ChapterZero

Problem solving for non-geniuses

Filed under: Mathematics — Alex @ 2:25 am 12/18/2004

I recall being told that a lot of the problems that one might encounter in a math competition and at first have no idea how to approach are actually quite easy one you know the trick. Here are some examples, taken shamelessly from Winning Solutions by Lozansky and Rousseau:

Prove that if are any two natural numbers, then $(2^a-1, 2^b-1) = 2^{(a,b)}-1$ .
Find a six digit number that is increased by a factor of 6 if one exchanges (as a block) the first three digits of its decimal expansion with the last three.
Show that if are positive integers, then
$\frac{(2m)!(2n)!}{m!n!(m+n)!}$ is an integer.