Post log

This is the post log. The main motivation to make such a page was to help other people to use this blog in the best possible way!

The archive widget and categories could possibly help, but this page is quite systematic. Moreover, I have given a basic preview of the blog post too!

This is always getting updated.

Total Posts: 47
Total problems discussed: 325

Miscellaneous Problem Discussions :

Top 10 problems of Week#1 I started a 4-week series, where I posted 10 problems each week, which had walkthroughs too.
Top 10 problems of Week#2
Top 10 problem of Week#3
Top 10 problems of Week#4
Week#5 This week, I posted two olympiad problems.
Geometry( Finally!!!) This was my first post that had Olympiad geometry in it! It had $13.$
High RMO level problems A collection of some nice diophantine which were overkilled by fancy NT theorems. It had $7.$
A Cute problem solution of a very cute Number theory problem I did. $1$ problem discussed..
Getting high with Geometry A collection of geometry problems with solutions from a few renowned books. $10$
Non-Geos are cool Started a series called Non-Geos are cool! The main motive is to get some cute non-geos that do not require that harsh theory! And is suitable for INMO level (Indian National Mathematical Olympiad). Featuring $5$ problems.
Non-geos are cool part 2 The second post of this series. Featuring $5$ problems.

Problems done in July Basically, at the end of each month, I will be sending the $5$ best problems I did in that month with the solutions. It might look a bit less to you, but I try to send $5$ problems each week. Doing the math, I will be sending $20$ problems in each month. With a level, probably equivalent to INMO (Indian National Mathematical Olympiad) or a bit higher.
Problems done in August Featuring some very cool Geos and cute polynomials! Featuring $10$ problems!
Geos are definitely cool Started a series called Geos are definitely cool! The main motive is to do some of the ISL geometries. And the series will increase in difficulty after each post! Collection $10$ ISLs.
Let's do problems only A pretty nice set of $20$ problems which are just INMO level.
Geos are my life support A set of $10$ nice geo problems explaining why geos are the best.
Number theory is my bias wrecker.html A set of nice $14$ IOQM- RMO level NTs
INMO problems $20$ past year INMOs which I did.
Birthday FEs I tried $11$ FEs
Number Theory revise part 1 I started a new series where I will revise the NT I learnt. This post has $13$ problems.
Number Theory revise part 2 Continuing with the above post, a collection of $11$ problems.
Number Theory revise part 3 Proved LTE and did $2$ problems..
Solving Random Problems before INMO Solved $9$ ISLs and shared $6$ sharygin.
Announcing the Unofficial TC A post where I discuss $13$ new problems of INMO~or INMO+ level!
Some problems in Olympiad Graph theory! A post where I discussed 16 graph theory problems. They are INMO or INM+ level :)

Notes/Problem Sets :

Symmetric Polynomials I wrote notes for Symmetric polynomials.
Newton Sums (Sym Poly part 2) This was the continuation of the Symmetric Polynomials post and talked about Newton sums.
My first own problem set This was my first problem set. All the problems are my own, though ideas are probably not new. The problems are of the level AMC-8 or AMC-10 level.
School notes and more problem sets The post where I sent more notes( handwritten). And some more problem sets compilation.
Vectors The basic introduction to vectors. It covers all the fundamentals!
Let's complex bash part 1 A series where I am posting my notes to "Complex number in geometry."
Let's complex bash part 2 The second post of the series.
Probability is global Olympiad Notes on probability with $17+18=35$ problems.
Expected value Continuing from "the probability is global", we do expected value along with $16$ problems.
Recurrence Relations Olympiad Notes on Recurrence Relations where almost each problem is s different topic and makes us learn something new.
We have $14$ problems to solve.
Graph theory part 1 Olympiad notes on Graph theory. In this post, I talk about Degrees, degrees sequence, etc with $8$ problems and many theorems.
Graph Theory part 2 Olympiad notes on Graph theory. In this post, I talk about the Euler path, Ore's Theorem, Hamiltonian, etc with $8$ problems and many theorems.
Graph Theory part 3 Olympiad notes on Graph theory. In this post, I talk about planar graphs with $6$ problems and theorems.
Graph Theory Part 4 Olympiad notes on Graph theory. In this post, I talk about Independent sets and covering sets

