IMO 2023 problem discussion

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Complete_quadrilateral

144 posts

Complete_quadrilateral

#1 h Jul 8, 2023, 1:48 PM • 2 Y

Y by Rounak_iitr, oolite

Get your frustrations out here.

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Complete_quadrilateral

144 posts

Complete_quadrilateral

#2 h Jul 8, 2023, 1:53 PM • 2 Y

Y by TheHimMan, Rounak_iitr

Redacted

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799786

1052 posts

799786

#3 h Jul 8, 2023, 2:24 PM • 1 Y

Y by steppewolf

As simple as MOHS scale:
P1: 0M
P2: 10-15M
P3: 25-30M

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Complete_quadrilateral

144 posts

Complete_quadrilateral

#4 h Jul 8, 2023, 2:27 PM • 1 Y

Y by Rounak_iitr

Redacted

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navi_09220114

479 posts

navi_09220114

#5 h Jul 8, 2023, 2:31 PM • 2 Y

Y by lambda5, Rounak_iitr

I get that the notion with "accessibility": Recently there are many new countries participating, and they want to have the easy problems being really accessible to them as well.

1) On one hand I support having easy problems being accesible, but I certainly think that doing that while having HM being at 7 points, really devalues its meaning. [something like 2016 p1/p4 would be really quite hard for an easy problem]

2) I will reapeat my thought from the other thread, of having 4 problems in 5 hours - basically having an easy p1, "hard p1/easy p2", "hard p2/easy p3" (like mine), and a hard p3 [harder than 2017 p3 could be good] that is reserved for the special prizes (?) and making perfect scorers truly outstanding [and of course HM not be 7 anymore]

Its also not too taxing when there are contests like IMTOT A-level, which is also 5 hours long and picking the 3 best questions, or some contests with 4 problems as well.

Bottom line is we don't want the IMO to be a "standard contest", and to have more ingenuity instead. Even the easy problems can be fun too without being an immediate routine exercise.

I also secretly hope that the IMO do get harder every year - it has been so since the very first IMO till somewhere in the 2010s, why can't it be harder now? Students also improve greatly over the years too, its time for the problems to catch up in difficulty as well before cutoffs like 23/29/34 becomes the norm.

Refering to hard problems, I thought of something like IMO 2017 P3 as an excellent example of what I was hoping for, just that I also hoped the students today can solve it much better than it was back then..

(I recall getting excited for more difficult problems in the next few IMOs after 2017, going even further than that, but alas... The closest in difficulty and beauty was probably 2020 P6)

Yeah, just a rant, and hoping to see some radical changes in the IMO for years to come. Eg: 2021 p2 is "great", but only if more students would embrace some analysis/differentiation as standard IMO context too, a direction that I do hope the IMO will go into. So as a little topology/geometry/analysis related problems, modulo the technicalities and sieve out the essential ideas instead. But I am not going to bother changing how the game works - I will merely enjoy creating more problems as my own hobby :>

[Also PS again about p3 being too easy: I sincerely think that my problem should be a p2/5 instead (in line with what I imagined how hard a future IMO should have been), but I am not the one who decided the difficulty in the shortlist, and the paper itself :p]

This post has been edited 11 times. Last edited by navi_09220114, Jul 8, 2023, 5:31 PM

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Complete_quadrilateral

144 posts

Complete_quadrilateral

#6 h Jul 8, 2023, 2:42 PM • 3 Y

Y by Miquel-point, RobertRogo, lambda5

Redacted

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khan.academy

634 posts

khan.academy

#7 h Jul 8, 2023, 2:59 PM

Y by

redacted..............

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kimyager

8 posts

kimyager

#8 h Jul 8, 2023, 3:31 PM

Y by

khan.academy wrote:

~~I totally agree, the IMO should only be done between the top 20 countries so getting a medal is hard.~~

CantonMathGuy wrote:

not a good opinion tbh

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c7h5n3o6_tnt

117 posts

c7h5n3o6_tnt

#9 h Jul 8, 2023, 4:13 PM

Y by

Actually many regional MO/TSTs are (much?) harder than IMO.
But this time's Day1 is really nonsence in my opinion

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carefully

240 posts

carefully

#10 h Jul 8, 2023, 4:25 PM • 7 Y

Y by JingheZhang, navi_09220114, khan.academy, Miquel-point, Schur-Schwartz, Rounak_iitr, ericxyzhu

For me, I think good IMO problems should be all medium, with almost the same difficulty (like 15M / 25M / 35M), so that students can freely choose to solve whichever problem in a subject they're good at.

