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Showing posts from January, 2021

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. Problem:  A convex quadrilateral $

Newton Sums (Sym Poly Part 2) Week#7

Well..  I was asked to prove Newton Sums, so since I latexed the proof, why not post it :P. Newton Sums: Consider a polynomial $P(x)$ of degree $n$, with roots $x_1,x_2,\dots,x_n$ $$P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0.$$  Define $p_d=x_1^d+\dots+x_n^d.$ Then we have,   $$a_np_1 + a_{n-1} = 0$$  $$a_np_2 + a_{n-1}p_1 + 2a_{n-2}=0$$ $$a_np_3 + a_{n-1}p_2 + a_{n-2}p_1 + 3a_{n-3}=0$$ $$\vdots$$ (Define $a_j = 0$ for $j<0$.) Proof : Note that $$p_1=x_1+x_2+\dots+x_n=\frac{-a_{n-1}}{a_n}\implies \boxed{a_n\cdot p_1+a_{n-1}=0}$$ Note that $$p_2=x_1^2+\dots+x_n^2=(x_1+\dots+x_n)^2-2(x_1\cdot x_2+x_1\cdot x_3+\dots+x_{n-1}\cdot x_{n})=$$ $$p_1 \cdot \frac{-a_{n-1}}{a_n}-2\cdot \frac{a_{n-2}}{a_n}\implies \boxed{p_2\cdot a_n+p_1\cdot a_{n-1}+2\cdot a_{n-2}=0}$$ Note that $$p_3=x_1^3+\dots+x_n^3=(x_1^2+\dots+x_n^2)(x_1+\dots+x_n)-(x_1\cdot x_2+x_1\cdot x_3+\dots+$$ $$ x_{n-1}\cdot x_{n})(x_1+x_2+\dots+x_n)+3(x_1\cdot x_2\cdot x_3+\dots+x_{n-2}\cdot x_{n-1}\cdot x_n)=$$ $$ p_

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

Week #5 ( a bit different?)

Well.. IOQM is too close, so I haven't been doing any Olympiad type problems lately :( . And continuously attempting mocks. I encountered a lot of nice problems, easy but cute , a bit hard too.. but then typing them was really not feasible for me right now! But then blog without regular updates is bad. Anyways I completed Alexander Remorov's projective geo Part 1 handout last week with JIB's, Rohan Bhaiya ,Sanjana's  and Serena's help. :P . And I did 2004 G8 ( it's in egmo too ). But still I found it cute.  So here fully motivating harmonic G8. I won't say it's easy, but yes killed by harmonic.  Problem(2004 G/8):  Given a cyclic quadrilateral $ABCD$, let $M$ be the midpoint of the side $CD$, and let $N$ be a point on the circumcircle of triangle $ABM$. Assume that the point $N$ is different from the point $M$ and satisfies $\frac{AN}{BN}=\frac{AM}{BM}$. Prove that the points $E$, $F$, $N$ are collinear, where $E=AC\cap BD$ and $F=BC\cap DA$. Here's

Summary of 2020

New year post!! Happy New year everyone!! ( Got inspired by  SnowPanda 's  and  anser 's blog post). Okie so first non math post :P. My grammar is really bad. I feel like 2020 by far has been the most productive year of mine and very unique too ( Just imagine not going to school , Non-dummy school students be like"yeaaahhhhh" ). The most important thing that happened was my INMO 2020. The most embarrassing yet if this incident wouldn't happen, then I wouldn't realise what hardwork and starving is ( okie sorry too philosophical ). Anyways here are some of the best experiences: a. Pr0 people aren't aliens ( exceptions: RG and JIB ) : So I joined OTIS this year, and this was such a nice decision. I seriously learnt a lot.  And before joining OTIS, I thought every person is so pr0 and everybody talks just math and no random thing. And my stereotypical image of Evan was like a strict guy, who only does math and math and math all day and talks only about math 

Top 10 Problems week#4 (it's late)

Okie!! Fine, I am late by 2 weeks but I was busy in OTIS submissions. And yayy!! I learnt how to use Evan.sty .  These problems are the exercises  from a Titu handout in  this website .  Here's the full  magazine , go to page 40's and one can find it :) So full Titu :P 10th position ( IMO shortlist, 1996) :  Suppose that $a, b, c > 0$ such that $abc = 1$. Prove that $$\frac{ab}{ab + a^5 + b^5} + \frac{bc}{bc + b^5 + c^5} + \frac{ca}{ca + c^5 + a^5} \leq 1. $$ Walkthrough:  Thanku  Rohan Bhaiya 😄   a.  $$\sum_{cyc} \frac{ab}{a^5+b^5+ab}\le \sum_{cyc}\frac{c}{a+b+c}=1.$$ b. Cross Multiplying, it is enough to show that $$a^2b+ab^2+abc\le a^5c+b^5c+abc . $$ c. Multiply $abc=1$ to each side and use Muirhead. 9th position (RMO, 2006): If $ a,b,c$ are three positive real numbers, prove that $$ \frac {a^{2}+1}{b+c}+\frac {b^{2}+1}{c+a}+\frac {c^{2}+1}{a+b}\ge 3$$ Walkthrough: a. Using Titu, get $$\frac {a^{2}+1}{b+c}+\frac {b^{2}+1}{c+a}+\frac {c^{2}+1}{a+b}\ge \frac{(a+b+c)^2 +