Skip to main content

Problems with meeting people!

Yeah, I did some problems and here are a few of them! I hope you guys try them!

Putnam, 2018 B3

Find all positive integers $n < 10^{100}$ for which simultaneously $n$ divides $2^n$, $n-1$ divides $2^n - 1$, and $n-2$ divides $2^n - 2$.

Proof

We have $$n|2^n\implies n=2^a\implies 2^a-1|2^n-1\implies a|n\implies a=2^b$$ $$\implies 2^{2^b}-2|2^{2^a}-2\implies 2^b-1|2^a-1\implies b|a\implies b=2^c.$$ Then simply bounding.

USAMO 1987

Determine all solutions in non-zero integers $a$ and $b$ of the equation $$(a^2+b)(a+b^2) = (a-b)^3.$$

Proof

We get $$ 2b^2+(a^2-3a)b+(a+3a^2)=0\implies b = \frac{3a-a^2\pm\sqrt{a^4-6a^3-15a^2-8a}}{4}$$ $$\implies a^4-6a^3-15a^2-8a=a(a-8)(a+1)^2\text{ a perfect square}$$ $$\implies a(a-8)=k^2\implies a^2-8a-k^2=0\implies \implies a=\frac{8\pm\sqrt{64+4k^2}}{2}=4\pm\sqrt{16+k^2}. $$ $$ 16+k^2=m^2\implies (m-k)(m+k)=16.$$

Now just bash.


USAMO 1988

Suppose that the set $\{1,2,\cdots, 1998\}$ has been partitioned into disjoint pairs $\{a_i,b_i\}$ ($1\leq i\leq 999$) so that for all $i$, $|a_i-b_i|$ equals $1$ or $6$. Prove that the sum $$ |a_1-b_1|+|a_2-b_2|+\cdots +|a_{999}-b_{999}|$$ ends in the digit $9$.

Proof

Note we just have to show that the answer is $-1 \mod 10.$ We will show that the answer is $1\mod 2,~~4\mod 5.$ Note that $a_i-b_i$ and $a_i+b_i$ have the same parity. \\ So the sum is $$\sum (a_1+b_i)\mod 2 \equiv \frac{1998\cdot 1999}{2}\mod 2.$$ And $|a_i-b_1|$ is always $1\pmod 5.$ Hence the sum is $999\cdot 1 \equiv 4\mod 5.$


IMO 2009 P1

Let $ n$ be a positive integer and let $ a_1,a_2,a_3,\ldots,a_k$ $ ( k\ge 2)$ be distinct integers in the set $ { 1,2,\ldots,n}$ such that $ n$ divides $ a_i(a_{i + 1} - 1)$ for $ i = 1,2,\ldots,k - 1$. Prove that $ n$ does not divide $ a_k(a_1 - 1).$

Proof

Assume not, then $a_ka_1 \equiv a_k \mod n.$ The given conditions can be written as, $a_i\cdot a_{i+1}\equiv a_i\mod n.$ So$$a_1\cdot a_2\equiv a_1\mod n$$$$a_2\cdot a_3\equiv a_2\mod n$$$$\vdots$$$$a_{k-1}\cdot a_k\equiv a_{k-1}\mod n$$ So we have$$a_1\equiv a_1\cdot a_2\equiv a_1\cdot a_2\cdot a_3\equiv \dots \equiv a_1\cdot\dots\cdot a_k\mod n.$$ Now, we again reduce, so we get$$ a_1\cdot (a_2\dots\cdot a_{k-2}\cdot a_{k-1})\cdot a_k\equiv a_1\cdot (a_2\dots\cdot a_{k - 2})\cdot a_k\equiv\dots\equiv a_1\cdot a_{k}\equiv a_k\mod n$$ So we get $a_1\equiv a_k\mod n.$ Which is not possible, since $ a_1,a_k\in{1,\dots n}$ and are distinct.


Iberoamerican 2020 P2

Let $T_n$ denotes the least natural such that $$n\mid 1+2+3+\cdots +T_n=\sum_{i=1}^{T_n} i$$Find all naturals $m$ such that $m\ge T_m$.