In such IMOs, gold cut-offs are high and bronze cut-offs are low. My favorite IMO is IMO 2005 (Gold 35, Bronze 12). Looking at the result, you can find people who solved almost any combination of problems (e.g. solving P3,6 but not P1,2,4,5).

However, many recent IMOs went the opposite way; the easy problems got easier, and the hard problems got harder (like 5M / 25M / 50M). Gold cut-offs were low and bronze cut-offs were high.

That's very bad for the competition because medals were basically determined by only two middle problems (and sometimes even one problem) instead of six problems. Luck also plays a larger part, depending on the subject of the middle problems, and whether you can solve this particular problem.

** See IMO 2017 for the extreme case, where gold or silver or bronze or nothing was basically decided on how many partials you got from P2 and P5. **

This post has been edited 2 times. Last edited by carefully, Jul 9, 2023, 10:29 AM

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Aiden-1089

285 posts

Aiden-1089

#11 h Jul 8, 2023, 4:59 PM • 2 Y

Y by MS_Kekas, Sumgato

As someone who has undergone literally no training for IMO, have knowledge of only basic, introductory number theory, and basically only starting my journey into these olympiads, I solved P1 in less than 10 minutes. I don't think I have solved any number theory problem on the IMO shortlist by myself, nor even on this very site. Considering that a layman such as me found this problem quite easy, I'm not sure how this problem made it into the shortlist, let alone the IMO.

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YIFANK

25 posts

YIFANK

#12 h Jul 8, 2023, 5:22 PM

Y by

Complete_quadrilateral wrote:

I personally think that P1 was way too trivial and just doesn't make sense having it on IMO. I believe that everyone who is able to get at least silver should instasolve all such problems (I instasolved it, and I didn't even qualify). I would say that it is maybe even better not having P1 at all if it is going to be this easy, it's just wasting most people's time and makes IMO more of a write-up so you don't lose a point competition than a math one. Or maybe I'm just salty cus I lost too many such points on TST.

Yea I instasolve it but I didn't even get a medal on USAMO (got 22 this year)

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steppewolf

351 posts

steppewolf

#13 h Jul 8, 2023, 5:26 PM • 3 Y

Y by khan.academy, Corella, Sumgato

The main issue that I have with such an easy P1 is the following: the problem should be accessible to untrained contestants, but this does not mean that the problem should be trivial. By giving a trivial problem at the IMO, you basically devalue the meaning of an honourable mention.

It is a much better practice to give a difficult problem that is based on getting the right ingenious idea instead of giving an almost free honourable mention.

Another problem that is caused by giving an overall easy IMO is that it also potentially makes it harder to distinguish between the top contestants and teams. If the gold medal requires almost a perfect score, this also decreases the value of a perfect score. An extreme example of this is when an entire team scores 42 points like last year.

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oolite

344 posts

oolite

#14 h Jul 8, 2023, 5:32 PM • 1 Y

Y by steppewolf

One property you'd hope for in a good IMO is that there are relatively few contestants on the borderline between medals.

For each of the past several years, here are the percentage number of contestants having the highest silver/bronze/no-medal score (unlucky) or the lowest gold/silver/bronze score (lucky).

edited to include IMO 2023 stats

2023 19.3% **************************************
2022 22.4% ********************************************
2021  9.2% ******************
2020 17.5% ***********************************
2019 13.2% **************************
2018 15.0% *****************************
2017 28.6% *********************************************************
2016 18.8% *************************************
2015 18.9% *************************************
2014 28.4% ********************************************************
2013 16.9% *********************************
2012 20.8% *****************************************
2011 23.1% **********************************************
2010 30.8% *************************************************************
2009 13.6% ***************************
2008 17.2% **********************************
2007 16.9% *********************************
2006 30.7% *************************************************************
2005  9.0% *****************
2004 16.7% *********************************
2003 16.9% *********************************
2002 18.0% ***********************************
2001 12.1% ************************
2000 15.4% ******************************

By that measure, last year was pretty bad (cf. @steppewolf's comment) and 2005 was exemplary (cf. post #10).

This post has been edited 1 time. Last edited by oolite, Aug 25, 2023, 5:18 PM
Reason: updated stats to include IMO 2023

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S.Das93

709 posts

S.Das93

#15 h Jul 8, 2023, 6:30 PM

Y by

I believe the decrease in difficulty level is the influx of LDCs participating in the IMO.

