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"Whatever happens, life goes on."

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beautifullll kid getting beautifulll food

Surprisingly, I am missing something serious from my toolkit. Quite an elegant result in fact, and as a result wasted 15 hours not solving 2012 ISL G8. So time to get immortalized via blog.

I was vaguely aware that power of point had some shenanigans from linpop, but had no idea there was a circle analog that used ratios instead of differences. It doesn't even make much sense initially apart from the fact that correspond to the case, and I'm pretty sure power of a point works terribly with inversion and homography so I'm shocked this works so elegantly with a projective condition. Anyways, there are a few ways to prove it, the simplest is just taking an isometry mapping the centers to the -axis, and then just observe that the resulting conic is of the form , so it must be a circle as it has at least two points on it.

I'm also honestly surprised by how hard it is to approach the G8 projectively. I tried a lot of stuff but it seems at the end of the day you have to just length chase and just use the classic ratio lemma / LoS + ceva/menelaus approach and somehow reframe it as a concurrence instead of collinearity. Brutal. There are a few projective solutions that require finding incredible observations or using Desargue's involution theorem, both of which I wasn't really in the mood to look for.

Actually, apparently there is a homography one liner on the thread, but not sure if it is correct. The polar condition looks like it might get distorted by homography.

And what a journey it's been. The 2023 IMO team selection cycle has finally ended, and results are now known. For those following the blog, the verdict is that I will not be going to IMO this year. I won't reveal names, but all of the people who did manage to qualify are competitors I greatly respect, who I know will earn the US a solid placement later this year. It's been a journey of many ups and downs, highs and lows, with dinner celebrations after success and comforting hugs after disappointing results. As with all things, journeys must come to an end, and it is time to begin thinking about next year's cycle once more.

Though the result is not the one I dreamed of, it still feels like a large weight has been lifted off my back. One of the toughest parts of the cycle was the five weeks I spent waiting after the end of USAMO, anxious about the inevitable announcement of results. It's obvious that you shouldn't worry about things outside your control, but when something truly means a lot to you, it can be hard to listen to your own words. I'm quite lucky, unlike others who have come before me, that I actually have another year, and another chance at IMO before I graduate. The next plane takes off in just two months, so there isn't even much of a wait. I will, of course, miss all the people I've gone on this year's journey with, but we'll meet soon in college (perhaps) anyways, so we can wait and see what happens.

Before leaving for the next ride though, I want to talk a quick bit about my experience this year. I've wanted to put updates on my blog about the TSTs this entire year, but I'm not the biggest fan of celebrating or despairing before the very end of everything: don't count your eggs before they hatch, as I've been told, so I kept everything in my memory banks until this moment so I could share everything at once. So, let's start from the very very beginning.

2022 MOP was the greatest summer camp experience I had ever experienced, even better than the first time I went to AMSP back in 6th grade. Everything about the experience was exciting and every morning I could look forward to an incredible day ahead, with fun and interesting classes and tractor to play at night. It was during 2022 MOP that my first in-person TSTSTs happened, a set of three tests and total nine questions, used to choose who would go on to compete for the coveted IMO spots as members of the TST group. The TSTSTs were mostly uneventful, though one highlight was the infamous TSTST 1, which nearly everyone fakesolved. I spent four hours on TSTST 1 and was super sad since I hoped to solve at least two problems, but when I came out, it turned out that everyone else had fake solved it and that the four hours were, funnily enough, very well spent. I produced one of the only constructive solutions, despite it being super ugly, and even the grader had beef with me for how annoying it was for them to grade. The other two TSTSTs were standard but unsurprising, and I ended up getting a distribution of 700770770: no partials to be seen, just a very solid distribution. Funnily enough this basic distribution with no p3s and no partials ended up earning me the top 4 spot on TSTST, likely due to the brutality of p1 and other random factors. The bad news is of course that TSTST score does not affect TST score, so any excessive success here wouldn't carry over in any way. Still, it was a good sign.

