### 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.

### 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

### 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

### Spoiled for choice

Hi! I'm Atul, in my last year of school and this is going to be my first blog post. For an introduction, I have a bronze at the IMO and a silver at the APMO this year. Apart from math, I also love reading, playing table tennis and making terrible jokes. Something I seem to miss out on saying often is that I love  cats , especially  colour ful ones. Anyway, let's begin! Like many things  i n math, the axiom of choice has a deceptively  s imple statement. All it says is - "Given a collection of  n onempty sets, it is possible to pick an element from each of them". That... sounds pretty obvious and something that just  should  be true. And well, it is. At least for a finite collection of sets, it is definitely true. With some thought, it's also clear how to do it if the collection is countably infinite. But what if the number of sets is just  extremely  huge? When the sets are uncountably infinite, that's when the problems begin. So far I've b e en rather ha