Hi, I am Emon and I am back this time to provide you with some ideas regarding the ''Inclusion and Exclusion Principle" , which is a really important topic that we should learn for olympiads. The I nclusion-Exclusion Principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets, which is given by the formula $$\left|A\cup B\right| = \left|A\right|+\left|B\right| - \left|A\cap B\right|.$$ where $A$ and $B$ are two finite sets and $|X|:=$ cardinality of set $X$, $i.e.$ the number of elements present in the set $X$. The proof of this formula can be understood directly from the above venn diagram. Additional Note (though a bit out of context). Let $\mathcal{S}$ be a discrete sample space. Then, if $A, B\subset \mathcal{S}$ are two events, then we may write $$\mathbb{P}(A\cup B)=\mathbb{P}(A)+\mathbb{P}(B)-\mathbb{P}(A\cap B),$$ which is equivalen...
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...