Proof

Note that all odd $m$ works. For $n=2^k$, it is not possible. Let $n=2^kp_1^{\alpha_1}\cdots p_l^{\alpha_l}$. We want $$M(M+1)\equiv 0\mod 2n. $$ And we want $M\le m.$ Note that such number satisfies $M(M+1)\equiv 0\mod 2^{k+1}\prod p_i^{\alpha_i}$ then $$ M\equiv 0 \text { or } 1 \mod 2^{k+1}, p_i^{\alpha_i}.$$ By CRT, we get that there are $2^{l+1}$ such numbers. Now, if any numbers are less than $m$ then we are done. Also, note that these numbers range from $0$ to $2m$. Note that if $$r(r+1)\equiv 0\mod 2m\implies (2m-1-r)(2m-1-r+1)\equiv 0\mod 2m. $$ So if any of these number $M>m,$ simply take $2m-1-M< m$ (and $M\ne 2m-1$)


IMO 2005 P6

Determine all pairs $(x, y)$ of integers such that $$1+2^{x}+2^{2x+1}= y^{2}.$$

Proof

Note that $y$ is odd. Clearly $(x,y)=(0,\pm 2)$ satisfies. Assume $x\ge 1$. Then by $\mod 8$, we get $x\ge 3.$ Note that $$2^x(2^{x+1}+1y^2-1=(y-1)(y+1).$$ Note that $$\gcd(y-1,y+1)=2.$$ We have two cases. 
Case 1: $v_2(y-1)=1,v_2(y+1)=x-1$ 
Proof: Then let $y=2^{x-1}\cdot k+1,~~k\text{ is odd}$ and $2^{x-1}=a$. We get $$2a(4a+1)=(ka+1)^2-1 \implies (8-k^2)a=2k-2.$$ We get that there is no such $k$. 
Case 2: $v_2(y+1)=1,v_2(y-1)=x-1$
Proof: So $y=k(2^{x-1})-1$ and $k$ is odd. Let $a=2^{x-1}$. We get $$2a(4a+1)=(ka-1)^2-1\implies 2+2k=(k^2-8)a.$$ Note that $k^2-8>2+2k$ for $k\ge 5$ and clearly $k>2$. So $k=3$ works. For $k=3$ we get $(x,y)=(4,\pm 23).$ So the answer is $\boxed{ (0,\pm 2),(4,\pm 23)}.$


Jappan 1996

Let $m,n$ be positive integers with $(m,n)=1$. Find $(5^m+7^m,5^n+7^n)$, given $m,n$ odd.

Proof

Let $\gcd(5^m+7^m,5^n+7^n)=d$. Now note that $$5^m\equiv -7^m\mod d$$ $$5^n\equiv -7^n\mod d.$$ So $$ \text{ord}_{d}(5/7)|2m, ~~\text{ord}_{d}(5/7)|2n\implies \text{ord}_{d}(5/7)=1\text{ or } 2.$$ So $$5^2\equiv 7^2\mod d,~~d|7^2-5^2=24.$$ But $$v_2(5^m+2^m)=2,~~v_3(5^m+7^m)\ge 1\implies \gcd(5^m+7^m,5^n+7^n)=12.$$


2006 N2

For $ x \in (0, 1)$ let $ y \in (0, 1)$ be the number whose $ n$-th digit after the decimal point is the $ 2^{n}$-th digit after the decimal point of $ x$. Show that if $ x$ is rational then so is $ y$.

Proof

Given $x$ is rational. So $x$ would be eventually periodic. Let the periodic length be $l$. \newline Now, note that if $\gcd(l,2)=1$, then $y$ will be eventually periodic $\mod \phi(l)$ as $\exists N$ such that we have $$2^n\equiv 2^{n+\phi(l)}\mod l\text{ for all } n>N$$ If $\gcd(l,2)=2$, then $y$ will be eventually periodic $\mod \phi(\frac{l}{v_2(l)})$ as $\exists N$ such that we have $$2^n\equiv 2^{n+\phi(\frac{l}{v_2(l)})}\mod l\text{ for all } n>N.$$


2017 EGMO P6

$ABC$ in its sides $BC,CA,AB$ are denoted by $G_1,G_2,G_3$ and $O_1,O_2,O_3$, respectively. Show that the circumcircles of triangles $G_1G_2C$, $G_1G_3B$, $G_2G_3A$, $O_1O_2C$, $O_1O_3B$, $O_2O_3A$ and $ABC$ have a common point.

Proof

Define $X$ as the anti-Steiner point of Euler line. Moreover, introduce $H$, $H_1$, $H_3$ analogously. We claim that all these circles pass through $X$. Note that $BXH_1H_3$ cyclic. So we have $$\angle H_1XH_3=\angle H_1BH_3=\angle H_1BH+\angle HBH_3=2(90-C)+2(90-A)=360-2(A+C)=2B.$$ As $H_1-G_1-O_1,H_3-G_3-O_3$ collinear, we have $$\angle O_1XO_3=2B=\angle G_1XG_3.$$ But note that, due to the reflection, we have $$\angle O_1BO_3=2B=\angle G_1BG_2\implies O_1XO_3B, G_1XG_3B \text{ cyclic}.$$


That's it for the math part. Here comes the non-math part!

Woah it's been months since I posted on this blog!!!!! Darn, there's a lotttt to update about my life! 
I (and Atul) were accepted for the Maths Beyond Limits camp which was going to be held in Poland. So for the Visa stuffs, I met Swastika and Atul!! 

