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Shrinking the diameter of Vitali Set
Claim: The diameter of Vitali Set V on [0, 1] can be shrinked as much as we please Proof: Let R be an equivalence relation defined on [0,1] such that x is related to y if x – y is rational. Let E be an equivalence class corresponding to the equivalence relation R. In order to […]
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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 preserves most elements of a subgroups in majority of cases, there exists some subgroups of \(S_4 \) which are ill […]
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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 […]