This is the first exercise from Thurston’s Three Dimensional Geometry and Topology Vol. 1. Which manifold is this? It is like an old trick. Try following the lines. There are actually 6 loops (circles) in this maze. Here is a color coded picture of it.

# Author: Ashani Dasgupta

Pursuing Ph.D. in Geometric Group Theory at University of Wisconsin, Milwaukee

## Direction of a vector field

Let \({ f: S^n \rightarrow S^n }\) be a map of degree zero. Show that there exists points \({ x, y \in S^n }\) with \({ f(x) = x }\) and \({ f(y) = – y}\). Use this to show that if F is a continuous vector field defined on the unit ball \({ D^n […]

## Round robin tournament

Problem : Suppose there are teams playing a round robin tournament; that is, each team plays against all the other teams and no game ends in a draw.Suppose the team loses games and wins games.Show that = Solution : Each team plays exactly one match against each other team. Consider the expression Since each team […]

This document is useful for current students. It contains teaching schedule for Number Theory 1. Overview: Number Theory 1 is an introductory module. It is useful for beginner math olympiad aspirants (preparing for AMC, AIME, ARML, Duke Math Meet etc.) Number systems Prime numbers Arithmetic and geometric sequences Mathematical Induction Divisibility techniques Arithmetic of remainders Modular […]

## Combinatorics Problem List for AIME

This is a collection of combinatorics and probability problems that have appeared in AIME. Two dice appear to be standard dice with their faces numbered from \(1\) to \(6\), but each die is weighted so that the probability of rolling the number \(k\) is directly proportional to \(k\). The probability of rolling a \(7\) with […]

## Orthogonality

Let ABC be a triangle and D be the midpoint of BC. Suppose the angle bisector of \(\angle ADC \) is tangent to the circumcircle of triangle ABD at D. Prove that \(\angle A = 90^o \) . (Regional Mathematics Olympiad, India, 2016)

## Construction of polynomials

The polynomial P(x) has the property that P(1), P(2), P(3), P(4), and P(5) are equal to 1, 2, 3, 4, 5 in some order. How many possibilities are there for the polynomial P, given that the degree of P is strictly less than 4? (Duke Math Meet 2013 Tiebreaker round) Discussion: Let \(P(x) = a […]

## 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.

1. Note that a complex number \(\alpha \) is algebraic if it is a root of a(nonzero) polynomial with integer coefficients. a) Sine we need \(7 + \sqrt [3]{2} \) to be root of an integer polynomial, there fore \((x – (7 + \sqrt [3]{2}) = (x-7 – 2^{\frac{1}{3}}) \) is one of the factors. […]

## 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 […]