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I encourage anyone who thinks these are easy high-school problems to try to solve some. They're published (including this year's) at https://www.imo-official.or

by tlb 1y ago

I encourage anyone who thinks these are easy high-school problems to try to solve some. They're published (including this year's) at https://www.imo-official.org/problems.aspx https://www.imo-official.org/problems.aspx. They make my head spin.

xpressvideoz 1y ago
I didn't know there were localized versions of the IMO problems. But now that I think of it, having versions of multiple languages is a must to remove the language barrier from the competitors. I guess having that many language versions (I see ~50 languages?) may make keeping the security of the problems considerably harder?
- dmurray 1y ago
  The problems are chosen by representatives from all the countries. So every country has someone who knows the full exam before the participants get it. Security is on the honour system, but it seems to mostly work.
- peteyreplies 1y ago
  iirc, the IMO system automatically translates the questions into 50 languages, after they are entered in English.
kappi 1y ago
[flagged]
- throw310822 1y ago
  I think you're joking, but you never know :)
- yongjik 1y ago
  All the past IMO problems are known to public and contestants practice on them. If solving an IMO problem is the simple matter of "looking at all the past problems and apply the same pattern," you'd expect human contestants to do a lot better.
  - kappi 1y ago
    I think you haven't gone thru AMC8, AMC10, AIME competitions. If you are so confident, try giving an unsolved math problem outside the high school math competitions.
    - saagarjha 1y ago
      It's still tough, though practice obviously helps.
mauriziocalo 1y ago
Related — these videos give a sense of how someone might actually go about thinking through and solving these kinds of problems: - A 3Blue1Brown video on a particularly nice and unexpectedly difficult IMO problem (2011 IMO, Q2): https://www.youtube.com/watch?v=M64HUIJFTZM https://www.youtube.com/watch?v=M64HUIJFTZM -- And another similar one (though technically Putnam, not IMO): https://www.youtube.com/watch?v=OkmNXy7er84 https://www.youtube.com/watch?v=OkmNXy7er84 - Timothy Gowers (Fields Medalist and IMO perfect scorer) solving this year’s IMO problems in “real time”: -- Q1: https://www.youtube.com/watch?v=1G1nySyVs2w https://www.youtube.com/watch?v=1G1nySyVs2w -- Q4: https://www.youtube.com/watch?v=O-vp4zGzwIs https://www.youtube.com/watch?v=O-vp4zGzwIs
- nmca 1y ago
  It takes Tim Gowers more than hour and a half to go through q4! (Sure, he could go faster without video. But Tim Gowers! An hour and a half!!)
- snewman 1y ago
  For people who prefer reading to watching videos, I wrote a detailed account of my process for solving one of last year's IMO problems, along with thoughts on how this relates to AI: https://secondthoughts.ai/p/solving-math-olympiad-problems https://secondthoughts.ai/p/solving-math-olympiad-problems
selfselfgo 1y ago
[dead]
bko 1y ago
I like watching youtube videos solving these problems. They're deceptively simple. I remember reading one: x+y=1 xy=1 The incredible thing is the explanation uses almost all reasoning steps that I am familiar with from basic algebra, like factoring, quadratic formula, etc. But it just comes together so beautifully. It gives you the impression that if you thought about it long enough, surely you would have come up with the answer, which is obviously wrong, at least in my case. https://www.youtube.com/watch?v=csS4BjQuhCc https://www.youtube.com/watch?v=csS4BjQuhCc
- Tainnor 1y ago
  This is slightly tedious to do by hand but there isn't really anything interesting going on in that problem - it's just solving a quadratic equation over the complex numbers.
  - roenxi 1y ago
    That isn't much of an argument; nothing in math is truly interesting if you take that approach. exp(i\pi)+1=0 could be said to be dis-interesting because it is just rotation on the complex plane. But it is the opposite - it is interesting because it turned out to be rotation on the complex plane but approached from summing infinite series. Similarly you can say that solving a quadratic over complex numbers is dis-interesting, but it is actually an interesting puzzle because it is trying its best to pretend it isn't a quadratic. In many ways succeeding, it isn't a quadratic - there is no "2" in it.
    - Tainnor 1y ago
      It's "not interesting" because no novel insight has to be used in order to solve this. It's immediately obvious how to solve it, just follow the textbook procedure. This is distinct both from other typical IMO problems that I've seen and from research mathematics which usually do require some amount of creativity. > exp(i\pi)+1=0 If your definition of "exp(i*theta)" is literally "rotation of the number 1 by theta degrees counterclockwise", then indeed what you quoted is a triviality and contains no nugget of insight (how could it?). It becomes nontrivial when your definition of "exp" is any of the following: - The everywhere absolutely convergent power series sum_{i=0}^\infty z^n/n! - The unique function solving the IVP y'=y, y(0)=1 - The unique holomorphic extension of the real-valued exponential function to the complex numbers Going from any of these definitions to "exp(i*\pi)+1=0" from scratch requires quite a bit of clever mathematics (such as proving convergence of the various series, comparing terms, deriving the values of sin and cos at pi from their power series representation, etc.). That's definitely not something that a motivated high schooler would be able to derive from scratch.
koakuma-chan 1y ago
How do those compare to leetcode hard problems?
- kenjackson 1y ago
  Depends on how hard, but the “average hard” leetcode problem is much easier. These will be more like the ACM ICPC level questions, which I’d put at the “hard hard” leetcode level (also this is a collegiate competition rather than high school, but with broader participation).

