Categories
Uncategorized

A SAGE experiment on Nested Approximate Subgroups

(this is a continuing write-up mostly for personal records) Spoiler Before you read on, here is what I have found curious so far (this section is constantly changing): A little computation reveals that though group action conjugation preservesĀ most elements of a subgroups in majority of cases, there exists some subgroups of \(S_4 \) which areĀ ill […]

Categories
Informative Articles

Number Theory in Math Olympiad – Beginner’s Toolbox

This article is aimed at entry level Math Olympiad (AMC and AIME in U.S. , SMO Junior in Singapore, RMO in India). We have complied some of the most useful results and tricks in elementary number theory that helps in problem solving at this level. Note that only with a lot of practice and conceptual […]

Categories
Informative Articles

Lifting the exponent and math olympiad number theory

In math olympiads around the world, number theory problems have many recurring themes. One such theme is the ‘LTE’ or lifting the exponent.

Categories
Informative Articles

Arithmetic of Remainders

Consider the two number: 37 and 52 What is the remainder when we divide 37 by 7? 2 of course. And 52 produces remainder 3 when divided by 7. Suppose we want to know the remainder when the product of 37 and 52 is divided by 7.

Categories
Informative Articles

Continuous Functions and open sets

Suppose f be a continuous function from X to Y (where X and Y are domain and range). If Y is a closed set (closed interval if we working in \(R^1 \) ) then can we say that the domain is also closed? There is a simple counter example. Suppose X = \(( -3pi,pi)\). Then […]

Categories
Informative Articles

Geometric Transformation and homothety for Math Olympiad 1

Categories
Informative Articles

Homothety 1

(This is a series of discussions on Homothety. It is largely derived from the Math Olympiad Classroom Discussion in Cheenta – www.cheenta.com) Homothety is a geometric transformation. It has a couple of synonyms: dilation and central similarity. A geometric transformation is a function. It can be thought of as a machine which takes in a […]

Categories
Informative Articles

Complex Numbers versus Projective Geometry – One problem, Two solutions

The Problem Suppose ABC is any triangle. D, E, F are points on BC, CA, AB respectively such that \((frac{BD}{DC} = frac{CE}{EA} = frac{AF}{FB})\). Prove that the centroids of triangles ABC and DEF coincide. A little Complex Number Let A, B, C be points on the Complex plane with complex coordinates a, b, c. The […]

Categories
Informative Articles

Is it a prime number?

353 is a prime number. So is 7919 (in fact it is the 1000th prime). There are 25 primes between 1 and 100. From 1 to 1000 there are 168 of them. It is difficult to check whether a number is prime or not. One simple method is to try and divide the number with […]

Categories
Informative Articles

The Dreams of Pythagoras

Pythagoras is famous. Even those who do not like mathematics, have heard of Pythagoras’ Theorem (regarding right angled triangles). He lived about 2500 years ago and did about 2500 wonderful things (well may be a little less) but all that is not the subject matter of this note. We want to talk about the shattered […]