Posted in category Algebra

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Occasionally, 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.)

Round 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.

Posted in category Algebra, Combinatorics

Some 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!

These 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.

These 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.

Posted in category Algebra

Notes from Heather Macbeth’s algebra lecture at the January 2009 camp.

(Updated, 12/4/2010)