Difference between revisions of "1991 AJHSME Problems/Problem 20"
5849206328x (talk | contribs) (Created page with '==Problem== In the addition problem, each digit has been replaced by a letter. If different letters represent different digits then <math>C=</math> <asy> unitsize(18); draw((-…') |
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{{AJHSME box|year=1991|num-b=19|num-a=21}} | {{AJHSME box|year=1991|num-b=19|num-a=21}} | ||
[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] | ||
| + | {{MAA Notice}} | ||
Revision as of 00:07, 5 July 2013
Problem
In the addition problem, each digit has been replaced by a letter. If different letters represent different digits then 
![[asy] unitsize(18); draw((-1,0)--(3,0)); draw((-3/4,1/2)--(-1/4,1/2)); draw((-1/2,1/4)--(-1/2,3/4)); label("$A$",(0.5,2.1),N); label("$B$",(1.5,2.1),N); label("$C$",(2.5,2.1),N); label("$A$",(1.5,1.1),N); label("$B$",(2.5,1.1),N); label("$A$",(2.5,0.1),N); label("$3$",(0.5,-.1),S); label("$0$",(1.5,-.1),S); label("$0$",(2.5,-.1),S); [/asy]](http://latex.artofproblemsolving.com/8/e/8/8e88799a4ccdbb6c37dec56b6b67f175e419db0b.png)
Solution
From this we have
![\[111A+11B+C=300.\]](http://latex.artofproblemsolving.com/2/a/9/2a9bc8582cda2534a8032588cda0209002437bad.png)
Clearly, 
. Since 
,
![\[111A > 201 \Rightarrow A\geq 2.\]](http://latex.artofproblemsolving.com/7/c/b/7cb3207cd37022d15ebdb849d7e8df2fea60c300.png)
Thus, 
and 
. From here it becomes clear that 
and 
.
See Also
| 1991 AJHSME (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 AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
