Problem solving in NT (day 2 week 1)

Well, I am finally free! It's more like I was busy with classes and covering up the NT4 course, dang it is hard! Finally got hand of convolutions etc!

ISL 2002 N3: Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$ . Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.

Proof: Now consider $d\mid p_1p_2\ldots p_n$ and $d'\mid p_1p_2\ldots p_n$ . Note that there is a prime $p|2^d+1|2^{p_1p_2\cdots p_n}+1$ and $p\nmid 2^{d'}+1$ by zsig for any $d, d'$ . Hence there are any $2^n$ distinct such $p$ ( as there are $2^n$ prime factors of $p_1p_2\cdots p_n$ ).

Since there are atleast $2^n$ distinct primes dividing $p_1p_2\cdots p_n \implies$ there are at least $2^{2^n}$ primes factors which is $>>4^n$ .

The exceptional case doesn't matter here.

[ISL 2000 N4]:Find all triplets of positive integers $(a,m,n)$ such that $a^m + 1 \mid (a + 1)^n$

Proof: Except for the exception of the Zsigmondy theorem, there exists $p$ such that $p|a^m+1$ and $p\nmid (a+1)$ but for $m>1$ , we get that $a^m+1\nmid (a+1)^n$ . So $m=1$ .

The exception case is $(a,m,n)=(2,3,n)$ and works for $(a,m,n)=(a,1,n)$

Comments

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

$k$ ...

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...

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 ...

Bio is Love..

Adios, everyone! Boards preparation at its peak :( However, I am not able to study how I used to. Every time I try to study for boards, I just keep thinking much about a topic, stare at the book, jam a song or just start doing procrastination by bookmarking random cute problems in HSO. It's been more than a year I have studied like with a focus on a book. My lappy is being a big distraction tbh. So after INMO score come out, I will just give my lappy for repair and say papa to bring it back home after June 2. Milk and Mocha I literally am taking 2 days to complete 1 bio chapter, some times even 3. The rate of my "slowness" is probably because I am like every 15 minutes checking discord to see if the INMO scores are out or not. So HBCSE, thank you for keeping me anxious. Funfact:- we must be grateful that there is an organisation that is conducting these national Olys. There are some countries where no Olys are being conducted. ( Same dialogue which mumma uses, but in p...

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...

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...

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 ...

Problems I did this week [Jan8-Jan14]

Yeyy!! I am being so consistent with my posts~~ Here are a few problems I did the past week and yeah INMO going to happen soon :) All the best to everyone who is writing! I wont be trying any new problems and will simply revise stuffs :) Some problems here are hard. Try them yourself and yeah~~Solutions (with sources) are given at the end! Problems discussed in the blog post Problem1: Let

$ABC$ be a triangle whose incircle

$\omega$ touches sides

$BC, CA, AB$ at

$D,E,F$ respectively. Let

$H$ be the orthocenter of

$DEF$ and let altitude

$DH$ intersect

$\omega$ again at

$P$ and

$EF$ intersect

$BC$ at

$L$ . Let the circumcircle of

$BPC$ intersect

$\omega$ again at

$X$ . Prove that points

$L,D,H,X$ are concyclic. Problem 2: Let

$ABCD$ be a convex quadrangle,

$P$ the intersection of lines

$AB$ and

$CD$ ,

$Q$ the intersection of lines

$AD$ and

$BC$ and

$O$ the intersection of diagonals

$AC$ and

$BD$ . Show that if

$\angle POQ= 90^\circ$ then

$PO$ is the bisector of

$\angle AOD$ ...

Sunaina thinks absurd

Search This Blog