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 $$|n-n^{\prime}|$$?

 2. (21 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. (42 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 $$|a-b|$$.

 4. (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 $$n$$.

 5. (28 p.) If the corresponding terms of two arithmetic progressions are multiplied we get the sequence 1440, 1716, 1848, ... . Find the eighth term of this sequence.

2005-2018 IMOmath.com | imomath"at"gmail.com | Math rendered by MathJax