by tlb 1y ago

xpressvideoz 1y ago

I didn't know there were localized versions of the IMO problems. But now that I think of it, having versions of multiple languages is a must to remove the language barrier from the competitors. I guess having that many language versions (I see ~50 languages?) may make keeping the security of the problems considerably harder?

dmurray 1y ago
The problems are chosen by representatives from all the countries. So every country has someone who knows the full exam before the participants get it. Security is on the honour system, but it seems to mostly work.
peteyreplies 1y ago
iirc, the IMO system automatically translates the questions into 50 languages, after they are entered in English.

kappi 1y ago

[flagged]

throw310822 1y ago
I think you're joking, but you never know :)
yongjik 1y ago
All the past IMO problems are known to public and contestants practice on them. If solving an IMO problem is the simple matter of "looking at all the past problems and apply the same pattern," you'd expect human contestants to do a lot better.
- kappi 1y ago
  I think you haven't gone thru AMC8, AMC10, AIME competitions. If you are so confident, try giving an unsolved math problem outside the high school math competitions.
  - saagarjha 1y ago
    It's still tough, though practice obviously helps.

mauriziocalo 1y ago

Related — these videos give a sense of how someone might actually go about thinking through and solving these kinds of problems: - A 3Blue1Brown video on a particularly nice and unexpectedly difficult IMO problem (2011 IMO, Q2): https://www.youtube.com/watch?v=M64HUIJFTZM https://www.youtube.com/watch?v=M64HUIJFTZM -- And another similar one (though technically Putnam, not IMO): https://www.youtube.com/watch?v=OkmNXy7er84 https://www.youtube.com/watch?v=OkmNXy7er84 - Timothy Gowers (Fields Medalist and IMO perfect scorer) solving this year’s IMO problems in “real time”: -- Q1: https://www.youtube.com/watch?v=1G1nySyVs2w https://www.youtube.com/watch?v=1G1nySyVs2w -- Q4: https://www.youtube.com/watch?v=O-vp4zGzwIs https://www.youtube.com/watch?v=O-vp4zGzwIs

nmca 1y ago
It takes Tim Gowers more than hour and a half to go through q4! (Sure, he could go faster without video. But Tim Gowers! An hour and a half!!)
snewman 1y ago
For people who prefer reading to watching videos, I wrote a detailed account of my process for solving one of last year's IMO problems, along with thoughts on how this relates to AI: https://secondthoughts.ai/p/solving-math-olympiad-problems https://secondthoughts.ai/p/solving-math-olympiad-problems

selfselfgo 1y ago

[dead]

bko 1y ago

I like watching youtube videos solving these problems. They're deceptively simple. I remember reading one: x+y=1 xy=1 The incredible thing is the explanation uses almost all reasoning steps that I am familiar with from basic algebra, like factoring, quadratic formula, etc. But it just comes together so beautifully. It gives you the impression that if you thought about it long enough, surely you would have come up with the answer, which is obviously wrong, at least in my case. https://www.youtube.com/watch?v=csS4BjQuhCc https://www.youtube.com/watch?v=csS4BjQuhCc

Tainnor 1y ago
This is slightly tedious to do by hand but there isn't really anything interesting going on in that problem - it's just solving a quadratic equation over the complex numbers.
- roenxi 1y ago
  That isn't much of an argument; nothing in math is truly interesting if you take that approach. exp(i\pi)+1=0 could be said to be dis-interesting because it is just rotation on the complex plane. But it is the opposite - it is interesting because it turned out to be rotation on the complex plane but approached from summing infinite series. Similarly you can say that solving a quadratic over complex numbers is dis-interesting, but it is actually an interesting puzzle because it is trying its best to pretend it isn't a quadratic. In many ways succeeding, it isn't a quadratic - there is no "2" in it.
  - Tainnor 1y ago
    It's "not interesting" because no novel insight has to be used in order to solve this. It's immediately obvious how to solve it, just follow the textbook procedure. This is distinct both from other typical IMO problems that I've seen and from research mathematics which usually do require some amount of creativity. > exp(i\pi)+1=0 If your definition of "exp(i*theta)" is literally "rotation of the number 1 by theta degrees counterclockwise", then indeed what you quoted is a triviality and contains no nugget of insight (how could it?). It becomes nontrivial when your definition of "exp" is any of the following: - The everywhere absolutely convergent power series sum_{i=0}^\infty z^n/n! - The unique function solving the IVP y'=y, y(0)=1 - The unique holomorphic extension of the real-valued exponential function to the complex numbers Going from any of these definitions to "exp(i*\pi)+1=0" from scratch requires quite a bit of clever mathematics (such as proving convergence of the various series, comparing terms, deriving the values of sin and cos at pi from their power series representation, etc.). That's definitely not something that a motivated high schooler would be able to derive from scratch.

koakuma-chan 1y ago

How do those compare to leetcode hard problems?

kenjackson 1y ago
Depends on how hard, but the “average hard” leetcode problem is much easier. These will be more like the ACM ICPC level questions, which I’d put at the “hard hard” leetcode level (also this is a collegiate competition rather than high school, but with broader participation).