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Ankoganit

3070 posts

Ankoganit

#101 h Jul 11, 2023, 7:05 PM • 29 Y

Y by PRMOisTheHardestExam, dangerousliri, khina, KST2003, egxa, vsamc, NO_SQUARES, ike.chen, Rg230403, Supercali, Mai-san, mira74, Justpassingby, Dem0nfang, MS_Kekas, lifeismathematics, Aryan-23, insertionsort, pavel kozlov, djmathman, greenturtle3141, sabkx, pieater314159, sargamsujit, nguyenducmanh2705, ATGY, A_Humanoid_Figure, cosinesine, Lhaj3

Regardless of whether you like this particular problem or not, can we all just take a step back and be respectful to problem authors and people who do happen to enjoy those problems? It's fine to express your negative opinions on a problem, but we need to draw the line at presenting them as facts.

"People being excited about P6 is such a shame, this is not classical geo" Are we seriously shaming people for enjoying a math problem? What kind of environment are we creating where a prolific problem author feels the need to publicly defend themselves (sarcastically, I hope) for creating something novel?

This post has been edited 1 time. Last edited by Ankoganit, Jul 11, 2023, 7:08 PM

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PRMOisTheHardestExam

409 posts

PRMOisTheHardestExam

#102 h Jul 11, 2023, 7:19 PM

Y by

people are really good in geometry nowdays they dont understand themselves

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dangerousliri

932 posts

dangerousliri

#103 h Jul 11, 2023, 7:24 PM

Y by

As result shows only 6 students solved problem 6 completely and 4 students got 6 points. So based on this I don't know how it was not suitable or how this was easier for its position.

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MarkBcc168

1595 posts

MarkBcc168

#104 h Jul 12, 2023, 12:38 AM • 1 Y

Y by GeoKing

As for the second point, the thought process (with ggb) that lead me into it kinda made me feel disappointed.
Spoliers
So, the key moment for me is just correctly guessing the second point, after which the problem comes nicely, but the contestants doesn't have a luxury of being able to verify or dispel their guess quickly, which is why this problem is very difficult.

On the other hand, I spent a significant amount of time trying to dream up some unconventional ways to solve the problem, for example by showing that the circumcenters are collinear. I even thought about using the known lemma that $A$ lies on the Neuberg cubic of $\triangle A_2BC$ , but that the circles just don't do anything. As a result, I gave up and used ggb to guess points, after which I figured out two points, one quickly after another.

Also, for the diagram, yes, it is possible in theory, but constructing three angles summing to $30^\circ$ is a tedious process, and the more constructions you do, the more accuracy you lose (which is crucial to solving the problem). Plus, intersecting two lines with a small angle can be quite detrimental to the accuracy of the diagram. I think getting an accurate diagram would be difficult, and I don't think I can guess claims from my diagram if I were taking it in the IMO.

Btw, I was told that hidden due to potential spoiler. In my opinion, this makes the problem significantly better because it allows for alternative synthetic solutions, but I have yet to think about this approach.

This post has been edited 4 times. Last edited by MarkBcc168, Jul 12, 2023, 1:11 AM

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IMD2

41 posts

IMD2

#105 h Jul 12, 2023, 1:54 AM

Y by

CantonMathGuy wrote:

rmtf1111 wrote:

P6 is very bad this year.

rmtf1111 wrote:

People being excited about P6 is such a shame, this is not classical geo,

Thank you for your messages. I was initially excited about seeing my problem on the IMO, but after reading your messages I realized how bad the problem was. If only you could've stopped the PSC or jury from choosing it... (And yes, you're right. It's not classical geo. I don't think anyone claimed it was.)

MarkBcc168 wrote:

Personally, I find this problem very disappointing and not the best choice for the IMO.

Sorry.

edit: For what it's worth, I think using GeoGebra significantly hurt me while trying to solve the problem: spoilers

Bro, your problem was great. Admits a wide variety of very different solutions spanning from synthetic to bashing while still being very difficult, if that doesn't make a problem excellent I don't know what does. Cons are what?, it gives advantage to people with experience in coaxiality?, so what?

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rmtf1111

698 posts

rmtf1111

#106 h Jul 12, 2023, 6:40 AM

Y by

@CantonMathGuy no disrespect whatsoever, your problem is a great proposal and congrats for it making the IMO. It is a fun problem and deserves its spot in the shortlist definitely.

