The Online Math Club Blog

The  European Girls' Mathematical Olympiad a.k.a EGMO 2022 just ended. Congrats to Jessica Wan from USA, Taisiia Korotchenko, and Galiia Sharafetdinova for the perfect scores! Moreover, the Indian girls brought home 4 bronze medals! By far, this is the best result the EGMO India Team has ever achieved! To celebrate the brilliant result, here's a compilation of EGMO 2022 solutions and motivations written by my and everyone's favorite IMOTCer Atul ! And along with that, we also have reviews of each problem written by everyone's favorite senior, Pranjal !  These solutions were actually found by Atul, Pranjal,  and Abhay  during the 3-hour live solve. In the live solve, they solved all the 6 problems in 3 hours 😍!!! Okie Dokie, I think we should get started with the problems! Enjoy! Problem 1:  Let $ABC$ be an acute-angled triangle in which $BC<AB$ and $BC<CA$. Let point $P$ lie on segment $AB$ and point $Q$ lie on segment $AC$ such that $P \neq B$, $Q \neq C$ and

Hiiiiiiiii everyone! Today I will be showing you a few combinatorics problems which are not very hard. Each one has a different idea. My aim is to introduce you to various ideas you can think of while trying combo problems and realize that combo is very fun!!!!  Oki without wasting much time, let's begin :P For those of you who want to try the problems before seeing the solution/ideas presented , here's a list of problems I have discussed below (note that I have modified a few questions, and some of them have answers to the original questions, so you can try reading the problems from AoPS instead if you're too worried on getting "spoiled" xD) 1) ISL 2009 C1 (a)  2) USAMO 1997 P1 3) IMO 2011 P4 4) USAMO 2002 P1 5) (adapted from) Canada 2018 P3 Rather than just giving out the solution, I will try to motivate how you come up with such a step, so that it helps to develop intuition :D Problem 1 [ISL 2009 C1 (a)]  :  Consider $2009$ cards, each having one gold side and

Math is all about ideas. Solving a problem in an Olympiad involves two steps: solving the problem (or progress) and writing it down .  It is therefore important to be able to express our ideas well. Additionally, one should be able to convey their ideas in a way that is easily understood by others. You need to keep in mind that the people who read your solutions or proofs are readers and not mind readers. And sometimes, they can not guess what you are thinking. So try to convey your thoughts through your answer sheet as clearly as you can. If you gained any non-trivial information during the process of solving a problem, write it in the answer sheet if you feel it helps prove the statement. The way you write your solutions and the number of details you skip will depend on the level of the exam you are taking. I would, for example, write some minute details in RMO that I would skip at the INMO level. Also, this post is for students who are completely new to olympiads and are trying to