Sunaina thinks absurd

So it's been 1 month since the INMO results were out! So here's my past 1 month's journey into problem-solving.  There hasn't been much in my life, although the Sharygin correspondence round results came out! And I qualified for the final round. I just made the cutoff though. I got 85 marks and the (unofficial) cutoff was 84 marks. I am one of the seven students selected for the final round and the only kid from India in my grade! They, however, are not inviting Indian students in the final round and have asked us to conduct the final round in India if an organization agrees. Monthly reflection: Till May 16, I tried the Awesome Math Application ( which I got accepted into woohoo! And I am taking courses in the second and third season hehe) Then I was completing 108 algebra problems along with MBL problems till June 1 June 1- June 10: Since I was feeling very guilty for not doing enough problems from the Sophie psets, I did a few Sophie psets problems. BTW Thankyou so mu...

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]: P...

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 ...

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}+\...

 This week was full algebra biased!! 😄  This is my first time trying inequalities, so this pure beginners level.   Do try all the problems first!! And if you guys get any nice solutions , do post in the comments section!  These problem uses only  Power mean Inequality   and  Titu's lemma . The First few problems happen to be  not problems, but tricks(?) which are extensively used..  Here are the walkthroughs of this week's top 10 Inequality problems! 10th position:  Prove that for any real $a>0$ , $a+\frac{1}{a}\ge 2$ Walkthrough: a.  only AM-GM 9th position: Prove that for any real  $a>0$ , $\frac{a}{1+a^2}\le \frac {a}{2a}$. Walkthrough: a. Only AM-GM  b. Use AM-GM to show that   $1+a^2\ge 2a $  8th position:  Prove that for any real $x,y>0$ ,$\frac{1}{x+y}\le \frac{1}{4x}+\frac{1}{4y}$ Walkthrough: a. AM-HM inequality (cute 💖)  7th position :  Prove that for any real positiv...