Next, came the winter TSTs. There were five months in between TSTST and TST, and I grinded the 2015, 2014, and 2013 ISL shortlists in preparation. On the day of the December TST, I felt like I was on a roll: nothing could possibly stop me, and I even exercised the morning of the competition and skipped all my classes for sleep. Sadly, I was met with my worst performance of the entire year that day: 700, missing a very basic, perhaps G4 level geometry on p2. The cruelest part was that I solved p2 just five minutes after the end of the test, feeling like a fool. Xax, my fellow Exonian taking all of these tests alongside me, also missed p2, and as the info came in, our prospects only grew dimmer. We were in the bottom third of the TST group. Did our journey really end the moment we left the landing ground? Was it all going to end just like that? Xax and I could only hug each other, praying that no one would sweep the day and push us too far down as a result.

Next was January TST, a test much like any other TST, with a nearly impossible p3 and tough p2. I had almost given up hope for IMO team before I was informed that I had gotten a 771. On its own, it was not an exceptional score, but it turned out that it was in fact the top score on the TST, thanks to the fact that almost everyone dropped partials on p2. Suddenly, I was pulled up from the bottom of the ladder to the top twelve scores. Although twelve was a long way away from any chance at IMO, and there were people who had swept day 1 and nearly swept day 2 who I had no chance competing against, a sparkle of hope was dangled in front of my eyes, and I was not going to let it go. That night, I thought a long time about my chances back at my dorm. December TST had been simply disastrous: it was a performance that would be difficult to recover from, and would likely have serious impacts on my IMO chance down the road. However, remembering TSTST, I knew I could put up a strong fight still. I may simply be unable to get IMO team no matter how hard I tried from here: but I was not about to give up. I was going to make it a hell of a fight for everyone else. For a brief moment, I ignored all the numbers and statistics and what not: if I were one of my competitors, I would be scared that I might make a huge comeback, given my performance on TSTST, and that alone was enough to push me to want more.

Then came RMM. To be honest, I did not except to make RMM. With a grim performance on December TST, I was a competitor with a big million-dollar dream but only a few hundred dollar bill in my pockets. From twelfth place, my TSTST score pushed me up the RMM list, and since selection was based both off of known TST and TSTST scores, somehow I barely got onto the list. I wasn't going to argue against it: I mean, it's a whole week off from school, and a lot of fun of course! In Romania, we ate a bunch of local cuisine, went to a vampire's old castle, met teams and famous AoPsers from around the world, and of course, we experienced our very first international math competition. Day 1 was quite a funny one: problems 1 and 2 were quite easy, but p3 was perhaps the hardest and most beautiful problem of the entire journey. No one solved it: in fact, not one even obtained the right answer. As a result, the jury made partial points very easy to obtain on the question, and sadly my solution was not worth even a single point. Many competitors earned up to the three. Later, it turned out that may day 1 was a distribution of 771: surprising, considering my claims in p3 were mostly misguided. But hey, if they're going to give free partial, I might as well take it. The top score was 773, just two points higher. I was just happy I got more than 770. Day 2 of RMM does not matter for the TST cycle, but I ended up with a rather disappointing 700, due to missing the key idea of p5 and not moving onto p6. However, it turned out many people struggled with p5 due to its unexpected difficulty, and actually several members of the US team skipped to p6 immediately and were actually able to earn points there. It was a little frustrating that the order of problems on both days had been screwed up, but the problems themselves were still high quality. The US team ended up winning RMM, and I came home with a silver medal: not great, but not terrible either: at least we didn't lose much score in this round.

