Difference between revisions of "1987 AIME Problems/Problem 8"
Mathgeek2006 (talk | contribs) m (→Solution 2) |
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Line 12: | Line 12: | ||
Flip all of the fractions for | Flip all of the fractions for | ||
− | <cmath>\begin{ | + | <cmath>\begin{array}{ccccc}\frac{15}{8} &>& \frac{k + n}{n} &>& \frac{13}{7}\\ |
− | 105n > 56 (k + n) > 104n\\ | + | 105n &>& 56 (k + n)& >& 104n\\ |
− | 49n > 56k > 48n\end{ | + | 49n &>& 56k& >& 48n\end{array}</cmath> |
Continue as in Solution 1. | Continue as in Solution 1. |
Revision as of 15:46, 13 March 2015
Contents
Problem
What is the largest positive integer for which there is a unique integer such that ?
Solution 1
Multiplying out all of the denominators, we get:
Since , . Also, , so . Thus, . is unique if it is within a maximum range of , so .
Solution 2
Flip all of the fractions for
Continue as in Solution 1.
See also
1987 AIME (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 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.