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 💜

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...

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...

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...

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...

IMO Shortlist 2021 C1

I am planning to do at least one ISL every day so that I do not lose my Olympiad touch (and also they are fun to think about!). Today, I tried the 2021 IMO shortlist C1. (2021 ISL C1) Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$. Suppose not. Then any $3$ elements $x,y,z\in S$ will be $(x,y)=(y,z)=(x,z)$ or $(x,y)\ne (y,z)\ne (x,z)$. There exists an infinite set $T$ such that $\forall x,y\in T,(x,y)=d,$ where $d$ is constant. Fix a random element $a$. Note that $(x,a)|a$. So $(x,a)\le a$.Since there are infinite elements and finite many possibilities for the gcd (atmost $a$). So $\exists$ set $T$ which is infinite such that $\forall b_1,b_2\in T$ $$(a,b_1)=(a,b_2)=d.$$ Note that if $(b_1,b_2)\ne d$ then we get a contradiction as we get a set satisfying the proble...

Symmetric Polynomials #week 6

Well... I haven't seen much symmetric polynomials in Olympiads, but still I am learning, because I found them cute. And I am basically using this blog as my notes :P What are symmetric polynomials? One can understand this with examples. If we are considering over 3 variables, $x_1,x_2,x_3$ then $$\sum_{sym}x_1^2\cdot x_2^3\cdot x_3=x_1^2\cdot x_2^3\cdot x_3+x_1^2\cdot x_3^3\cdot x_2+x_2^2\cdot x_1^3\cdot x_3+x_2^2\cdot x_3^3\cdot x_1+x_3^2\cdot x_1^3\cdot x_2.$$ See? $3!$ terms! Let's take one more example with again over 3 variables, $x_1,x_2,x_3$ then $$\sum_{sym}x_1^2\cdot x_2^2= x_1^2\cdot x_2^2+x_1^2\cdot x_3^2+x_2^2\cdot x_1^2+x_2^2\cdot x_3^2+x_3^2\cdot x_1^2+x_3^2\cdot x_2^2$$ Wait.. why 2 times ? So basically what happens in symmetrictric sums, is we go through all $n!$ possible permutations. So, here we have $a^2\cdot b^2\cdot c^0$ as like the "general" form type, right? Now, list down all the $3!=6$ permutations of $x_1,x_2,x_3$, and put them in the gene...

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 ...

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...

Some Geometry Problems for everyone to try!

These problems are INMO~ish level. So trying this would be a good practice for INMO! Let $ABCD$ be a quadrilateral. Let $M,N,P,Q$ be the midpoints of sides $AB,BC,CD,DA$. Prove that $MNPQ$ is a parallelogram. Consider $\Delta ABD$ and $\Delta BDC$ .Note that $NP||BD||MQ$. Similarly, $NM||AC||PQ$. Hence the parallelogram. In $\Delta ABC$, $\angle A$ be right. Let $D$ be the foot of the altitude from $A$ onto $BC$. Prove that $AD^2=BD\cdot CD$. Note that $\Delta ADB\sim \Delta CDA$. So by similarity, we have $$\frac{AD}{BD}=\frac{CD}{AD}.$$ In $\Delta ABC$, $\angle A$ be right. Let $D$ be the foot of the altitude from $A$ onto $BC$. Prove that $AD^2=BD\cdot CD$. Let $D\in CA$, such that $AD = AB$.Note that $BD||AS$. So by the Thales’ Proportionality Theorem, we are done! Given $\Delta ABC$, construct equilateral triangles $\Delta BCD,\Delta CAE,\Delta ABF$ outside of $\Delta ABC$. Prove that $AD=BE=CF$. This is just congruence. N...

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

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