Sunaina thinks absurd

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 $k$...

 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...

Welcome back! I tried the previous year INMO problems. And yes, I did a lot of them, here are a few! Here are the problems! ( they are kept in random order) [2008 P2]:  Find all triples $ \left(p,x,y\right)$ such that $ p^x=y^4+4$, where $ p$ is a prime and $ x$ and $ y$ are natural numbers. [some INMO]: Find all $m,n\in\mathbb N$ and primes $p\geq 5$ satisfying $$m(4m^2+m+12)=3(p^n-1).$$ [2015 P3]  Find all real functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x^2+yf(x))=xf(x+y)$. [2005 P6]:  Find all functions $f : \mathbb{R} \longrightarrow \mathbb{R}$ such that\[ f(x^2 + yf(z)) = xf(x) + zf(y) , \]for all $x, y, z \in \mathbb{R}$. [2012 P6] Let $f : \mathbb{Z} \to \mathbb{Z}$ be a function satisfying $f(0) \ne 0$, $f(1) = 0$ and $(i) f(xy) + f(x)f(y) = f(x) + f(y)$ $(ii)\left(f(x-y) - f(0)\right ) f(x)f(y) = 0 $ for all $x,y \in \mathbb{Z}$, simultaneously. $(a)$ Find the set of all possible values of the function $f$. $(b)$ If $f(10) \ne 0$ and $f(2) = 0$, find ...

Continuing from the previous post ( which is  here  ). This is a bit short post. I think this post is not at all helpful in olys but then let's do it cause I find them interesting :P. Here are my notes.. Book referred: Daniel A Marcus Graph theory Chapter H: Planar Graphs: A planar graph is a graph that can be represented by a diagram with no edge cross. Such a diagram is called a plane diagram. For $K_4$ is a planar graph. Regions formed by a graph:  The planar graph divides the plan into regions  Non- planar graph: A non-planar graph is a graph that always has a edge cross.  Regional Degrees: The regions are boundaries by edges. The number od edges each region is boundaries with is called regional degrees. Subdivision: A graph that is obtained by inserting vertices of degree $2$ is an already existing edge. Regional Degree theorem: Let $G$ be a connected graph and let $r_1,r_2,\dots $ be the degrees of the regions in any plane diagram of $G.$ Then the sum o...

Heyy! Welcome Back to my blog!  Today we are doing Graph Theory. This post is a continuation of a previous post which if you are interested in here . I somehow always get scared when someone mentions Graph theory. I have to overcome the fear. So here we go! I am writing some important theorems, terms, problems, etc that I learnt from the GT books! In short, notes. Book referred: Daniel A Marcus, Graph Theory, A Problem Oriented Approach

Heyy! Welcome Back to my blog!  Today we are doing Graph Theory. I somehow always get scared when someone mentions Graph theory. I have to overcome the fear. So here we go! I am writing some important theorems, terms, problems, etc that I learnt from the GT book.  In short, Notes. Book referred: Daniel A Marcus, Graph Theory, A Problem Oriented Approach