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Domain of Function is Closed Interval

Source: 2013 AIME I Problem 8

March 15, 2013
algebrafunctiondomainlogarithmsmodular arithmeticAMCtrigonometry

Problem Statement

The domain of the function f(x)=arcsin(log⁡m(nx))f(x) = \text{arcsin}(\log_{m}(nx)) is a closed interval of length 12013\frac{1}{2013}, where mm and nn are positive integers and m>1m > 1. Find the remainder when the smallest possible sum m+nm+n is divided by 10001000.
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