Kolkata


When I was in Kolkata, I went to ISI Kolkata! Woah it is such a beautiful campus! And I met Neena Gupta there and talked for a good 45 mins and she explained to me how important it is to continuously work no matter where I am studying. She was so down to earth and is so good at conversations ( I was nervous the whole time and I was screaming inside AAAAAAAAAAAAAAAAAAAAAAAA)

Kolkata and Guwahati, weather-wise don't have much difference. As in, both cities have almost same humidity etc. Even the people there, they talk in bangla so yeah I can survive in Kolkata because I understand Bangla and the food style is also similar. 

The buildings are huge though, like the area near the Kolkata airport is so so so well developed, very beautiful! 

Since I was in Kolkata, I and Swastika decided to meet up in the Eco park. Her house is quite close to Eco park and Kolkata airport is close too ( We had flight the same night). 

And WOW. ECO PARK IS HUGE. And it's so pretty! I and Swastika were lost in park, it's so huge! We talked a lot and her brother is insanely smart and cute!

A few photo dumps! 









Bangalore


Bangalore was hectic. We had planned to stay two nights one day in Bangalore. Anyways, Atul and I decided to meet and come to my place. 

HE IS SO TALL!!!!!!!! ( LIKE HE IS) 

He played chess with Suhan and during the match, I and my cousin were so bored.  But then we decided to go to Dominoes and yeah that was fun!!! I still haven't accepted the fact that someone can like onions on pizzas ( onions in general are just nasty).. We talked about a lot of stuffs ( all non math) and yeah the pizzas were great! Our talks were mainly me talking, him listening and then interrupting me and ask me to eat my pizza. (Yes I talk a lot when I am with the right person)

We then decided to just walk back from the dominos place to my home and then I realised dogs are scary and Atul is scarier. THERE WAS A DOG IN OUR WAY. WHAT'S THE BEST THING TO DO? CHANGE OUR LANE right? WHAT THIS GUY DID WAS DRAG ME TO THE DOG'S WAY..

Then I realised Atul's plan was not to meet me but was to take revenge because of how much I annoyed him. Oki oki Just kidding, Atul is a very nice person and the bestest friend I ever had. 

We, unfortunately, didn't take any selfies ( Your fault Atul, you toh are not careless nah, I know I am very careless, then why did you not remind me for taking selfie) but we had one pic. Let me add that and then end this section! 




We both are not looking the best but it's fine


Technotholon

I and Ananda met Rushil, Krutarth and Debayu!! So a few photo dumps!





And we even watched stand-up comedy where we had to pretend we were long-lost siblings!! 


EGMOTC


It was a fun experience meeting new people! Writing EGMOTSTs were stressful too! We had late night dance parties, gossips about so many random stuffs! 

I and Gunjan were room mates! ( Gunjan's reaction was memorable cause the whole pre camp, she was like "I want a single room, Sunaina", guess it's faith?) We actually met in the airport and darn she is so pretty!!! Perfect example of beauty with brain! 

We had to find our cab and it was so hard and plus Gunjan was talking to the cab driver and she heard " Green Indian Flag".. ( For context, near the Chennai Airport is huge Indian flag where all Taxi drivers stand)

On our way going to CMI, we were talking with few of our friends ( Malay, Pranav and Atul). When we reached CMI, I met Rohan Bhaiya! And then Aditi di! 
My first reaction to CMI was "Ooo warm colours and greenery".. CMI has blackboards every where ( students can discuss about problems legit anywhere).

We then met Yashwi and Bhavya, who became roomates! Sanjana came the next day ( and OMG she is so tiny). Sanjana got the single room! Gunjan wanted single room but I emotionally convinced her to stay in our room.. 

We met Ananya di, Rohinee di, Prateek bhaiya and Sahil bhaiya the next day!

Apart from our first night where we simply slept, rest all the nights we( I and Gunjan) talked so so so much! Gunjan has such a fun personality and we had a lot of common friends so both of us could understand each other!

From Dec 2, our classes started and omg they were so tiring, I got so busy that I wasn't able to text people back.. We had classes from 8:30 to 6 ( with lunch break) and then zeta time from 10pm to whatever time we wanted. 

In one zeta time, Rohan bhaiya and Sahil bhaiya somehow convinced most of us that Reals are countable and I was like " NO REALS ARE NOT COUNTABLE" but then even I started questioning my life. 

I missed out on the second zeta time but from my friends reaction, I could see they were questioning their life.. It was on " Axiom of choice". 

The classes were interactive, not heavy on theory and a lot of problems. Rohan bhaiya had a lot to talk on Ramsey theory, which for sure blown our mind! "Edges in multidimensional and proving theorems on those dimensions".. Fancy RG ( but yes the ideas were nice)

All the classes and professor interactions were so fun! 

