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A mathematician’s bookshelf

A mathematician’s bookshelf is probably more informative than his resume. The idea of ‘book’ has been recently challenged by the advent of technology. Outstanding authors such as Hatcher (of ‘Algebraic Topology’ fame) prefers to keep an electronic copy of his book. This electronic copy is updated from time to time.

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Some Beautiful Books

Straight Lines and Curves by Vasiliyev N. B. Vasilyev was the chief architect of Mathematical Olympiads in Soviet Union. This gem from erstwhile Soviet Union’s publication, explores loci of points in plane and space. The entire discussion is aided by geometric intuition. The authors occasionally use algebraic tools to augment the ideas. The holistic nature […]

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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 […]

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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.

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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.

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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 […]

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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 […]

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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 […]

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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 […]