Returning from RMM, there was little to no break: we were going to have APMO and USAMO back to back: essentially half of all the TST selection points in a span of three weeks. The variance was going to be huge. Since APMO happened during Exeter's spring break, I had to take it at a local professor's classroom. It was a different location, but I knew I had to make up the lost points on the December TST here if I wanted to even have a remote chance at IMO by the time USAMO rolled around. So in the weeks leading up to APMO, I grinded the short time limit of APMO: I mocked five past APMOs, getting 35, 35, 14, 35, 35 each. I knew I was capable of earning a full score (35) and sweeping the test, but it was clear that there was a serious element of luck involved. On test day, I quickly swept through the first four problems within ninety minutes: then came p5, which was surprisingly hard for an APMO problem: after a long think, I managed to find the key idea in the final hours and barely finished my writeup. With that, all I could do is pray and hope that the scan went through correctly. Later (after USAMO), it turned out I had earned the miracle: a 35 full score, scaled down to 21 in context of the TST selection, one of only two perfect scores from within the TST group. It was exactly the push I needed to have a chance at IMO, and even I myself was shocked. For the first time since the beginning of the TST cycle, I entered into the top six placements thanks to the incredible performance: all the remained was USAMO.

Sitting down for USAMO, Xax was surprisingly talkative before the test, completely different from his usual solemn and focused demeanor, with headphones on, three protractors, five compasses, and seven bottles of water. I think both of us realized that we were reaching the end. Every win or loss at this point would be like life and death. That's why when we finished the test, both of us holding a perfect score, you could not have captured the happy look on Xax's face. However, as info came in, the elation quickly died away: nearly everyone near the IMO line had also gotten a perfect score, sweeping day 1. No gains and no loss had been made by anyone: the test was simply far too easy to make a sizable impact on the crowd. And so, all our hopes and expectations for day 1 moved to day 2 instead. Only two questions remained:

Could we sweep USAMO? And more importantly, could we even afford not to at this point?

USAMO day 2 was a mixed bag. A quick and slick p4 and p5, but p6 was surprisingly challenging. Involving two constructed reflections and an inversion, together with a linearity of power of a point argument, it required a lot of careful steps to reach the end. Unfortunately, I did not make the observations necessary for unlocking the problem during the test, and took an L. Initially we were not even sure if it was an L, but it turns out that nearly every one of the people near the IMO line had managed to sweep the USAMO: that is, solve every problem perfectly on the test. Missing p6 was not just missing a sweep, it meant losing a critical seven points in a very very tight field. And just like that, at the very end, I was 7th on the list, a breath away from the cutoff.

And just like that, all of the tests from this year's cycle are wiped away into history. Truth be told, it's an odd feeling, almost like landing a plane: you're relieved to hit the ground again, but soon longing once again to fly and achieve more and begin the redemption arc. The last day of 2022 MOP feels like it were only a month ago in my mind, and yet already the summer of 2023 is quickly approaching. I look back for regrets, but I don't really have any. I guess I could work a little more on finding the confidence to use the geometric method of cases while in-contest, but that's more psychological than practical. I guess I could grind some CTST 3s, but it will require a long-term commitment. Hopefully I've grown and learned enough this year to give me a better chance in next year's cycle, but you can never say with these kinds of things. The funny thing is, before TSTSTs this year, I did not even dream of getting 2023 IMO team: I was hoping to do somewhat ok this year, and when all the seniors would leave for college, I would fight for IMO in 2024. With my kind of a greedy mindset, achievements like this are always bittersweet: to arrive on the very edge of 2023 IMO, only to not cross over it. Who wouldn't feel a bit of pain there? Would you call someone who survived a lightning strike lucky for surviving, or do you call them unlucky for even being hit in the first place?

With the landing of the plane, it's also time to beginning to think of my future outside of math contests. Colleges await me after this summer, and I can no longer keep certain questions unanswered. Where to go? What to do? Likely as that time rolls around, I will also become less and less active on AoPs (as I already am). In the meantime, I'm returning to ordinary life, catching up on schoolwork, and all that. US history at Exeter is undisputedly the hardest class in the school, and I still have to survive it. I'm going to RSI this summer, so maybe I will end up doing some kind of research, or perhaps I might get pulled into some other field like ML or physics, who knows. All left to the winds.

I don't know

So have this instead: 8/5p1p/4p1k1/4P3/2Rp2P1/2b2K1P/8/8 w - - 2 4

White to play, defeat stockfish, and prove you are the best endgame player.

