Difference between revisions of "1987 AIME Problems/Problem 1"
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== Problem == | == Problem == | ||
| − | An ordered pair <math>\displaystyle (m,n)</math> of non-negative | + | An [[ordered pair]] <math>\displaystyle (m,n)</math> of [[non-negative]] [[integer]]s is called "simple" if the [[addition]] <math>\displaystyle m+n</math> in base <math>\displaystyle 10</math> requires no [[carrying]]. Find the number of simple ordered pairs of non-negative integers that sum to <math>\displaystyle 1492</math>. |
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== Solution == | == Solution == | ||
| − | + | Since no carrying over is allowed, the range of possible values of any digit of <math>m</math> is from <math>0</math> to the respective [[digit]] in <math>1492</math> (the values of <math>n</math> are then fixed). Thus, the number of [[ordered pair]]s will be <math>(1 + 1)(4 + 1)(9 + 1)(2 + 1) = 2\cdot 5\cdot 10\cdot 3 = 300</math>. | |
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== See also == | == See also == | ||
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{{AIME box|year=1987|before=First Question|num-a=2}} | {{AIME box|year=1987|before=First Question|num-a=2}} | ||
Revision as of 16:49, 15 February 2007
Problem
An ordered pair 
of non-negative integers is called "simple" if the addition 
in base 
requires no carrying. Find the number of simple ordered pairs of non-negative integers that sum to 
.
Solution
Since no carrying over is allowed, the range of possible values of any digit of 
is from 
to the respective digit in 
(the values of 
are then fixed). Thus, the number of ordered pairs will be 
.
See also
| 1987 AIME (Problems • Answer Key • Resources) | ||
| Preceded by First Question |
Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
