# Number Theory

 1. (4 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 $$a + b + c$$.

 2. (34 p.) Let $$f$$ be a function defined along the rational numbers such that $$f(\tfrac mn)=\tfrac1n$$ for all relatively prime positive integers $$m$$ and $$n$$. The product of all rational numbers $$0< x< 1$$ such that $f\left(\dfrac{x-f(x)}{1-f(x)}\right)=f(x)+\dfrac9{52}$ can be written in the form $$\tfrac pq$$ for positive relatively prime integers $$p$$ and $$q$$. Find $$p+q$$.

 3. (34 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 $$1000$$.

 4. (16 p.) Let $$\tau (n)$$ denote the number of positive divisors of $$n$$, including 1 and $$n$$. Define $$S(n)$$ by $$S(n)=\tau(1)+ \tau(2) + \dots + \tau(n)$$. Let $$a$$ denote the number of positive integers $$n \leq 2008$$ with $$S(n)$$ odd, and let $$b$$ denote the number of positive integers $$n \leq 2008$$ with $$S(n)$$ even. Find $$|a-b|$$.

 5. (10 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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