Some problems from Indian National Mathematics Olympiad, 2014

I just want to add a couple of problems from INMO 2014 that I solved this morning. The first problem was slightly less tricky, and just involved pairing divisors with each other in the most obvious way. However, the second problem is quite devious, and is more of an existence proof than writing down anContinue reading “Some problems from Indian National Mathematics Olympiad, 2014”

Exploring Indian Mathematical Olympiads

Indian math olympiad questions are famous (infamous?) for being very analytical. There mostly do not exist any clever one line proofs. Multiple cases have to be analyzed and exhaustively eliminated before arriving upon the correct answer. I tried solving problems from the Indian National Mathematics Olympiad, 2013 today. My solutions are different (lengthier, and henceContinue reading “Exploring Indian Mathematical Olympiads”

An interesting Putnam problem on the Pigeonhole Principle

The following problem is contained in the book “Putnam and Beyond” by Gelca, and I saw it on stackexchange. I’m mainly recording this solution because it took me longer than usual to come up with the solution, as I was led down the wrong path many a time. Noting what is sufficient for a blockContinue reading “An interesting Putnam problem on the Pigeonhole Principle”

Proving that the first two and last two indices of the Riemann curvature tensor commute

I’ve always been confused with the combinatorial aspect of proving the properties of the Riemann curvature tensor. I want to record my proof of the fact that . This is different from the standard proof given in books. I have been unable to prove this theorem in the past, and hence am happy to writeContinue reading “Proving that the first two and last two indices of the Riemann curvature tensor commute”

Thinking about a notorious Putnam problem

Consider the following Putnam question from the 2018 exam: Consider a smooth function such that , and and . Prove that there exists a point and a positive integer such that . This is a problem from the 2018 Putnam, and only 10 students were able to solve it completely, making it the hardest questionContinue reading “Thinking about a notorious Putnam problem”

Putnam 2010

The Putnam exam is one of the hardest and most prestigious mathematical exams. Every year, more than 4,000 students, including math olympiad medalists from various countries, attempt the exam. The median score, almost every year, is 0. Each correctly answered question is worth 10 points I often find myself trying to solve old Putnam problems,Continue reading “Putnam 2010”

Lie derivatives: a simple idea behind a messy calculation

I want to write about Lie derivatives. Because finding good proofs for Lie derivatives in books and on the internet is a lost cause. Because they have caused me a world of pain. Because we could all do with less pain. In all that is written below, we assume that all Lie derivatives are beingContinue reading “Lie derivatives: a simple idea behind a messy calculation”

Coming to grips with Special Relativity

Contrary to popular opinion, Special Relativity is not a more specialized, more involved part of General Relativity. It is the easier of the two Relativity theories, involving only thought experiments and Linear Algebra. However, despite having been exposed to ideas from this theory right from school, and also taking an advanced course (and doing well)Continue reading “Coming to grips with Special Relativity”