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Showing posts from August, 2022

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

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