Direction of a vector field

Let $latex { f: S^n \rightarrow S^n }&fg=000000$ be a map of degree zero. Show that there exists points $latex { x, y \in S^n }&fg=000000$ with $latex { f(x) = x }&fg=000000$ and $latex { f(y) = - y}&fg=000000$. Use this to show that if F is a continuous vector field defined on the… Continue reading Direction of a vector field


Number Theory 1 Teaching Schedule

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… Continue reading Number Theory 1 Teaching Schedule

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 $latex 1$ to $latex 6$, but each die is weighted so that the probability of rolling the number $latex k$ is directly proportional to $latex k$. The probability of… Continue reading Combinatorics Problem List for AIME

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. Legendary mathematician Terence Tao religiously publishes… Continue reading A mathematician’s bookshelf