## IMO 1981, Question 3

I had a great time solving the following question a couple of nights back. More so because I’d failed to solve this question in the past. This question is completely unapproachable if you try to use algebra. However, generating simple examples helps. Let me try and write down some solutions to : they are DoContinue reading “IMO 1981, Question 3”

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”

## A beautiful generalization of the Nesbitt Inequality

I want to discuss a beautiful inequality, that is a generalization of the famous Nesbitt inequality: (Romanian TST) For positive , prove that Clearly, if , then we get Nesbitt’s inequality, which states that . This is question 14 on Mildorf’s “Olympiad Inequalities”, and its solution comprises finding a factor to multiply this expression with,Continue reading “A beautiful generalization of the Nesbitt Inequality”

## A small note on re-defining variables to prove inequalities

I just want to record my solution to the following problem, as it is different from the one given online. For positive real numbers, prove that This has a fairly straight forward solution using Cauchy-Schwarz inequality, which for some reason I did not think of. The way that I solved it is that I re-definedContinue reading “A small note on re-defining variables to prove inequalities”

## Some solutions

I solved two math problems today. The solutions to both were uniquely disappointing. The first problem was the first problem from IMO 1986: Let be a number that is not or . Prove that out of the set , we can select two different numbers such that is not a square. A quick check wouldContinue reading “Some solutions”