# Number Theory

 1. (32 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. (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.

 3. (32 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 $$p+q$$.

 4. (19 p.) Let $$a,b,c$$ and $$d$$ be positive real numbers such that $$a^2+b^2-c^2-d^2=0$$ and $$a^2-b^2-c^2+d^2=\frac {56}{53}(bc+ad)$$, Let $$M$$ be the maximum possible value of $$\frac {ab+cd}{bc+ad}$$ ,If $$M$$ can be expressed as $$\frac {m}{n}$$,$$(m,n)=1$$ then find $$100m+n$$

 5. (6 p.) Find the least positive integer $$n$$ such that when its leftmost digit is deleted, the resulting integer is equal to $$n/29$$.

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