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Number Theory
1.
(3 p.)
\( n \) is an integer between 100 and 999 inclusive, and \( n^{\prime} \) is the integer formed by reversing the digits of \( n \). How many possible values are for \( nn^{\prime} \)?
2.
(38 p.)
Find the largest possible integer \( n \) such that \( \sqrt n + \sqrt{n+60} = \sqrt m \) for some nonsquare integer \( m \).
3.
(30 p.)
Let \( 0 < a < b < c < d \) be integers such that \( a \), \( b \), \( c \) is an arithmetic progression, \( b \), \( c \), \( d \) is a geometric progression, and \( d  a = 30 \). Find \( a + b + c + d \).
4.
(7 p.)
How many pairs of integers \( (x,y) \) are there such that \( x^2y^2=2400^2 \)?
5.
(19 p.)
Find the sum of all positive integers of the form \( n = 2^a3^b \) \( (a, b \geq 0) \) such that \( n^6 \) does not divide \( 6^n \).
20052017
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