | That makes 36. But <math>10a+b>25</math>, so we subtract all the extraneous pairs: <math>(1,2), (1,3), (2,3), (1,4), (2,4), (1,5), (2,5), (1,6), (1,7), (1,8),</math> and <math>(1,9)</math>. <math>36-11=\boxed{025}</math> | ||
== See also == | == See also == | ||
{{AIME box|year=2002|n=I|num-b=2|num-a=4}} | {{AIME box|year=2002|n=I|num-b=2|num-a=4}} | ||
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| + | [[Category:Intermediate Combinatorics Problems]] | ||
Revision as of 09:49, 1 July 2008
Problem
Jane is 25 years old. Dick is older than Jane. In 
years, where is a positive integer, Dick's age and Jane's age will both be two-digit number and will have the property that Jane's age is obtained by interchanging the digits of Dick's age. Let 
be Dick's present age. How many ordered pairs of positive integers 
are possible?
Solution
Let Jane's age years from now be 
, and let Dick's age be 
. If 
, then 
. The possible pairs of 
are:
![\[(1,2), (1,3), (2,3), (1,4), (2,4), (3,4), \dots , (8,9)\]](http://latex.artofproblemsolving.com/9/2/2/92256dff9de809c0b89282a9d66c65e8c43c3215.png)
That makes 36. But 
, so we subtract all the extraneous pairs: 
and 
. 
See also
| 2002 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
