The unexpectedly hard windmill question (2011 IMO, Q2)

The famous (infamous?) “windmill” problem on the 2011 IMO
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The author of this problem was Geoff Smith. You can find the full list of problems considered for the IMO that year, together with their solutions, here:

You can find data for past IMO results here:

Viewer-created interactive about this problem:

And another:

I made a quick reference to “proper time” as an example of an invariant. Take a look at this minutephysics video if you want to learn more.


These animations are largely made using manim, a scrappy open-source python library:

If you want to check it out, I feel compelled to warn you that it’s not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind.

Music by Vincent Rubinetti.
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If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then “add subtitles/cc”. I really appreciate those who do this, as it helps make the lessons accessible to more people.


3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe:

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Alexander Levakin

I suppose there was a hot discussion between professors about whether to include the problem to list or not.


This video is clearly giving us how to approach any kind of math problem… awesome 😀

Most of the problems, while still being proofs don't generally involve invariants. in a time constrained envi- you tend to stick to your usual route. In addition, some problems are way easier using a topological proof, which pre-uni students are not exposed to. That being said, these are usually more intuitive, if they were intended to be inside 3 dimensions.

omar shrara

I'm really having a hard time understanding (poincaré conjecture & its solution).. and I hope U explain (Ricci flow) in the easiest way possible.. plz
Because I cannot sleep easily every night, due to a headache that this concepts gives to me 😢😭🌊

Erme BoXi

The reason is that when a dot turns into a pivot, it always goes to the opposite end

Samuel Shelley

This is where math gets confusing.
It feels like the problem tells you an interesting scenario to think about and then tells you to give an answer.
What is the question even asking me to do?

Brandon N

It evaded children. To assume that this problem would evade adults in a similar way is fallacious. Some thing may happen developmentally that makes the unique logic of this problem easier (or harder) for adults to arrive at.

Calhoon Williams

I sucked at math in school and now i want to learn so much more about it

Joseph Lupton

You're one of the teachers on Khan Academy. I can recognize your voice.

Neelansh Guptaa

I’m 20 years old and I regret that I wasn’t able to watch videos like these when I was 15 and still in high school taking maths. Who knows what I’d be like now if I’d fallen in love with maths then just like I did after watching this now.

Jack McLane

Maths is easy if you have done the same piece of maths a hundred times. This is why maths lectures are so annoying. There is the bored lecturer that simply cannot remember what might be difficult on this – in his or her view – mind numbingly trivial kindergardener stuff. And the students are pissed off by lecturers that say stuff like "It's clear to see how this operation transforms the n-dimensional hypercube into a toroid of n+1 dimensions." (exemplary) NO! IT'S NOT clear to see!!! ^^
I've never had a math teacher or professor that was actually good at teaching maths. This is why I enjoy such great maths vids on youtube!


12:49 – "Turn it around in your head a couple times…"

I see what you did there. Clever.

Ethan Dcosta

As a former Olympiad participant i got a littlemore then halfway through but i was unable to complete it.


I want to hear the clicking sounds more, like, no commentary just this windmill stuff going. It's relaxing

No mention of God who is omniscient and infinite.. try to wrap your little finite dust-made brain around that.

Thuong Vo

But how can you prove that there will always be a line that splits the set S into two equal groups?


Can we appreciate the thumbnail for 3.14 seconds? It’s the only picture I hear 👂 ( in Ross Gellers voice: PIVOTTTTT) 😂❤️ New subscriber ! I have math videos I’m trying to figure out. Now I’ll just say PIVOTTTTT LOL

Christino Sim

I competed in the Olympiad in 2018 in China. It was split into 3 parts. Short answer, essay, and problem solving. Short answer was just finding an answer, but had 20 or so questions. Essay needed an explanation and was 15 or so questions. Problem solving was more focused on visuals and had 7 questions. I was so trash at short answer and essay, but problem solving got me a bronze medal.

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