About

Hewo! My name is Sunaina Pati, and I am a first year undergraduate student in CMI.

I started this blog to discuss my math ideas, to share my experiences, my own notes(both school and olympiad) and many more. I would call this blog, heavily mathematics-biased. I do post a few non-math posts. Moreover, this blog is where I put a lot of problems I solved. So I post the solutions here, to verify my solution and practice proof writing.

Also, these problems are really perfect and not heavy theory-based. The difficulty range is however quite large. In general, it's IMO P1 level or higher.

Though there is no guarantee about when I will post, so if you are interested, then do subscribe 😄, so that you don't miss out on anything new!

Here's a little about me:

I would describe myself as a math enthusiast, who likes learning new theories and applying them to various problems.
I am a three times IMOTC qualifier and EGMO 2023 silver medalist.
My MBTI type is INTJ-T.
I love solving Olympiad math problems and teaching math to other kids.
I run a very cool math club called The Philomath club.
I don't have a favourite area in Olympiad mathematics, however, I tend to solve some of the geometry problems in the test, thanks to the amazing book EGMO.
Apart from doing olympiad math, I usually spend my time studying school books, watching k-dramas, reading webtoons and making digital art.
I love listening to K-POP ( TXT, Enhypen, BTS, Seventeen, Exo, G-Idle, Kard, Mamamoo, Black Pink, New Jeans, Lessarafim, NCT, Stray Kids, Red Velvet, Eric Nam, Woostein, Woosung, Sunmi, Somi) and Indian Indies( Western Ghats and Zaeden are my favs)

Total problems discussed: 330
Total Posts: 52

Some fun FAQs:

0. How should I use this blog?

Ans: I won't be biased. You can subscribe to this blog if you want to (you can unsubscribe whenever you want to)!

The content I will be sharing is probably me writing solutions to ISLs and many more Math contest problems. And maybe rate them according to me. I personally prefer walkthroughs ( And I agree it's easier to write too, but I need to practice my proof-writing too! )
You can simply binge-read my posts! ~~It's quite fun to see me suffering~~
I am a human. I won't post only math. I will be posting non-math too. The non-math content is pretty random. Sometimes I just post cat memes or sometimes some serious talk. But I assure you, this won't be boring.
If you are a high-school student who wants to do good on Boards and in the olympiads too. I understand how much pressure you must be going through, so don't worry about the school. I have made notes for a few of the chapters of Class 10, which I used for myself! They are pretty nice ( sufficient enough for scoring 95% I guess?). Read them before you read the book. After reading the notes, read the book. You are set!

I think you can refer to my Humanities area and Bio notes! They are actually pretty good. Rest depends on how good you can "bend and write the answers with beauty."
You can get them on my Notes and Problem Sets page.
I have a Post log page. 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!

1. What's your favourite K-POP band?

Ans: Currently Tomorrow x Together. I really love them. However musically, I am really into Enhypen!

2. What's your favourite colour?

Ans: Aquamarine and ochre. Bright yellow and pastel shades too.

3. What's your Zodiac sign?

Ans: Capricorn ( And I am a true Capricorn)

4. What's your favourite genre in music?

Ans: K-PoP, J-PoP and Indian Indies.

Lastly, I am Sunaina Pati in MSE, Jelena_ivanchic in AOPS, Sun_Prime#7400 in Discord and Sunaina Pati in Linkedin.

See y'all soon!

Sunaina 💜

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...

How to prepare for RMO?

"Let's wait for this exam to get over".. *Proceeds to wait for 2 whole fricking years! I always wanted to write a book recommendation list, because I have been asked so many times! But then I was always like "Let's wait for this exam to get over" and so on. Why? You see it's pretty embarrassing to write a "How to prepare for RMO/INMO" post and then proceed to "fail" i.e not qualifying. Okay okay, you might be thinking, "Sunaina you qualified like in 10th grade itself, you will obviously qualify in 11th and 12th grade." No. It's not that easy. Plus you are talking to a very underconfident girl. I have always underestimated myself. And I think that's the worst thing one can do itself. Am I confident about myself now? Definitely not but I am learning not to self-depreciate myself little by little. Okay, I shall write more about it in the next post describing my experience in 3 different camps and 1 program. So, I got...

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...

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...

Birthday Functional Equations problems

Heyoo!!! Birthday FEs!!!!!! $11$ FEs!! Also I would be posting solutions to RG's FE handout, I am done with 10 prs :P!! Problem: Find all functions $f :\Bbb R \rightarrow \Bbb R$ such that $$2f (x) - 5f (y) = 8, \forall x, y \in \Bbb R$$ Solution: $$2f(x)-5f(y)=8$$ $$\implies 2f(x)-5f(x)=8$$ $$\implies f(x)=\frac{-8}{3}, \text{ a constant function }$$ We did this in Rohan Bhaiya's FE class..Oh btw the EGMO camp is sooo niceee! I am loving it!! It's such a big deal to be able to train and attend the camp with EGMO team members! Problem: Find all functions $f :\Bbb R \rightarrow \Bbb R$ such that $$f (x) + xf (1 -x) = x, \forall x\in \Bbb R.$$ Solution: $$f(x)+xf(1-x)=x$$ $$f(1-x)+(1-x)f(x)=1-x$$ This is actually in the linear equations in two variable form! $$x+ay=a$$ $$y+bx=b$$ Anyways, $$f(x)+xf(1-x)=x$$ $$xf(1-x)+f(x)(1-x)x=(1-x)x$$ $$ \implies f(x)(x-x^2)-f(x)=-x^2\implies f(x)=\frac{-x^2}{x-x^2-1}=\frac{x^2}{x^2-x+1}$$ But verifying, this doesn't work. Problem: ...

