1987 AIME Problems/Problem 8
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 | |
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