## Posts

### Problems I did this week [Jan8-Jan14]

Yeyy!! I am being so consistent with my posts~~ Here are a few problems I did the past week and yeah INMO going to happen soon :) All the best to everyone who is writing!  I wont be trying any new problems and will simply revise stuffs :) Some problems here are hard. Try them yourself and yeah~~Solutions (with sources) are given at the end! Problems discussed in the blog post Problem1: Let $ABC$ be a triangle whose incircle $\omega$ touches sides $BC, CA, AB$ at $D,E,F$ respectively. Let $H$ be the orthocenter of $DEF$ and let altitude $DH$ intersect $\omega$ again at $P$ and $EF$ intersect $BC$ at $L$. Let the circumcircle of $BPC$ intersect $\omega$ again at $X$. Prove that points $L,D,H,X$ are concyclic. Problem 2: Let $ABCD$ be a convex quadrangle, $P$ the intersection of lines $AB$ and $CD$, $Q$ the intersection of lines $AD$ and $BC$ and $O$ the intersection of diagonals $AC$ and $BD$. Show that if $\angle POQ= 90^\circ$ then $PO$ is the bisector of $\angle AOD$ and
Recent posts

### Problems I did this week #1[Jan1-Jan8]

Random thoughts but I think these days I am more into Rock? Like not metal rock but pop/indie rock. Those guitars, drums, vocals everything just attracts me.  The Rose, TXT, N.Flying, The western ghats, Seventeen, Enhypen and Woosung are my favourites currently.  My current fav songs are: Oki a few problems I did this week! Tuymaada 2018 Junior League/Problem 2 A circle touches the side $AB$ of the triangle $ABC$ at $A$, touches the side $BC$ at $P$ and intersects the side $AC$ at $Q$. The line symmetrical to $PQ$ with respect to $AC$ meets the line $AP$ at $X$. Prove that $PC=CX$. Proof: Note that $$\angle CPX=\angle APB=\angle AQP=\angle XQC\implies PQCX\text{ is cyclic}.$$ So $$\angle XPC=\angle AQP=\angle CXP.$$ We are done. EGMO 2020 P1 The positive integers $a_0, a_1, a_2, \ldots, a_{3030}$ satisfy$$2a_{n + 2} = a_{n + 1} + 4a_n \text{ for } n = 0, 1, 2, \ldots, 3028.$$ Prove that at least one of the numbers $a_0, a_1, a_2, \ldots, a_{3030}$ is divisible by $2^{2020}$.

### Trying to go beyond my comfort level?

Wrapping up year 2022 with a few problems I did in the past week :)  Edge colouring with $n$ colours $K_n$  Prove that if the edges of $K_n$ are coloured with $n$ colours, then some triangle has its edges of different colours.  Proof: We use induction. For $n=3$ it is true. Now, suppose it is true for $n=1,\dots, l$. We will show it is true for $n=l+1$. Now, consider $k_{l+1}$ with vertex $v_1,\dots,v_{l+1}$. Consider the $k_l$ with vertex $v_2,\dots, v_{l+1}$. Now note that the colours used in that $k_l$ are max $l-1$ colours (since by induction, if $l$ colours then we get a triangle with the property given.) But since $l+1$ colours are used, at least two of the edges $v_1v_2,v_1v_3,\dots v_{l+1}$ are coloured with $2$ colours which are not used in $k_{l}$. WLOG say $v_1v_2,v_1v_3$, then $v_1v_2v_3$ is a triangle with edges of diff colour.  Russia 2011 Grade 10 P6 Given is an acute triangle $ABC$. Its heights $BB_1$ and $CC_1$ are extended past points $B_1$ and $C_1$. On these extens

### Announcing the Unofficial TC 2022! ( Month 1 problem solving )

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