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Showing posts from December, 2021

Let's do problems only

Welcome back to this blog!  My blog completed it's 1-year :D i.e 29 Nov. I am so happy that this blog grew so much and it didn't die! It also crossed 10k views! Thanks a lot! Enjoy the problems. It's more of a miscellaneous problem set with the level being INMO or less. So here are $20$ INMO level problems. Problems:  Problem 1: [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).$ Problem 2[ USEMO 2021 P4]:  Let $ABC$ be a triangle with circumcircle $\omega$, and let $X$ be the reflection of $A$ in $B$. Line $CX$ meets $\omega$ again at $D$. Lines $BD$ and $AC$ meet at $E$, and lines $AD$ and $BC$ meet at $F$. Let $M$ and $N$ denote the midpoints of $AB$ and $ AC$. Can line $EF$ share a point with the circumcircle of triangle $AMN?$ Problem 3 [ 2006 n5]: Prove th

Graph Theory: Planar Graphs

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 of $$r_1+r_2+r_3+\dot

Recurrence relations + Christmas Blog

  Hey everyone! Welcome back to my blog! Today's topic is Recurrence relations!  The handouts/ books I have referred to are GRAMOLY's recurrence in combinatorics,  Recursion in the AIME  , Combinatorics: A problem-oriented approach and principles and techniques in Combinatorics. Recursion is as cute as a rabbit ( not really) Here are my notes! Recurrence Relation: A recurrence relation is a formula or rule by which each term of a sequence can be determined using one or more of the earlier terms. Problem 1: For each of the following sequences find a recurrence pattern. a. $1,10,100,\dots $ b. $1,3,6,10,\dots $ c. $1,2,6,24,120,\dots $ d. $1,1,2,3,5,8,13,\dots$ e. $0,1,9,44, 265, \dots $ Solution:  a. $$a_n=10\cdot a_{n-1}$$ b. $$a_n=a_{n-1}+n$$ c. $$a_n=n\cdot a_{n-1}$$ d. This is the Fibonacci sequence . We have $$F_0=1,F_1=1, F_2=2,\dots F_n=F_{n-1}+F_{n-2}$$ e. This is the derangement number . We have $$ D_n= n \cdot D_{n-1}+(-1)^n.$$ We also have $$D_n=(n-1)(D_{n-1}+D_{n-2})

Global Probability: Expected Value

  Hey, welcome back! This blog post is a continuation of my previous blog post which you can read over  here  . These are just notes and problems.  The handouts/ books I referred to are Evan Chen's  Probability handout  , AOPS introduction to Counting and probability, Calt's  Expected value handout , brilliant and this  IIT Delhi handout. I am reading expected value because it's a prerequisite to the Otis combo unit Global. Hence the name Global Probability." Probability is Global Expected Value:    The expected value is the sum of the probability of each individual event multiplied by the number of times the event happens. It is denoted as $\Bbb E$ $[x]$ We have $$\Bbb E[x]=\sum x_n P(x_n)$$ where $x_n$ is the value of the outcome and $P(x_n)$ is the probability that $x_n$ occurs. Problem 1: What is the expected value of the number that shows up when you roll a fair $6$ sided dice? Solution: Since it's a fair dice, we get each outcome to have equal probability

Probability is Global + Are (a,b) and [a,b] the same length?

 Hey everyone! Welcome back to my blog!  The number of diagonals are $\binom {12}{2}-30=36$ Today we have another topic which I am very scared of, PROBABILITY! I think to overcome this fear, I should start from basics. So, I refer AOPS's introduction to counting and probability. I did till chapter seven and thought to do the rest problems in this blog as notes. The book's pirated copy is available in zlib btw ( I don't think so I should have said this, but I think it's fine). I did the harder problems and wrote the solutions in Xournal. If you want to see them, here is the  pdf . Also both the pdf and following post has amc/ioqm type problems! So enjoy!! Problem 1: $n$ coins are flipped simultaneously flipped. The probability that at most one of them shows tails is $\frac{3}{16}.$ Find $n.$ Solution: We have $$\frac{n+1}{2^n}=\frac{3}{16}\implies 16(n+1)=3\cdot 2^n \implies n\ge 4. $$  But the RHS grows very fast, so $n=5$ is the only solution. Intersection of i

Geos are my life support! ft life update

Just a compilation of $10$ very nice and hard Geo problems and solutions :P. Without diagrams ( cause I am a lazy person). Problem 1[ China TST]:  Let $E$ and $F$ be the intersections of opposite sides of a convex quadrilateral $ABCD$. The two diagonals meet at $P$. Let $O$ be the foot of the perpendicular from $P$ to $EF$. Show that $\angle BOC=\angle AOD$. Proof:  Define $S=AC\cap EF,~~T=FE\cap BD.$ Note that $$(E,R;D,C)=(S,P;A,C)=-1 .$$ Since $\angle POS=90,$ by Right Angles and Bisectors harmonic lemma, we get $OP$ bisecting $\angle COA.$  Similarly , we get $$((B,D;P,T)=-1. $$ Since $\angle POT=90,$ by Right Angles and Bisectors harmonic lemma, we get $OP$ bisecting $\angle BOD.$  Problem 2[ISL 2002]:  The incircle $ \Omega$ of the acute-angled triangle $ ABC$ is tangent to its side $ BC$ at a point $ K$. Let $ AD$ be an altitude of triangle $ ABC$, and let $ M$ be the midpoint of the segment $ AD$. If $ N$ is the common point of the circle $ \Omega$ and the line $ KM$ (distinct f