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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

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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

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This week was full Number Theory and algebra biased!! ðŸ˜„ 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

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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

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