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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 :


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)

Popular posts from this blog

My experiences at EGMO, IMOTC and PROMYS experience

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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...
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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....
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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 ...
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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...
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Number Theory Revise part 1

I thought to revise David Burton and try out some problems which I didn't do. It's been almost 2 years since I touched that book so let's see! Also, this set of problems/notes is quite weird since it's actually a memory lane. You will get to know on your own! I started with proving a problem, remembered another problem and then another and so on! It was quite fun cause all these questions were the ones I really wanted to solve! And this is part1 or else the post would be too long. Problem1: Prove that for $n\ge 1$  $$\binom{n}{r}<\binom{n}{r+1}$$ iff $0\le r\le \frac{n-1}{2}$ Proof: We show that $\binom{n}{r}<\binom{n}{r+1}$ for $0\le r\le \frac{n-1}{2}$ and use the fact that $$\binom{n}{n-r}=\binom{n}{r}$$ Note that $\binom{n}{r}= \frac{n!}{r!(n-r)!}, \binom{n}{r+1}=\frac{n!}{(r+1)!(n-r-1)!}$  Comparing, it's enough to show that $$\frac{1}{n-r}<\frac{1}{r+1}\text{ or show } n-r>r+1$$ which is true as $0\le r\le \frac{n-1}{2}$ Problem2: Show that the exp...
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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...
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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$ ...
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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...
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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...
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