Difference between revisions of "1987 AIME Problems/Problem 8"
Pinkmuskrat (talk | contribs) (→Solution) |
Pinkmuskrat (talk | contribs) (→Solution) |
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Flip all of the fractions for | Flip all of the fractions for | ||
| − | <math></math>\begin{align*}\frac{15}{8} > \frac{k + n}{n} > \frac{13}{7} | + | <math></math>\begin{align*}\frac{15}{8} > \frac{k + n}{n} > \frac{13}{7}\\ |
| − | < | + | <math>105n > 56 (k + n) > 104n\\ |
</math>49n > 56k > 48n\end{align*}<math></math> | </math>49n > 56k > 48n\end{align*}<math></math> | ||
Revision as of 15:26, 15 March 2011
Problem
What is the largest positive integer 
for which there is a unique integer 
such that 
?
Solution
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
$$ (Error compiling LaTeX. Unknown error_msg)\begin{align*}\frac{15}{8} > \frac{k + n}{n} > \frac{13}{7}\\
$105n > 56 (k + n) > 104n\$ (Error compiling LaTeX. Unknown error_msg)49n > 56k > 48n\end{align*}$$ (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 | ||