Problems done in August

Welcome back! So today I will be sharing a few problems which I did last week and some ISLs. Easy ones I guess. Happy September 2021! Problem[APMO 2018 P1]: Let $ABC$ be a triangle with orthocenter $H$ and let $M$ and $N$ denote the midpoints of ${AB}$ and ${AC}$. Assume $H$ lies inside quadrilateral $BMNC$, and the circumcircles of $\triangle BMH$ and $\triangle CNH$ are tangent. The line through $H$ parallel to ${BC}$ intersects $(BMH)$ and $(CNH)$ again at $K$, $L$ respectively. Let $F = {MK} \cap {NL}$, and let $J$ denote the incenter of $\triangle MHN$. Prove that $FJ = FA$. Proof: By angle chase, we get $\angle FKL=\angle FLK.$ Hence $KL||MN\implies FM=FN.$ And we get $\angle MFN=2A\implies F$ is circumcentre if $(AMN)\implies FA=FM=FN.$ And we get $\angle MHN=180-2A$ Hence $MFHN$ is cyclic. By fact 5, $ME=FJ=FN\implies FJ=FA.$ Problem[Shortlist 2007 G3]: Let $ABCD$ be a trapezoid whose diagonals meet at $P$....

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...

Some NMTC sub-junior level Problems

Well.. Many people don't know but I was a part of STEM's Horizon ( Now, I have left them due to boards etc.) BTW STEM's Horizons is really great! And anyone interested in Olympiad math should join it! More info about it in below (make sure to check it out!). So here are some problems I sent to them. They are fairly easy, and most of them are repetitive ideas. But they are my first sets of problems ( I know UMO was there but still..) The solutions will be posted in another blog posts. You guys can type out sols in the comments sections too :) Problems:- 1. What is maximum possible number dividing $x^2+x+1$ and $x^5 +x^4 +x^3 + 3x^2 +2 x +4$ for all $x\in \Bbb{N}$ 2. Let $P$ be the sum of all $x$ and $y$ satisfying $45^x-2^x=2021^y.$ What is the last two digits of $p^2+p+1.$ 3. What is the greatest value of $r$ such that $3^r$ is factor of $10^{2022}-8^{674}$. 4. Find all possible tuples $ (x,y,l)$ such that $\frac{x}{100}=\frac{20}{y}=\frac{5}{l}.$ 5. Consider the following...

Just spam combo problems cause why not

This post is mainly for Rohan Bhaiya. He gave me/EGMO contestants a lot and lots of problems. Here are solutions to a very few of them. To Rohan Bhaiya: I just wrote the sketch/proofs here cause why not :P. I did a few more extra problems so yeah. I sort of sorted the problems into different sub-areas, but it's just better to try all of them! I did try some more combo problems outside this but I tried them in my tablet and worked there itself. So latexing was tough. Algorithms "Just find the algorithm" they said and they died. References: Algorithms Pset by Abhay Bestrapalli Algorithms by Cody Johnson Problem1: Suppose the positive integer $n$ is odd. First Al writes the numbers $1, 2,\dots, 2n$ on the blackboard. Then he picks any two numbers $a, b$ erases them, and writes, instead, $|a - b|$. Prove that an odd number will remain at the end. Proof: Well, we go $\mod 2$. Note that $$|a-b|\equiv a+b\mod 2\implies \text{ the final number is }1+2+\dots ...

IMO Shortlist 2022 C1

Today we shall try IMO Shortlist $2022$ C1. A $\pm 1$-sequence is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and$$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$ We claim that the answer is $\boxed{506}$. $506$ is the upper bound. Just consider the sequence $$+1,-1,-1,+1,+1,-1,-1,+1\dots,-1,-1,+1,+1,-1.$$ Here $1, -1, -1, 1$ is repeated $505$ times and $1,-1$ is concatted to it. Now,our sequence would be $a_1,a_3,a_4,a_5,a_7,\dots$ which on summing would give $506$. And clearly, this would give the upper bound. Now, we show that $506$ is attainable by every sequence. WLOG there are at least $1011$ positive numbers in the sequence. Then we choose $+1$ whenever we can. Let the sequence be $c_1,b_1,\dots, c_n,b_n$ where $c_i$ are ...

Sunaina thinks absurd

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