Non-math :

Summary of 2020 A new year post and summary of 2020.
IOQM results and predicted INMO I talked about my IOQM and predicted INMO paper.
Cute cats part 1 One of the most loved series of all time. It's really fun compiling some cute cat pics and adding some relatable yet fun captions.
Cute cats part 2 This was very special since I wrote it for 5k views special.
Our cute Siblings Post is dedicated to all the elder siblings who have to go through this painful period :(.
Reflecting on past Post where I explore what stuffs went wrong ( and also had discussion related to INMO)

Solving Random ISLs And Sharygin Solutions! And INMO happened!!

Some of the ISLs I did before INMO :P [2005 G3]: Let

$ABCD$ be a parallelogram. A variable line

$g$ through the vertex

$A$ intersects the rays

$BC$ and

$DC$ at the points

$X$ and

$Y$ , respectively. Let

$K$ and

$L$ be the

$A$ -excenters of the triangles

$ABX$ and

$ADY$ . Show that the angle

$\measuredangle KCL$ is independent of the line

$g$ Solution: Note that

$\Delta LDK \sim \Delta XBK$ and

$\Delta ADY\sim \Delta XCY.$ So we have

$\frac{BK}{DY}=\frac{XK}{LY}$ and

$\frac{DY}{CY}=\frac{AD}{XC}=\frac{AY}{XY}.$ Hence

$\frac{BK}{CY}=\frac{AD}{XC}\times \frac{XK}{LY}\implies \frac{BK}{BC}=\frac{CY}{XC}\times \frac{XK}{LY}=\frac{AB}{BC}\times \frac{XK}{LY}$

$\frac{AB}{LY}\times \frac{XK}{BK}=\frac{AB}{LY}\times \frac{LY}{DY}=\frac{AB}{DL}$

$\implies \Delta CBK\sim \Delta LDK$ And we are done. We get that

$\angle KCL=360-(\angle ACB+\angle DKC+\angle BCK)=\angle DAB/2 +180-\angle DAB=180-\angle DAB/2$ Motivation: I took a hint on this. I had other angles but I did...

Some problems in Olympiad Graph theory!

Hello there! It has been a long time since I uploaded a post here. I recently took a class at the European Girls' Mathematical Olympiad Training Camp 2024, held at CMI. Here are a few problems that I discussed! My main references were Po-Shen Loh's Graph theory Problem set (2008), Adrian tang's Graph theory problem set (2012) and Warut Suksompong's Graph Cycles and Olympiad Problems Handout and AoPS. I also referred to Evan Chen's Graph theory Otis Problem set for nice problems! Text Book Problems which are decent A connected graph

$G$ is said to be

$k$ -vertex-connected (or

$k$ -connected) if it has more than

$k$ vertices and remains connected whenever fewer than

$k$ vertices are removed. Show that every

$k$ -connected graph of order atleast

$2k$ contains a cycle of length at least

$2k$ . We begin with a lemma. Prove that a graph

$G$ of order

$n \geq 2k$ is

$k$ connected then every 2 disjoint set

$V_1$ and

$V_2$ of

$k$ distinct vertices each, there exist

$k$ ...

