Archive for the ‘Algebra’ Category

Limits, continuity and completeness

Wednesday, April 8, 2009 22:09
Posted in category Algebra

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.)

Polynomials, pure mathematics, Princeton Companion

Thursday, April 2, 2009 13:23
Posted in category Algebra, Links

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.

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Recurrence relations

Tuesday, January 27, 2009 17:06
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!

Symmetric polynomials

Monday, January 26, 2009 15:56
Posted in category Algebra

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.

Functional equations

Sunday, January 25, 2009 1:36
Posted in category Algebra

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.

Convex functions

Wednesday, January 14, 2009 21:08
Posted in category Algebra

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

(Updated, 12/4/2010)