### 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.

### Constructions in Number Theory

Hi, I am Emon, a ninth grader, an olympiad aspirant, hailing from Kolkata. I love to do olympiad maths and do some competitive programming in my leisure hours, though I take it seriously. I had written INOI this year. Today, I would be giving you a few ideas on how to Construct in Number Theory . Well, so we shall start from the basics and shall try to dig deeper into it with the help of some important and well-known theorems and perhaps, some fancy ones as well. Okay, so without further delay, let's start off... What do we mean by "Constructions"? If noticed, you can see that you often face with some problems in olympiad saying, "... Prove that there exists infinitely many numbers satisfying the given conditions" or "... Prove that there exists a number satisfying the above conditions." These are usually the construction-problems .  For example, let's consider a trivial example : Problem. Prove that there exist infinitely many integers $a$ such th

### EGMO solutions, motivations and reviews ft. Atul, Pranjal and Abhay

The  European Girls' Mathematical Olympiad a.k.a EGMO 2022 just ended. Congrats to Jessica Wan from USA, Taisiia Korotchenko, and Galiia Sharafetdinova for the perfect scores! Moreover, the Indian girls brought home 4 bronze medals! By far, this is the best result the EGMO India Team has ever achieved! To celebrate the brilliant result, here's a compilation of EGMO 2022 solutions and motivations written by my and everyone's favorite IMOTCer Atul ! And along with that, we also have reviews of each problem written by everyone's favorite senior, Pranjal !  These solutions were actually found by Atul, Pranjal,  and Abhay  during the 3-hour live solve. In the live solve, they solved all the 6 problems in 3 hours 😍!!! Okie Dokie, I think we should get started with the problems! Enjoy! Problem 1:  Let $ABC$ be an acute-angled triangle in which $BC<AB$ and $BC<CA$. Let point $P$ lie on segment $AB$ and point $Q$ lie on segment $AC$ such that $P \neq B$, $Q \neq C$ and

### Q&A with experts about bashing

Heyy everyone! From the title, you can see that we are going to talk about  BASH.  Yesss, we are going to discuss whether you really need to learn bash or not?  Let me first introduce myself, I am Pranav Choudhary, a 10th grader from Haryana(India). I like to do geo and combo the most :PP. Oki so let's begin! For those of you who don't know what bashing is, lemme give you a brief introduction first. Bashing is basically a technique used to solve Geometry Problems. In general, when you try a geo problem you might think of angles, similarities, and some other techniques (e.g. Inversion, spiral similarity etc). All of which are called synthetic geometry. But sometimes people use various other techniques called "bash" to solve the same problems.  Now there are different kinds of bashing techniques, e.g. - Coordinate bash, Trig Bash, Complex bash, Barycentric Coordinates. Let me give you a brief introduction to each of them.  Coordinate Bash : You set one point as the orig