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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

### 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