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 block

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 write

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 question

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,