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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

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0 replies
jlacosta
May 1, 2025
0 replies
Sequence of rational numbers
mojyla222   2
N 10 minutes ago by Assassino9931
Source: Iran 2024 3rd round number theory exam P1
Given a sequence $x_1,x_2,x_3,\cdots$ of positive integers, Ali proceed the following algorythm: In the i-th step he markes all rational numbers in the interval $[0,1]$ which have denominator equal to $x_i$. Then he write down the number $a_i$ equal to the length of the smallest interval in $[0,1]$ which both two ends of that is a marked number. Find all sequences $x_1,x_2,x_3,\cdots$ with $x_5=5$ and such that for all $n\in \mathbb N$ we have
$$
a_1+a_2+\cdots+a_n= 2-\dfrac{1}{x_n}.
$$
Proposed by Mojtaba Zare
2 replies
mojyla222
Aug 27, 2024
Assassino9931
10 minutes ago
Generic Real-valued FE
lucas3617   6
N 22 minutes ago by Audreyma0321
$f: \mathbb{R} -> \mathbb{R}$, find all functions where $f(2x+f(2y-x))+f(-x)+f(y)=2f(x)+f(y-2x)+f(2y)$ for all $x$,$y \in \mathbb{R}$
6 replies
lucas3617
Apr 25, 2025
Audreyma0321
22 minutes ago
Another FE
M11100111001Y1R   2
N 34 minutes ago by AblonJ
Source: Iran TST 2025 Test 2 Problem 3
Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that for all $x,y>0$ we have:
$$f(f(f(xy))+x^2)=f(y)(f(x)-f(x+y))$$
2 replies
M11100111001Y1R
Today at 8:03 AM
AblonJ
34 minutes ago
Midpoint in a weird configuration
Gimbrint   1
N 34 minutes ago by Beelzebub
Source: Own
Let $ABC$ be an acute triangle ($AB<BC$) with circumcircle $\omega$. Point $L$ is chosen on arc $AC$, not containing $B$, so that, letting $BL$ intersect $AC$ at $S$, one has $AS<CS$. Points $D$ and $E$ lie on lines $AB$ and $BC$ respectively, such that $BELD$ is a parallelogram. Point $P$ is chosen on arc $BC$, not containing $A$, such that $\angle CBP=\angle BDE$. Line $AP$ intersects $EL$ at $X$, and line $CP$ intersects $DL$ at $Y$. Line $XY$ intersects $AB$, $BC$ and $BP$ at points $M$, $N$ and $T$ respectively.

Prove that $TN=TM$.
1 reply
Gimbrint
May 23, 2025
Beelzebub
34 minutes ago
No more topics!
IMO 2023 problem discussion
Complete_quadrilateral   108
N Jul 15, 2023 by YIYI-JP
Get your frustrations out here.
108 replies
Complete_quadrilateral
Jul 8, 2023
YIYI-JP
Jul 15, 2023
IMO 2023 problem discussion
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Complete_quadrilateral
144 posts
#1 • 2 Y
Y by Rounak_iitr, oolite
Get your frustrations out here.
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Complete_quadrilateral
144 posts
#2 • 2 Y
Y by TheHimMan, Rounak_iitr
Redacted
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799786
1052 posts
#3 • 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
#4 • 1 Y
Y by Rounak_iitr
Redacted
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navi_09220114
487 posts
#5 • 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]
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Complete_quadrilateral
144 posts
#6 • 3 Y
Y by Miquel-point, RobertRogo, lambda5
Redacted
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khan.academy
634 posts
#7
Y by
redacted..............
This post has been edited 2 times. Last edited by khan.academy, May 3, 2024, 11:49 AM
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kimyager
8 posts
#8
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
#9
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
241 posts
#10 • 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
302 posts
#11 • 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
#12
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
#13 • 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
#14 • 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

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
#15
Y by
I believe the decrease in difficulty level is the influx of LDCs participating in the IMO.
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