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Number Theory
1.
(27 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 \( a + b + c + d \).
2.
(20 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)
3.
(3 p.)
Let \( n \) be the largest positive integer for which there exists a positive integer \( k \) such that \[ k\cdot n! = \frac{(((3!)!)!}{3!}.\] Determine \( n \).
4.
(41 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 \( ab \).
5.
(6 p.)
How many pairs of integers \( (x,y) \) are there such that \( x^2y^2=2400^2 \)?
20052019
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