The point I wanted to make is that while this problem can be solved in very nice synthetic ways, it is very easy to bash (if you have seen the right thing). Congrats to @MarkBcc and @tastymath for finding nice synthetic solutions, yet the reality of this problem is that it is very approachable if you know how linearity of PoP/coaxiality lemma work. My first instinct was just PoP bash, and after you figure this out (which IMHO is not very hard to think of if you know about the technique), then the problem becomes rather simple. The solutions on the thread speak just about that. And to be fair, PoP techniques or any other standard techniques shouldn't be seen as something to diminish the quality of a problem, but the issue is that for this problem PoP was pretty much everything needed. I used Coaxiality Lemma in 2019 for P6, yet there it was just a simple step alongside a long solution.

Maybe I am wrong and I just like PoP too much. However, it would be interesting to know how many of the 10 participants who solved the problem bashed it this way. My guess would be pretty high

Again, @CantonMathGuy, absolutely no offense, this is a nice problem, but I am not sure how fit it is for the IMO.

This post has been edited 1 time. Last edited by rmtf1111, Jul 12, 2023, 6:40 AM

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hliu1

939 posts

hliu1

#107 h Jul 12, 2023, 12:12 PM

Y by

@above at least three solves were synthetic, and at most six

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anantmudgal09

1980 posts

anantmudgal09

#108 h Jul 12, 2023, 12:33 PM • 27 Y

Y by lifeismathematics, Supercali, Rg230403, Uagu, NO_SQUARES, qwedsazxc, insertionsort, pavel kozlov, GeoKing, djmathman, khan.academy, somebodyyouusedtoknow, greenturtle3141, Plasma_Vortex, kamatadu, PRMOisTheHardestExam, pieater314159, Diorite, channing421, Mahdi_Mashayekhi, starchan, SatisfiedMagma, sargamsujit, math_comb01, axolotlx7, A_Humanoid_Figure, EpicBird08

I apologise for the long post, but I have to say, I find this entire thread very disappointing and in very poor taste. I do not intend to target anyone's comment specifically, or offend anyone who expressed views different from mine, since I understand that the IMO holds a lot of importance to many people and it's natural to then have strong views.

People have long called every single IMO problem "easy" or "trivial" or any moniker that makes everyone involved look bad --- contestants who can't solve it, and those who do end up solving it, as if they did not do much work; and even the proposers, the PSC and IMO jury who have all spent ridiculous amounts of effort in making, selecting and publishing the final list of problems.

A few anecdotes I can tell from the jury meetings will tell you that almost everyone found P6 really hard: most of the leaders did not solve P6 and only a handful of observer As solved it. The problem was voted for having a beautiful statement and an entirely synthetic solution that is most in spirit of IMO geometry.

My personal taste in geometry gravitates towards a mix of synthetic and projective observations and conjuring new points so it suffices to say there are other problems from the long-list I like better than P6; but my personal taste is not how I would judge the suitability of a problem for the IMO: it is much more preferable to not have the key ideas be all or any of projective/DDIT/moving points. Thus, regardless of my own tastes, I advocated very strongly for P6. This choice also underlies knowledge of the actual IMO shortlist, which most of the people dissatisfied with the problem simply don't have, so it's strange to me that people who dislike the problem jump to hard conclusions instead of mild curiosity about the process.

I would also like to congratulate the author; and I believe I speak for a majority of the jury when I say that we all absolutely love your problem and thank you for making it a wonderful P6. I personally consider it even prettier than my own P6 geometry, even if it differs from my own taste.

Regarding the issue of "easy" problems or "devaluing" HM or "search for talent" or "trainable vs talent-seeking" problems or "more representation for strong countries"; it seems to me to be a grave difference (and error) in perspective. I have always viewed the IMO as the event where we celebrate people who love doing mathematics, and we do that by bringing tons of them together in some mildly competitive environment to give them the extra nudge they need to shine and bring their A-game. If this becomes about "filtering" for the best ones, or any sort of survival-of-the-talented contest, it'll be gravely disappointing and would also inevitably just die out. To me, more important than identifying talent is increasing access: the identification will then just be a byproduct. As such, making the contest more appealing for many people to try seems like a good strategy to me, regardless of what you want to optimise among talent-search or having fun (and I suspect both are strongly correlated anyway).

