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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...
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LMAO Revenge

Continuing the tradition of past years, our seniors at the Indian IMO camp(an unofficial one happened this year) once again conducted LMAO, essentially ELMO but Indian. Sadly, only those who were in the unofficial IMOTC conducted by Pranav, Atul, Sunaina, Gunjan and others could participate in that. We all were super excited for the problems but I ended up not really trying the problems because of school things and stuff yet I solved problem 1 or so did I think. Problem 1:  There is a   grid of real numbers. In a move, you can pick any real number  ,  and any row or column and replace every entry   in it with  .  Is it possible to reach any grid from any other by a finite sequence of such moves? It turned out that I fakesolved and oh my god I was so disgusted, no way this proof could be false and then when I was asked Atul, it turns out that even my answer was wrong and he didn't even read the proof, this made me even more angry and guess wha...
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Functional Equations 101

Let's get to the math:  Let there be two sets $X$ and $Y$. A function  from $X$ to $Y$ denoted as $f \colon X \to Y$ is assigning a value in $Y$ for every element in $X$. We say that $X$ is the domain of the function $f$ and $Y$ is the range.  A function $f \colon X \to y$ is said to be injective if $f(x) = f(x^{\prime}) \implies x = x^{\prime}$ To put it in a more abstract way, if there is some $a \in Y$ then there is at most one $b \in X$ such that $f(b) = a$ holds true.  A function is said to be surjective when for any $a \in Y$ there is at least one $b \in X$ such that $f(b) = a$ holds true.  A function is bijective if for every $a \in Y$ there is exactly one $b \in X$ such that $f(b) = x$. Bijective functions are basically functions which are both injective and surjective.  Bonus: A function $f \colon X \to X$ is known as an involution if $f(f(x)) = x \; \forall  x \in X$  As an exercise, the readers should try to prove that every function th...
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