My experiences at EGMO, IMOTC and PROMYS experience

Yes, I know. This post should have been posted like 2 months ago. Okay okay, sorry. But yeah, I was just waiting for everything to be over and I was lazy. ( sorry ) You know, the transitioning period from high school to college is very weird. I will join CMI( Chennai Mathematical Institue) for bsc maths and cs degree. And I am very scared. Like very very scared. No, not about making new friends and all. I don't care about that part because I know a decent amount of CMI people already. What I am scared of is whether I will be able to handle the coursework and get good grades T_T Anyways, here's my EGMO PDC, EGMO, IMOTC and PROMYS experience. Yes, a lot of stuff. My EGMO experience is a lot and I wrote a lot of details, IMOTC and PROMYS is just a few paras. Oh to those, who don't know me or are reading for the first time. I am Sunaina Pati. I was IND2 at EGMO 2023 which was held in Slovenia. I was also invited to the IMOTC or International Mathematical Olympiad Training Cam...

Introduction

Hey Everyone!! This is my first Blog post. So let me give a brief introduction about myself. I am Sunaina Pati. I love solving Olympiad math problems, learning crazy astronomical facts , playing hanabi and anti-chess, listening to Kpop , love making diagrams in Geogebra and teaching other people maths 😊 . I love geometry , number theory and Combinatorics . I am starting this blog to keep myself a bit motivated in doing studies 😎 . Right now, I am planning to write walkthroughs on some of the best problems I tried over the week which can refer for hints 'cause solutions contain some major spoilers and one learns a lot while solving the problem on his own rather than seeing solutions . Also, there will be some reviews about Kpop songs, study techniques, my day to day lifestyles,exam reviews and ofc some non-sense surprises 😂. I am planning to try posting every week on Sundays or Saturdays ( most probably) ! Though there is no guarantee about when I ...

Geometry ( Finally!!!)

This is just such an unfair blog. Like if one goes through this blog, one can notice how dominated Algebra is!! Like 6 out of 9 blog post is Algebra dominated -_- Where as I am not a fan of Algebra, compared to other genres of Olympiad Math(as of now). And this was just injustice for Synthetic Geo. So this time , go geo!!!!!!!!!!! These problems are randomly from A Beautiful Journey through Olympiad Geometry. Also perhaps I will post geo after March, because I am studying combi. Problem: Let

$ABC$ be an acute triangle where

$\angle BAC = 60^{\circ}$ . Prove that if the Euler’s line of

$\triangle ABC$ intersects

$AB$ and

$AC$ at

$D$ and

$E$ , respectively, then

$\triangle ADE$ is equilateral. Solution: Since

$\angle A=60^{\circ}$ , we get

$AH=2R\cos A=R=AO$ . So

$\angle EHA=\angle DOA.$ Also it's well known that

$H$ and

$O$ isogonal conjugates.

$\angle OAD =\angle EAH.$ By

$ASA$ congruence, we get

$AE=AD.$ Hence

$\triangle ADE$ is equilateral....

New year with a new beginning! And a recap of 2024..and all the best for INMO 2025!

Hi everyone! Happy New Year :) Thank you so much for 95k+ views!!! How was everyone's 2024? What are everyone's resolutions? ( Do write down in the comment section! And you can come back 1 year later to see if you made them possible!). A Better Mathematician Well, technically a theoretical computer scientist. I am so grateful to be allowed to study at CMI where I can interact with so many brilliant professors, access the beautiful library and obviously discuss mathematics ( sometimes non math too ) with the students. And this year, I want to learn more mathematics and clear my fundamentals. I have become much worse in math actually. And hopefully, read some research papers too :) And discuss a lot of mathematics with other people. However, with that whole depressing 2024 year, I have lost a lot of my confidence in mathematics. And to be a better mathematician, I should gain the confidence that I can be a mathematician. And well, I am working on...

