Students who begin studying for math competitions have a tendency to be very competent at algebra, trig and anything calculus-like, mainly because that is what they have studied for years on end through school.  However, in geometry and number theory, especially anywhere in North America, my students, even the brighter ones, have quite a few more challenges ahead of them.  The lack of number theory or geometric repetition in their daily math routines often leaves the students frustrated and confused when forced to tackle anything that is outside their immediate sphere of knowledge.

To combat this of course, we need to expose the students to as many problems as possible, giving them time to solve them, but not so long that they abandon the subject.  Here are a few that I find interesting.  One easy, the other more challenging.

One: If p is a prime number greater than 5, find the sum of the divisors of 15p.

Two: If p and q are positive integers such that \frac{p}{q} = 1 + \frac{1}{2} - \frac{2}{3} +\frac{1}{4} + \frac{1}{5} - \frac{2}{6} + \frac{1}{7} + \frac{1}{8} - \frac{2}{9}+ \cdots + \frac{1}{478}+\frac{1}{479} - \frac{2}{480}, show that p is divisible by 641.

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