Skip to main content


Showing posts from August, 2021

Geos are definitely cool

So, I  did G1s. Here are solutions to a few of them. I think they were really INMO level. Not much theory was needed. It's true that almost everything was angle chase-able. Although $2017$ G$1$ was an exception, as it had pappus. But the rest were nice. The hardest would also be $2017$ G$1$. It's not hard-hard, but hard in terms of theoretically. The second hardest would probably be $2018$ G$1.$  Anand had proof with no words. I will add the image here. One should definitely check!  Then, the third hardest would probably be $2009$ G$1.$ The right construction was hard for me to guess. Then $2005$ G1. And then $2007$G$1,$ since the right use of similarities were required.   Also I, fortunately(?) didn't use any ggb , so I would recommend making your own diagrams.  And before starting, I would like to thanks every one of you who comes here almost daily and views this blog! I hope this blog is helpful to you in some way. Yeah, we crossed 6k views! 💖 So here are 5+1 problems,

Non geos are cool part 2 and Mailchimp!

Welcome back! I have now a Mailchimp email subscriber widget! That is, people can now subscribe to this blog again and get email updates when I post a blog post!  To subscribe, just go to the right side of the blog (it should be visible, if not click those three bars thing) A few things to note is that If you have already subscribed to this blog through Feedburner, then there is no need to subscribe again, until you want two email notifs about my blog. Also, the "following" option doesn't give you updates, it's more like supporting the blogger and inform the reader about how many people love the blog! One can always unsubscribe to this blog. So do subscribe, it's free.  Well, I practically don't get anything from this blog :P. But it's fun sharing about your day to day life and seeing a lot of people coming to this blog! And I am back with 5 more cute Non-geos!   So here are some 5 Non-geos that were cool! 😎 I think I should say them as NTs. Problem:  Det

Non geos are cool!

  Hi! I am posting 5 quick, cute and simple non-geo questions! I have posted their solutions too. The level is probably a good  INMO level. Do try them! I actually gsolved 4 of them with PC ! All of them are from AOPS's Old highschool Olympiad forum. Problem1: In the following triangular table $$0,~1,~~2\dots 1958$$ $$1,~~3,~~5\dots 3915$$ $$\dots \dots $$ each number (except for those in the upper row) is equal to the sum of the two nearest numbers in the row above. Prove that the lowest number is divisible by $1958$ Solution:  I actually messed up the calculation a bit (posted at MSE), so thanks to everyone who helped me :) Well, $1958$ surely isn't special. So let's prove for any $$a_1,a_2,a_3,\dots a_n$$ triangular table. So we have $$a_1~~a_2~~a_3~~a_4\dots \dots~~a_n $$ $$ \downarrow$$ $$(a_1+a_2)~~(a_2+a_3)~~(a_3+a_4)~~(a_4+a_5)\dots \dots ~~(a_{n-1}+a_n) $$ $$ \downarrow$$ $$(a_1+2a_2+a_3)~~(a_2+2a_3+a_4)\dots\dots ~~( a_{n-2}+2a_{n-1}+a_n)$$ $$ \downarrow$$ $$(a

Problems done in July

 July was LIOG month. Just LIOG was done, and then we had MMC ( till July 14). SO from the last two weeks, I was doing LIOG only. And I did some non-geo too! Also, I am not adding all the problems I did. A few only in fact. One can go through my apps post ( Jelena_ivanchic ) Here are a few problems and solutions/walkthroughs: Problem:   Let $A_1$ be the intersection of tangent at $A$ to the circumcircle of a triangle $\Delta ABC$ with sideline $BC$. Similarly define $B_1,C_1$. Show, that $A_1,B_1$ and $C_1$ are collinear Walkthrough:  Use this $\Delta ACA_1\sim \Delta BAA_1.$ So we get$$\frac{AC}{BA}=\frac{CA_1}{AA_1}=\frac{AA_1}{BA_1}$$$$ \implies \frac{CA_1}{AA_1}\cdot \frac{AA_1}{BA_1} =\frac{AC^2}{BA^2}$$$$\implies  \frac{AC^2}{BA^2}=\frac{CA_1}{BA_1}. $$Now, apply menelaus on $ABC.$ Problem:  Let $\Gamma$ be a circle and let $B$ be a point on a line that is tangent to $\Gamma$ at the point $A$. The line segment $AB$ is rotated about the center of the circle through some angle to