Orders and Primitive roots

Theory We know what Fermat's little theorem states. If

$p$ is a prime number, then for any integer

$a$ , the number

$a^p − a$ is an integer multiple of

$p$ . In the notation of modular arithmetic, this is expressed as

$a^{p}\equiv a{\pmod {p}}.$ So, essentially, for every

$(a,m)=1$ ,

${a}^{\phi (m)}\equiv 1 \pmod {m}$ . But

$\phi (m)$ isn't necessarily the smallest exponent. For example, we know

$4^{12}\equiv 1\mod 13$ but so is

$4^6$ . So, we care about the "smallest" exponent

$d$ such that

$a^d\equiv 1\mod m$ given

$(a,m)=1$ . Orders Given a prime

$p$ , the order of an integer

$a$ modulo

$p$ ,

$p\nmid a$ , is the smallest positive integer

$d$ , such that

$a^d \equiv 1 \pmod p$ . This is denoted

$\text{ord}_p(a) = d$ . If

$p$ is a primes and

$p\nmid a$ , let

$d$ be order of

$a$ mod

$p$ . Then

$a^n\equiv 1\pmod p\implies d|n$ . Let

$n=pd+r, r\ll d$ . Which implies

$a^r\equiv 1\pmod p.$ But

$d$ is the smallest natural number. So

$r=0$ . So

$d|n$ . Show that

$n$ divid...

Let's complex bash Part 1

I have to learn complex bash. And almost everyone knows that I am notes taking girl so thought why not make a post on complex bash ( so that I don't get emotionally demotivated lol).😇 There wasn't any need for learning complex bash, but it was in my dream checklist i.e " To learn a bash." And since I am not loaded with exams, I think it's high time to learn Bash and new topics. Also if anyone from the "anti-bash" community is reading, sorry in advance and R.I.P. Notes:- 1. Complex numbers are of the form

$z=a+ib,$ where

$a$ and

$b$ are real numbers and

$i^2=-1.$ 2. In polar form,

$z=r(\cos \theta+~~i\sin\theta)=~~re^{i\theta},$ where

$r=~~|z|=~~\sqrt{a^2+b^2},$ which is called the magnitude. 3. Here we used euler's formula i.e

$\cos \theta+~~i\sin\theta=~~e^{i\theta}.$ 4. The

$\theta$ is called the argument of

$z,$ denored

$\arg z.$ (

$\theta$ can be considered in

$\mod 360$ and it is measured anti-clockwise). 5. The complex conjugate of

$z$ is ...

How to prepare for INMO

Since INMO is coming up, it would be nice to write a post about it! A lot of people have been asking me for tips. To people who are visiting this site for the first time, hi! I am Sunaina Pati, an undergrad student at Chennai Mathematical Institute. I was an INMO awardee in 2021,2022,2023. I am also very grateful to be part of the India EGMO 2023 delegation. Thanks to them I got a silver medal! I think the title of the post might be clickbait for some. What I want to convey is how I would have prepared for INMO if I were to give it again. Anyway, so here are a few tips for people! Practice, practice, practice!! I can not emphasize how important this is. This is the only way you can realise which areas ( that is combinatorics, geometry, number theory, algebra) are your strength and where you need to work on. Try the problems as much as you want, and make sure you use all the ideas you can possibly think of before looking at a hint. So rather than fixing time as a measure to dec...

Reflecting on past

INMO Scores are out!! I am now a two times INMO awardee :) I got 16|0|1, so 17 in total! Yes, 16 in P1 T_T. I was thinking I would lose marks because of the way I wrote. Lemme tell ya'll what happened that day but first I should share a few thoughts I had before the exam. My thoughts Honestly, my preparation for INMO was bad. In fact, I should say I didn't work hard at all. As I have said earlier, I had lost all my hopes for INMO and Olympiads as a whole after EGMO TSTs happened. Art by Jelena Janic EGMO TSTs i.e European Girl's Mathematical Olympiad Team selection Tests 2022. Literally my thoughts after EGMO TSTs I feel very ashamed to share but I got 1 mark in my EGMO TSTs. Tests in which I literally gave my whole life. I did so many ISLs ( like SO MANY), I mocked EGMO 2021 TST where my score was 28/42 and I perfected Day 2. 1 mark in the TST just showed my true potential. There are way better people than me in olys. A friend even said to me, "If I wouldn't...

Sunaina thinks absurd

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