The Online Math Club Blog

Hello everyone!  Congratulations to everyone who attempted the RMO 2024. As you might know, we had an amazing livesolve of the paper with Archit, Adhitya, Abel and Kanav which you can check out  here . We also have question wise video solutions to all the problems, thanks to Nanda, Om and Shreya!  We had a lot of people interested in solutions for the KV/JNV paper, which is what this blog post will be about. Without further ado, let's get started! Problem 1:  Find all positive integers $x,y$ such that $202x+4x^2=y^2$. Solution:  Notice that $y>2x$. Let $y=2x+k$ for some integer $k>0$. Thus, the given equation reduces to $$202x=4xk+k^2\implies x=\frac{k^2}{202-4k}\cdots (1)$$ This tells us that $202-4k|k^2,$ or that $101-2k|2k^2\implies 101-2k|101k$. However, since 101 is a prime, $\gcd(101-2k,\,101)=1\implies 101-2k|k$ or that $101-2k|2k\implies 101-2k|101\implies k=50$. Substituting in $(1)$, we get that $x$ must be $$\frac{50^2}{202-4(50)}=50\cdot 25=12...

Hellooooooo people I am Rishad. I am a 12th grader. Passionate about pursuing a career in mathematics. I like doing and studying all types of math. Hobbies: Listening to music and doing absolutely nothing and of course reading articles about math(mainly) and related fields( physics, computer science, cryptography..... not chemistry, of course). Okay, now moving on to the objective of this blog. Here, I will be presenting to you the solutions for the Indian Statistical Institute( Objective Paper) 2022. Q1.Any positive real number $x$ can be expanded as:  $x=a_{n}\cdot 2^{n} + a_{n-1}\cdot 2^{n-1}+ a_{n-2}\cdot 2^{n-2}+.........+a_{-1}\cdot 2^{-1}+a_{-2}\cdot 2^{-2}.....$  for some $n\geq0$ , where each $a_{i}  \in  \{0,1\}$. In the above-described expansion of 21.1875, the smallest positive integer k such that $a_{k}\neq0$ is:  Motivation/Overview: Okay, so this is a question based on a binary representation of numbers. For a number in base b the general represe...

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...

Hi fellow people and cats :D.. Welcome to the first blogpost of the OMC blog. The Online Math Club is an initiative to reach high school students interested in math and give them a platform to learn more and interact with others! By the way, I am Sunaina and you will be reading my thoughts :P I hope you all are doing well and are enjoying math! You probably guessed what type of content today's post will have! So without further ado, lemme begin. We all remember the first time we heard about the term "Olympiad math" ( Let's not consider competitions like SOF, simply consider HBCSE's Olympiads, USA(J)MO or the equivalents in other countries). What were your reactions? Lemme share my reactions when I heard about them for the first time, " WHAT?!!? HOW AM I SUPPOSED TO GIVE THE SAME EXAM AN 11th GRADE KID IS GIVING." (I found out about olys in grade 8 btw, but started seriously preparing in grade 9 :P) Haha anyways do comment what your first thoughts were! S...