# Number Theory

 1. (43 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 $$p+q$$.

 2. (3 p.) How many pairs of integers $$(x,y)$$ are there such that $$x^2-y^2=2400^2$$?

 3. (10 p.) Determine the number of positive integers with exactly three proper divisors each of which is less than 50. (1 is a proper divisor of every integer greater than 1)

 4. (25 p.) It is given that $$181^2$$ can be written as the difference of the cubes of two consecutive positive integers. Find the sum of these two integers.

 5. (17 p.) Find the largest possible integer $$n$$ such that $$\sqrt n + \sqrt{n+60} = \sqrt m$$ for some non-square integer $$m$$.

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