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Basic Combinatorics

Hello people its Rishad again. Yes I'm still alive. Now I am back here with another post (after an eternity).  This post is just about very fun questions from combinatorics. This post is not going to have really "high level" content. For some this might be just fun puzzle solving (experienced olympiad people) while for those who are starting off and who have little exposure to olympiad combinatorics will have exposition to some really fun mathematics. The questions are just sort of puzzles and not much mathematical knowledge is required to solve them.  Some warmup before we go to solving the main set of questions. I will just be providing the answers to these questions you have to do these questions on your own.  Q1 . A number of bacteria are placed in a glass. One second later each bacterium divides in two, the next second each of the resulting bacteria divides in two again, et cetera. After one minute the glass is full. When was the glass half-full? ANSWER : 59 SECOND
Recent posts

Some Wandering Through Orthic Triangles

 Hi, I am Emon and I it's been long since I posted last. Today I will try to give you some ideas on how to work with a special type of triangle, known as " Orthic Triangles ". In this post, I will mainly focus on problem-solving, but still, let me first give you some ideas on what exactly it is and what properties does it have... Definition (Orthic Triangle). Let $ABC$ be a triangle and let $D, E, F$ be the foot of the perpendiculars from $A, B, C$ to $BC, CA$ and $AB$, respectively. Then, $\triangle DEF$ is known as the orthic triangle of $\triangle ABC$. Lemma $1$ (Orthic Triangle). If $\triangle DEF$ is the orthic triangle of $\triangle ABC$ with orthocenter $H$, then the following conditions are satisfied : $(i)$ $AEHF$ is a cyclic quadrilateral with circumdiameter $AH$. $(ii)$ $BCEF$ is a cyclic quadrilateral with circumdiameter $BC$. $(iii)$ $H$ is the incenter of $\triangle DEF$. Lemma $2$. $\angle ABE = \angle ADE$ and $\angle ACF=\angle ADF$. (We can prove this w

A phase I'll never forget...

 It was the night of 8th August 2021, when we had the closing ceremony of CAMP (A math camp organized by Ritam Nag, Aatman Supkar and Rohal Goyal). My eyes were a bit wet realizing that it is ending and I won't have those 3 - 4 hours of fun every night which I had the whole past week. Thinking that time about all the memories I made in the camp from joining the game nights to getting frank with my seniors, it was probably one of the best experiences I had in any of the math related thing I did. I was fascinated by the organization of this camp and was just wondering if I was good enough to ever do something like this in future, but I never really got enough courage or time to do anything like this until recently. Fast forwarding to the time after INMO results, me and many of my friends were very sad that IMOTC has been cancelled by HBCSE and we couldn't meet each other for another year. Well, jokingly in a meet around that time, Sunaina told me to ask Rohan if they can organize

Top 10 problems of the week!

This week was full Number Theory and algebra biased!! 😄 Do try all the problems first!! And if you guys get any nice solutions, do post in the comments section! Here are the walkthroughs of this week's top 5 Number Theory problems! 5th position (1999 JBMO P2):  For each nonnegative integer $n$ we define $A_n = 2^{3n}+3^{6n+2}+5^{6n+2}$. Find the greatest common divisor of the numbers $A_0,A_1,\ldots, A_{1999}$. Walkthrough:  It doesn't require a walkthrough, I wrote this here, cause it's a cute problem for the person who has just started Contest math :P a. What is $A_0$? b. Find out $A_1$. c. Show that $\boxed{7}$ is the required answer! 4th position (APMO, Evan Chen's orders modulo a prime handout):   Let $a,b,c$ be distinct integers. Given that $a | bc + b + c, b | ca + c + a$ and $c | ab + a + b$, prove that at least one of $a, b, c$ is not prime. Walkthrough:  Fully thanks to  MSE  ! (Also one should try MSE, it has helped me a lot, ofc it's more tilted to Coll

Whose Game?

 In this blog post, we will talk about Combinatorial games, how to quantify positions, and analyse games. Most of this is based on the beginning chapters of the comprehensive book Winning Ways for your Mathematical Plays by Berlekamp, Conway and Guy. What is a Game? You might be familiar with multiple 'games', all with greatly varied meanings. For example, a  game could refer to a wild animal hunted for sport (This post isn't about those!). We will talk about games that are defined by the following (restrictive) rules: There are just 2 players, often named Left and Right. There are several well-defined positions and a starting position. There are clearly defined rules specifying the moves each player can make from a position, and the resulting positions are called options. Left and Right move alternately in gameplay. In the normal play convention, the player unable to move loses. In misere  play, the last player to move loses. The rules ensure that the game ends and one p

Trust Issues

Hi, I am Debayu and this is my second blogpost !! We start off with the following problem.  Problem: Find the sum of all $b$ such that $$2n+1 \mid \binom{2n}{b} -1$$ for all $n$ such that $2n \geq b$. At first glance, this looks somewhat innocent and its probably some problem from some random computational contest, some may this can be what the IOQM 10 markers will look like this year. hmmm so what do we do here? Let's check what happens when $b$ is $2$. So we have that for all $n$, $$2n+1 \mid \binom{2n}{2}$$ $$\iff 2n+1 \mid n(2n-1)-1=2n^2-n-1$$ $$ \iff 2n+1 \mid |2n^2-n-1-n(2n+1)| = |-2n-1|$$ which is true so yay $2$ works. Now we might want to check $b=3$, this means $$2n+1 \mid \binom{2n}{3}$$ $$\iff  2n +1 \mid \frac{n(2n-1)(2n-2)}{3}$$ $$\iff 6n+3 \mid 4n^3-6n^2+2n$$ from here, it is easy to see that if $n$ is even, the divisibility wont hold so $3$ doesn't work. Similarly one can check that $4$ doesn't work either and one might guess that the only possible $b$ is $2

Algorithms, or Mathematics?!

Hi everyone! In my last blog, I spoke about a couple of very interesting algorithmic techniques and how they can be used to solve a variety of very interesting problems. This blog however, is going to be completely different.   When we’re young we begin by learning the steps to add – we’re given the rules and we must learn to repeat them – no questions asked. Why does carry forward work the way it does? Well, no questions asked. To some extent, it is important to know how exactly to do a certain set of things while first learning maths. We may not be able to get anywhere in a subject if we’re unable to learn a few basic rules and know how to use them. However, after a certain point it is important to bring in the spirit of mathematical thinking within each student too – something missing in almost every form of school math education. Mathematical miseducation is so common, we wouldn’t even see it. We practically expect a math class to look like repetition and memorisation of disjointed

Mobius Inversion

The word 'Mobius' would instantly remind the readers of the Mobius strip, the famous object used to attract fellow students to the wonders of topology. But the discoverer of this object, August Ferdinand Mobius was also responsible for the creation of another amazing thing, the theory of Mobius inversions. It is a formula in number theory that answers the following question: Given two functions $f$ and $g$ on the set of naturals, if it is known that $$f(n)=\sum_{d\vert n}g(d),$$ how do you express $g$ in terms of $f$? Actually, the Mobius inversion formula that is applied in number theory is a special case of a more general theorem in the branch of Order theory. Without any doubt, we explore the general theorem first! For that, we need to build up some theory though... POSETS: A poset is a short word for 'partially ordered sets'. It is a set $P$ equipped with a binary relation $\leq$ that satisfies the following three conditions: $\forall a\in P$, $a\leq a$ (Reflexivity