# Number Theory

 1. (52 p.) Suppose $$m$$ and $$n$$ are positive integers with $$m> 1$$ such that the domain of the function $$f(x) = \text{arcsin}(\log_{m}(nx))$$ is a closed interval of length $$\frac{1}{2013}$$. Let $$S$$ be the smallest possible value of $$m+n$$. Find the remainder when $$S$$ is divided by $$1000$$.

 2. (6 p.) Let $$a$$, $$b$$, $$c$$ be positive integers forming an increasing geometric sequence such that $$b-a$$ is a square. If $$\log_6a + \log_6b + \log_6c = 6$$, find $$a + b + c$$.

 3. (20 p.) Find the largest possible integer $$n$$ such that $$\sqrt n + \sqrt{n+60} = \sqrt m$$ for some non-square integer $$m$$.

 4. (4 p.) How many pairs of integers $$(x,y)$$ are there such that $$x^2-y^2=2400^2$$?

 5. (16 p.) If the corresponding terms of two arithmetic progressions are multiplied we get the sequence 1440, 1716, 1848, ... . Find the eighth term of this sequence.

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