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Tag Archives: basic
Symmetric Polynomials (II)
When we move on to n=3 variables, we now have, as basic building blocks, These are just the coefficients of in the expansion of . Once again, any symmetric polynomial in x, y, z with integer coefficients can be expressed as a polynomial … Continue reading
Symmetric Polynomials (I)
[ Background required: knowledge of basic algebra and polynomial operations. ] After a spate of posts on non-IMO related topics, we’re back on track. Here, we shall look at polynomials in n variables, e.g. P(x, y, z) when n = 3. Such … Continue reading
What is Curvature? (I)
[ Background required: calculus, specifically differentiation. ] In this post, we will give a little background intuition on the definition of curvature. One possible approach is given in wikipedia, ours is another. Note that this is not IMO-related (my apologies … Continue reading
Modular Arithmetic Deluxe Edition
[ Background required: standard modular arithmetic. ] Consider the following two problems: Problem 1. Prove that if p > 2 is prime, then when is expressed in lowest terms , m must be a multiple of p. Problem 2. Prove that if … Continue reading
Posted in Notes
Tagged basic, congruence, modular arithmetic, notes, number theory, rational numbers
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Thoughts on a Problem II
The following problem caught my eye: (USAMO 1997 Q3) Prove that for any integer n, there is a unique polynomial Q(X) whose coefficients all lie in the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and Q(-2) … Continue reading
Estimating Sums Via Integration
Background required : calculus, specifically integration By representing a sum as an area, it is often possible to estimate its size by approximating it with the area underneath a curve. For example, suppose we wish to compute the sum . … Continue reading
Matrices and Linear Algebra
Background recommended : coordinate geometry Here I thought I’d give an outline of linear algebra and matrices starting from a more axiomatic viewpoint, instead of merely giving rules of computation – the way it’s usually taught in school. The materials … Continue reading
Sample Problem Solving + Homework Hints
In this post, I’ll talk about basic number theory again. But I’ll still assume you already know modular arithmetic. 🙂 In the first part, there’ll be some sample solutions for number theoretic problems, some of which were already presented in … Continue reading
Number Theory Homework (2 Weeks)
Homework problems for 5 Nov 2011: Let a1, a2, … be a series recursively defined as follows: a1 = 20, a2 = 11, and for n ≥ 1, an+2 is the remainder when an+1 + an is divided by 100. … Continue reading
Thoughts on a Problem
From time to time, I may post my experience in solving a specific problem including some wrong twists and turns I’ve taken, as well as the motivation for some rather cryptic constructions. Warning : since this post documents my thought … Continue reading
Mathematics and Such 