We had a dance party where even Ananya di joined us! ( And we danced in a lot of Bollywood songs)

On Dec 4, we went to Marina Mall, and brought a lot of stuffs and chocolates for Bhavya ( cause it was her birthday!) and then we had a practice test! I got very lucky in my practice test and scored 7|7|1. Problems were graded out of 7 here btw. Rg said I could have arbitrated and could have got more marks for P3 but it's fine.. I was happy only..

Dec 5-9 we were having classes and then I was going to the library to study ( BTW CMI Library is great)..
We girls were having so much fun!! 

Then we had our TSTs happen and I got my score as 3|0|3 for day 1 and 1|6|10 for day 2  TSTs were graded out of 10. 

P6 was proposed by Atul!!! On Day 2, Once I submitted my solutions to Rohan bhaiya and I was happy that I fully solved P6, Rohan Bhaiya said "guess who proposed P6?" and I was like "who?" and he said "Your best friend". And I was surprised as hell. 

Because I had told him to not propose for this year's TSTs as it's embarrassing to not solve your own best friend's problem. ( Atul you betrayed me )

Anyways, congrats Atul for the problem, it's a nice problem!

We played pictionary and codenames. Oh yes, we also had Estimathon and "solve it in your mind". These were so fun!  We talked about K-Pop too in a lecture! 





We were just too lazy to click pictures actually



Thanks to Rohan bhaiya for treating us to such delicious food. The EGMO TC was just 11 days camp but it made all of us so close.

Thanks to EGMOTC girls for taking care of me when I was sick!

Thanks to Rohan bhaiya, CMI, PD sir, the proposers and all the instructors for making EGMO TC possible! 

Thanks to Malay, Pranav, Sid, and Swastika for wishing me luck and being nice friends. Thankyou to Krutarth, Ananda and Rushil too ( even though they forgot to wish me luck) for gsolving with me so much!


And obviously, Atul, thank you for listening to me even though you were busy with College stuff, having so many  conversations, calling me whenever I needed you,  being such a nice friend, motivating me to have confidence and thanks for believing in me . Although you weren't physically there in the camp with me, you still played such an important part in the camp for me..( Oki too much emotional stuffs, but yes you are the best, dum kid. )







oh btw, yes, I did make the EGMO team. 
Labels:

Comments

  1. Gunjan AggarwalDecember 17, 2022 at 9:27 AM

    slay girl, all the best for egmo!!

    ReplyDelete
  2. Klavi3rJanuary 8, 2025 at 11:13 PM

    Hiyo, I believe the problem labeled 2005 IMO6 is 2006 IMO4.
    High quality post as always though, great job!

    ReplyDelete

Post a Comment

Popular posts from this blog

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 $k$...
2 comments
Read more

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 ...
20 comments
Read more

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...
10 comments
Read more

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!). And.. What about me?  A Better human being Well, I want to become a better human being this year compared to last year. From a very young age, my father has been saying to me, "It does not matter if you are a good mathematician, but you should be a nice human being." As a teenager, I never took the statement seriously. Well, all that mattered to me was to do good mathematically. Why should I care about other people's feelings? These were all my thoughts in high school.  So I ended up saying a few hurtful statements without realising that they were hurtful.  I never actually cared throughout my high school. You know, the world is too big, if I hurt person A, no worries, I will move on to person B and start a new friendship! As a res...
5 comments
Read more

INMO Scores and Results

Heya! INMO Results are out! Well, I am now a 3 times IMOTCer :D. Very excited to meet every one of you! My INMO score was exactly 26 with a distribution of 17|0|0|0|0|9, which was a fair grading cause after problem 1, I tried problem 6 next. I was hoping for some partials in problem 4 but didn't get any.  I am so so so excited to meet everyone! Can't believe my olympiad journey is going to end soon..  I thought to continue the improvement table I made last year! ( I would still have to add my EGMO performance and also IMO TST performance too) 2018-2019[ grade 8]:  Cleared PRMO, Cleared RMO[ State rank 4], Wrote INMO 2019-2020[ grade 9]:  Cleared PRMO, Cleared RMO[ State topper], Wrote INMO ( but flopped it) 2020-2021[grade 10]:  Cleared IOQM, Cleared INMO [ Through Girl's Quota] 2021-2022[grade 11]:  Wrote EGMO 2022 TST[ Rank 8], Qualified for IOQM part B directly, Cleared IOQM-B ( i.e INMO) [Through general quota],  2022-2023 [grade 12]:  Wrote E...
Post a Comment
Read more

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...
7 comments
Read more

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....
3 comments
Read more

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$ ...
Post a Comment
Read more

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...
7 comments
Read more

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 ...
4 comments
Read more