Difference between revisions of "1987 AIME Problems/Problem 8"
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== Problem == | == Problem == | ||
What is the largest positive integer <math>n</math> for which there is a unique integer <math>k</math> such that <math>\frac{8}{15} < \frac{n}{n + k} < \frac{7}{13}</math>? | What is the largest positive integer <math>n</math> for which there is a unique integer <math>k</math> such that <math>\frac{8}{15} < \frac{n}{n + k} < \frac{7}{13}</math>? | ||
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== Solution 1== | == Solution 1== | ||
Multiplying out all of the [[denominator]]s, we get: | Multiplying out all of the [[denominator]]s, we get: | ||
Revision as of 22:16, 12 March 2012
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
\begin{align*}\frac{15}{8} > \frac{k + n}{n} > \frac{13}{7}\\
105n > 56 (k + n) > 104n\\
49n > 56k > 48n\end{align*} (Error compiling LaTeX. Unknown error_msg)
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 | ||
