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Showing posts from May, 2022

The Art of Double Counting

Hey everyone I'm Bratin and I'm a class 10 student hailing from Kolkata. I've been into math contests for a while now and enjoy combinatorics and geometry the most. In this post I'll dive into the topic of double counting by working through some problems and hopefully be able to showcase how to employ the technique when the need arises for the same.    Put simply, double counting is basically counting in two different ways (duh). The main goal is to count a single quantity in $2$ different ways. There really is no theoretical concepts attached to it. It's just basically elementary combinatorics with ingenuity since you need to figure out what are the $2$ "ways" in which you can consider counting. Below are few problems to get you started. I've discussed the solutions to all of the problems mentioned, with motivation/comments. It is however advisable that the readers try their hand independently at them before moving on to read the solutions.   IMO 1987

Linear Algebra in Graph Theory

Hi, I am Ameya Vikrama Singh, a college student with a recreational interest in mathematics. I was involved in Olympiad math in school, and these days pursue it in leisure. Here we will talk about two interesting applications of Linear Algebra in Graph Theory. It is highly recommended that you have some familiarity with Linear Algebra, such as the definition of linear independence, rank, rank-nullity theorem, determinants, and some idea of eigenvalues. 3b1b's playlist is a great place to start, and you do not require very rigorous insights into these concepts. Some familiarity with the definitions in graph theory will help too. $G(n,e)$ denotes a simple graph $G$ with $n$ vertices and $e$ edges, all graphs in this post are simple, though you may extend the results beyond them. I encourage you to ponder upon the why? checkpoints to follow the post. Here we present a few definitions: Fig 1: Sample Graph Adjacency Matrix $A$: The adjacency matrix associated with a simple graph $G(n

ISI 2022 Objective Solutions

Hellooooooo people I am Rishad. I am a 12th grader. Passionate about pursuing a career in mathematics. I like doing and studying all types of math. Hobbies: Listening to music and doing absolutely nothing and of course reading articles about math(mainly) and related fields( physics, computer science, cryptography..... not chemistry, of course). Okay, now moving on to the objective of this blog. Here, I will be presenting to you the solutions for the Indian Statistical Institute( Objective Paper) 2022. Q1.Any positive real number $x$ can be expanded as:  $x=a_{n}\cdot 2^{n} + a_{n-1}\cdot 2^{n-1}+ a_{n-2}\cdot 2^{n-2}+.........+a_{-1}\cdot 2^{-1}+a_{-2}\cdot 2^{-2}.....$  for some $n\geq0$ , where each $a_{i}  \in  \{0,1\}$. In the above-described expansion of 21.1875, the smallest positive integer k such that $a_{k}\neq0$ is:  Motivation/Overview: Okay, so this is a question based on a binary representation of numbers. For a number in base b the general representation is  $a_{n}\cdot

Constructions in Number Theory

Hi, I am Emon, a ninth grader, an olympiad aspirant, hailing from Kolkata. I love to do olympiad maths and do some competitive programming in my leisure hours, though I take it seriously. I had written INOI this year. Today, I would be giving you a few ideas on how to Construct in Number Theory . Well, so we shall start from the basics and shall try to dig deeper into it with the help of some important and well-known theorems and perhaps, some fancy ones as well. Okay, so without further delay, let's start off... What do we mean by "Constructions"? If noticed, you can see that you often face with some problems in olympiad saying, "... Prove that there exists infinitely many numbers satisfying the given conditions" or "... Prove that there exists a number satisfying the above conditions." These are usually the construction-problems .  For example, let's consider a trivial example : Problem. Prove that there exist infinitely many integers $a$ such th