Difference between revisions of "2013 AMC 12B Problems/Problem 14"
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Revision as of 22:41, 22 November 2013
- The following problem is from both the 2013 AMC 12B #14 and 2013 AMC 10B #21, so both problems redirect to this page.
Problem
Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is 
. What is the smallest possible value of ?
Solution
Let the first two terms of the first sequence be 
and 
and the first two of the second sequence be 
and 
. Computing the seventh term, we see that 
. Note that this means that and must have the same value modulo 8. To minimize, let one of them be 0; WLOG assume that 
. Thus, the smallest possible value of is 
; since the sequences are non-decreasing 
. To minimize, let 
. Thus, 
.
See also
| 2013 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 13 |
Followed by Problem 15 |
| 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 | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
