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

# Blog

## 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… Continue reading Round robin tournament

## 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

## Orthogonality

Let ABC be a triangle and D be the midpoint of BC. Suppose the angle bisector of $latex \angle ADC $ is tangent to the circumcircle of triangle ABD at D. Prove that $latex \angle A = 90^o $ . (Regional Mathematics Olympiad, India, 2016) Discussion: Since DE is tangent to the circle at D,… Continue reading Orthogonality

## 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 $latex P(x) =… Continue reading Construction of polynomials

## 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