Limits, continuity and completeness
Wednesday, April 8, 2009 22:09Occasionally, in contest problems, it helps to have a careful understanding of real numbers and real-valued functions. But what, exactly, is a real number? These Auckland squad training lecture notes by Heather Macbeth outline some basics.
(Update, 19/4/09: several errors fixed.)
Polynomials, pure mathematics, Princeton Companion
Thursday, April 2, 2009 13:23Round about sixth form one learns that every polynomial can be factorized, as a product of linear factors. Why? Well, here’s a polynomial, see. It’s probably a cubic with integer coefficients — after all, most nontrivial polynomials that one encounters are. You play with it until you discover a root, likely by looking at integer factors of the highest and lowest coefficients. Then you polynomial-divide through by the linear factor which that root gives you, and get a quadratic, whose roots there’s a formula for finding. Tada!
Of course, there’s a problem with this algorithm: it depends on figuring out how to break down your polynomial into only linear and quadratic factors.
Recurrence relations
Tuesday, January 27, 2009 17:06Some notes and problems on finding and solving recurrence relations. Read these if you’ve ever wondered how to find a formula for the Fibonacci sequence!
Symmetric polynomials
Monday, January 26, 2009 15:56These notes by Arkadii Slinko explain how to extract information from symmetric polynomials of a set of variables, and how to break any symmetric polynomial down into a few simple ones. The final section gives some applications to triangle geometry.
Functional equations
Sunday, January 25, 2009 1:36These notes by Arkadii Slinko cover techniques — some standard, some exotic — for solving functional equations: groups of substitutions, commutativity, the Cauchy functional equation.
Solutions to some of the problems are available, and can be obtained by writing to nzmathsolymp@gmail.com.
Convex functions
Wednesday, January 14, 2009 21:08Notes from Heather Macbeth’s algebra lecture at the January 2009 camp.
(Update, 24/1/2009: some typos fixed.)
