Difference between revisions of "2013 AIME I Problems/Problem 8"
(Created page with "== Problem 8 == The domain of the function f(x) = arcsin(log<math>_{m}</math>(''nx'')) is a closed interval of length <math>\frac{1}{2013}</math> , where ''m'' and ''n'' are posi...") |
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== Solution == | == Solution == | ||
| + | The domain of the arcsin function is [-1, 1], so -1 <math>\le</math> log<math>_{m}</math>(''nx'') <math>\le</math> 1. | ||
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| + | <math>\frac{1}{m} \le nx \le m</math> | ||
| + | |||
| + | <math>\frac{1}{mn} \le x \le \frac{m}{n}</math> | ||
| + | |||
| + | <math>\frac{1}{mn} - \frac{m}{n} = \frac{1}{2013}</math> | ||
Revision as of 20:01, 15 March 2013
Problem 8
The domain of the function f(x) = arcsin(log
(nx)) is a closed interval of length 
, where m and n are positive integers and m > 1. Find the remainder when the smallest possible sum m + n is divided by 1000.
Solution
The domain of the arcsin function is [-1, 1], so -1 
log(nx) 1.
