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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 \[ f\left(\dfrac{xf(x)}{1f(x)}\right)=f(x)+\dfrac9{52}\] 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 \( x^2y^2=2400^2 \)?
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 nonsquare integer \( m \).
20052019
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