One of my favorite random facts is that and which are reverses of each other. This doesn't generalize as and aren't reverses of each other, though they do look alike. Some other cases satisfy this too like and but is already reverse of so this doesn't count, it is a trivial case.

Anyone know the next few nontrivial reverse square pairs? I'm curious to see what is possible, since somehow I've never seen any other nontrivial reverse pairs except this one. If someone could code something to search these it would be cool.

Rigorously: Find natural such that and are reverses (no leading 0s) of one another and and are not reverses of one another.

Just speedran the entire 2016 USA December TST in minutes, seconds.

Noticed the inductive reclaim to in 10 seconds, proposed and solved the case where is a single swap in 3 minutes, solved problem in 4 minutes, 28 seconds flat. Half a page used, no more, no less.

We're on good pace here. Looking for properties now. 14 minutes, tested inversions through all of , swapping incircle with excircle or excircle with mixtilinear incircle, no progress. Need to switch tracks, recently I noticed by biggest shortcoming is when I get stuck on a one-way train of thought.

I look for angles: find collinear where is the reflection of over the perpendicular bisector of , this is huge, is solved, now we just need to solve .

25 minutes. are ugly to work with, they have no properties. What if... I reverse my thinking. What if we instead give the conditions collinear and , which are quite nice to work with the inversion in , and instead from is on the -excircle, which has no properties? The inversion looks very promising.

29 minutes, 34 seconds. Finished, is the reflection of the -extouch point of the angle bisector, key is that the -excircle is fixed under the reflection. Next! Another page used.

OMG, OMG, I've never been on this pace before, we have 4 hours to solve P3! We have got this! Ok but calm down, don't waste this opportunity. Need to be serious, don't throw the pace. Let's read quick.

Oh god, seems hard and hard to understand. Why should this transform have any nice properties at all? Let's try linear roots, huh, even they don't make any sense... I don't get it, don't get it... Ok wait, let's reset. Instead of experimentation, let's look for elegance. First, let's look at the constant case. Easy. Now linear case? It still doesn't make sense. We might be stuck. Ok, let's come at another angle. , which is not , so a direct congruence doesn't work either. Dang! For now, let's assume always.

40 minutes. Hm, just went to the bathroom. What if we establish some kind of substitution congruence! If we substitute out for in polynomials of that form , we can cancel the constant term, which is exactly kind of what we do in the simple gcd Euclidean algorithm, we cancel out the constant term. Maybe there is something here. Ok, trying, but we are halting because how do we actually know we can divide out factors of ? If we rewrite the Euclidean algorithm to cancel out the constant term instead of the largest term, we would need some way to decrement the degree.

45 minutes. Wait, we have a strong claim, and it might just be strong enough to be equivalent to the problem, warranting complete dedication to proving it! It's that we can delete extra factors of when has no factor of . Hm. Well , that's obvious. So it comes down to proving . How do you even start? Well there must be something unique about the factors of polynomials which can be written in the form. Well by subtitution we must have that . Wait, that's obvious, it's by freshman's dream! Hm. Well, actually we have... wait, we can make a stronger claim! That all such polynomials, by removing the , must have only single roots! This is incredibly strong, but can we actually prove it?

47 minutes, 24 seconds. Yes, we can! for roots if ! This should imply no double roots, doesn't this just finish? Let's check quick! Only one page used on this problem too!

GG! New PB!

This technique can be evaluated in two ways depending on the person:

Case 1: You are good at algebra, specifically trig problems. Then you don't need this technique, it will only slow you down and distract you.
Case 2: You are not good at trig. Then this technique might solve some simple trig, but you will never learn to develop your own skills, so it will harm you long term.

Therefore this technique is ALWAYS NEGATIVE EV, and will harm your math skills. Still interested?

Example: Find . Plugging into the formula, we get this is , finishing.

Note: The maximum error in the approximation is 0.0016, or around . That's right. On desmos, you cannot see the difference between the graphs because this error is not humanly perceptible.

Probably 2017 ISL C8.

After reading the solution, I too wanted to become a butterfly and fly away.