# 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 $$|a-b|$$.

 5. (6 p.) How many pairs of integers $$(x,y)$$ are there such that $$x^2-y^2=2400^2$$?

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