Difference between revisions of "2003 AMC 12B Problems/Problem 8"
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==See Also== | ==See Also== | ||
| − | {{AMC12 box|year=2003|ab=B|num-b=7|num-a=9 | + | {{AMC12 box|year=2003|ab=B|num-b=7|num-a=9}} |
{{AMC10 box|year=2003|ab=B|num-b=12|num-a=14}} | {{AMC10 box|year=2003|ab=B|num-b=12|num-a=14}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 00:24, 5 January 2019
- The following problem is from both the 2003 AMC 12B #8 and 2003 AMC 10B #13, so both problems redirect to this page.
Problem
Let 
denote the sum of the digits of the positive integer 
. For example, 
and 
. For how many two-digit values of is 
?
Solution
Let 
and 
be the digits of ,
![\[\clubsuit(\clubsuit(x)) = a + b = 3\]](http://latex.artofproblemsolving.com/f/f/4/ff44b47ffecb595f4a6536063ead47ab9c218244.png)
Clearly can only be 
or 
and only 
and 
are possible to have two digits sum to.
If sums to , there are 3 different solutions : 
If sums to , there are 7 different solutions: 
The total number of solutions is 
See Also
| 2003 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 7 |
Followed by Problem 9 |
| 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 | |
| 2003 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 12 |
Followed by Problem 14 | |
| 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 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
