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### Numbers, and that too IMAGINARY!!

Hi, I am Emon and yes, I am gonna write something today, after a long period of time, maybe on some imaginary numbers and in turn, complex numbers?

Okay, so let's look back into the history of complex numbers which first evolved mainly in the country of Italy...

Back in the $16$th century, a famous Italian mathematician, Niccolo Fontana Tartaglia posed the following problem in a journal :

Can you find a number $x$ such that $x^3+px=q$, where $p, q$ are given numbers?

In fact, he had a secret formula to this question : $$\boxed{x=\sqrt[3]{\frac{q}{2}+\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}} + \sqrt[3]{\frac{q}{2}-\sqrt{\frac{q^2}{4}+\frac{p^3}{27}}}}$$

Later, Italian Mathematician, Gerolamo Cardano proposed the same problem, with the only modification $p\mapsto -p$ :
Can you find a number $x$ such that $x^3=px+q$, where $p, q$ are given numbers?

He went on to give a second problem by asking :
"Can you divide $10$ into two parts such that their product is $40$?"

The people of those days, having no idea of imaginary numbers whatsoever said, "No, it's not possible!".
But, Cardano said, "Why? Just consider two numbers $5+\sqrt{-15}$ and $5-\sqrt{-15}$."

A few more problems involving the same idea were proposed by some other mathematicians as well. Mathematician Rafael Bombelli, another Italian, proposed a problem: Find $x$ such that $x^3=15x+4$.
Now, everyone, at very first sight answered, "$4!$". Bombelli announced that the answer was correct, but he, at the same time, gave a really intriguing solution, which involved the secret formula of Tartaglia...
Solution of Bombelli. By the secret formula of Tartaglia, we get, \begin{aligned} x &=\sqrt[3]{2+\sqrt{-121}}+\sqrt[3]{2-\sqrt{-121}}\\ &= 2+\sqrt{-1}+2-\sqrt{-1}\\ &= 4.\end{aligned}
So, basically, the solutions are exactly the same...

Later on, $\sqrt{-1}$ was defined as $i$, and was originally coined by Rene Descartes in the $17$th century and in the future, got wide acceptance from Leonhard Euler in the $18$th century and Augustin-Louis Cauchy and Carl Friedrich Gauss in the $19$th century. The imaginary number was combined with a real number to form the complex number $z=a+ib$, where $a, b\in \mathbb{R}$ and $i=\sqrt{-1}$. Hence, the largest set of numbers $\mathbb{C}$, overtaking the reals $\mathbb R$, came into being and the era of complex numbers started.

### 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 what? I was not alone, Krutarth too fakesol

### The importance of "intuition" in geometry

Hii everyone! Today I will be discussing a few geometry problems in which once you "guess" or "claim" the important things, then the problem can easily be finished using not-so-fancy techniques (e.g. angle chasing, power-of-point etc. Sometimes you would want to use inversion or projective geometry but once you have figured out that some particular synthetic property should hold, the finish shouldn't be that non trivial) This post stresses more about intuition rather than being rigorous. When I did these problems myself, I used freehand diagrams (not geogebra or ruler/compass) because I feel that gives a lot more freedom to you. By freedom, I mean, the power to guess. To elaborate on this - Suppose you drew a perfect  diagram on paper using ruler and compass, then you would be too rigid on what is true in the diagram which you drew. But sometimes that might just be a coincidence. e.g. Let's say a question says $D$ is a random point on segment $BC$, so maybe

### Edge querying in graph theory

In this post, I will present three graph theory problems in increasing difficulty, each with a common theme that one would determine a property of an edge in a complete graph through repeated iterations, and seek to achieve a greater objective. ESPR Summer Program Application: Alice and Bob play the following game on a $K_n$ ($n\ge 3$): initially all edges are uncolored, and each turn, Alice chooses an uncolored edge then Bob chooses to color it red or blue. The game ends when any vertex is adjacent to $n-1$ red edges, or when every edge is colored; Bob wins if and only if both condition holds at that time. Devise a winning strategy for Bob. This is more of a warm-up to the post, since it has a different flavor from the other two problems, and isn't as demanding in terms of experience with combinatorics. However, do note that when this problem was first presented, applicants did not know the winner ahead of time; it would be difficult to believe that Bob can hold such a strong