Finally, I want to say that it seems many of the posters are former IMO contestants. It is natural to feel strongly about the IMO and to feel a strong need for increase in difficulty or quality no matter how unbalanced the effects might be, and I understand that that comes from a good place of care about the event itself. But I think it is important to realise that the IMO is about the contestants who actually take the test; and not us bystanders who once did. Just as difficulty and quality are important, so is the accessibility and fun aspect of solving these problems, and we should perhaps not let our personal biases affect that too much.

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MyobDoesMath

51 posts

MyobDoesMath

#109 h Jul 12, 2023, 3:41 PM • 1 Y

Y by khan.academy

For those still interested in the original topic discussed here, with the published results there are also published statistics, of whose I will only refer to the mean value of given points per problem (for more details, check http://www.imo-official.org/year_statistics.aspx?year=2023):

P1 had a mean value of 5.845, making it the easiest P1 since 2017 and one of the all-time easiest. That being said, over the past 20 years one can find quite a few P1/4s whose mean value was above 5.5 - I leave it up to you to decide whether it was too easy, I would argue that it fits in the upper part of the range of difficulties over the last years.

P2's mean value was around 3.162 which also is above average for P2/5 position, but still fits into the range (and is especially massively lower than 2022's 4.306). Also notice that these high numbers often (but not only) occur if P1 is easy, I would say that is natural - less time spent on P1 means more time for the other problems.

P3's average was 1.256. This is indeed the upper end of P3/6's difficulty over the last years. Still, the same notice as for P2 holds (especially since P2 was Geometry, so there were people who skipped it, looking at the results).

So was day 1 relatively easy? Yes. Was it too easy? - Well, I leave that to more knowledged people to decide.

For day 2:

P4's average was around 4.717, which is definitely not the hardest P4 ever, but is not easier than usual. I would go as far as to say, this is a pretty average difficulty.

P5's mean value is about 2.417, which also seems pretty averaged, maybe a little above average, concerning there were some P5s with really low scores.

And last but not least, P6 fits with an average score of 0.275 into the range of harder P3/6s.

You may have your own opinion on how to interpret these facts. I don't think I can contribute much more to this discussion than recycling the arguments from above, something I don't want to do. But I agree strongly on not blaming the students or problem creators for anything - there is a reason each participant was selected for the competition and there is a reason each problem was chosen by the PSC and whether or not you like these decisions, you should not blame someone for them who is, in fact, not responsible.

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Therealway

48 posts

Therealway

#110 h Jul 13, 2023, 11:05 AM • 1 Y

Y by Iveela

@anantmudgal09 I do not think all the comments are "disappointing", having these discussions is what can help to make progress, and diverse perspectives help to enrich the olympiad landscape. Also, the system aformentioned was meant to increase "accessibility" to students by allowing more student from strong countries to participate, if you have seen how students work in those strong countries, but only to be elminated due to competition, it can be understood why this discussion is in place and why it's valid, and how it, in fact, allow more people to enjoy the world stage this competition has provided since 1959.

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LoloChen

479 posts

LoloChen

#111 h Jul 13, 2023, 3:42 PM • 3 Y

Y by GeoKing, PRMOisTheHardestExam, vfed

This post has been edited 3 times. Last edited by LoloChen, Apr 10, 2025, 3:56 AM

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Iblamemyself

44 posts

Iblamemyself

#112 h Jul 13, 2023, 8:33 PM • 1 Y

Y by PRMOisTheHardestExam

Quote:

I even thought about using the known lemma that $A$ lies on the Neuberg cubic of $\triangle A_2BC$

See the thing is... basically nobody else would ever have thought to do that.

Meanwhile the thought process you are describing as taking minutes would take hours for many solid national olympiad qualifiers. It's great you are so skilled at geometry but it has clearly warped your perception of what a good imo problem looks like.

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aidensharp

362 posts

aidensharp

#113 h Jul 13, 2023, 9:20 PM

Y by

https://www.imo-official.org/country_individual_r.aspx?code=USA

How is @dottedcalculator so orz

This post has been edited 1 time. Last edited by aidensharp, Jul 13, 2023, 9:20 PM

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xeroxia

1137 posts

xeroxia

#114 h Jul 15, 2023, 5:31 AM

Y by

Why the host country(Japan) did not release the official solution booklet?
The site https://imo2023.jp/en/problems/ is terrible.
Also where is the official Shortlist Booklet of 2022?

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YIYI-JP

2178 posts

YIYI-JP

#115 h Jul 15, 2023, 5:48 AM

Y by

Because it's under construction.
I have the IMO 2021 Shortlist Booklet but apparently there is no 2022 Shortlist Booklet.

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