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Educators in many countries have found that preparing highly motivated students for national and international mathematical olympiads can be useful in training them in mathematical ways of thinking.

Successfully answering the questions in such competitions is said to require only a pre-university level of mathematical knowledge. But it often requires an advanced use of it. Moreoever, it's made easier by familiarity and practical experience with topics that aren't on many pre-university syllabuses in any country - or even, for that matter, at university level.

My question is whether there are any international mathematical olympiad-style competitions at a lower level. If so, what are they, what are the age restrictions, at how low a level do they start (and what does that mean), and how useful have mathematics educators found them?

I'm especially interested in any international competitions in which candidates' and countries' scores are made public and can be ranked, as is true for the IMO.

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    $\begingroup$ I think there's something called Kangaroo. There might be more. $\endgroup$
    – Sue VanHattum
    Jan 15 at 17:12
  • $\begingroup$ Thanks Sue. I know a bit about Kangaroo but not much. I will edit the question. $\endgroup$
    – tell
    Jan 15 at 18:39
  • $\begingroup$ Can you clarify what you mean by "lower level"? Is it "easier" or "for younger ages"? $\endgroup$
    – shoover
    Jan 18 at 21:24

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I know that at least in Germany the qualification for the International Math Olympiad consist of an entire pyramid of math competition exams at different levels and age groups. The lowest level is open to all and you qualify for the next higher level by doing well enough. All of them are in a format very similar to the IMO, very small number of problems, several hours long written exam, no knowledge beyond the age apropriate school knowledge is required.

The problems and solutions are all available online (in German) and range from 3rd grade (8 to 9 year olds) to 11/12 year of school (16 to 18 years old) and have 4 levels: school, regional, statewide, federal with a significantly increasing level of difficulty.

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I'm not sure if this qualifies as "IMO-style", but there is the International Math Youth Challenge https://iymc.info/en/, founded in 2018. Regarding age limit, the FAQ state

"All students (high school or university) are allowed to participate. Students from all grades and all countries are invited to participate! However, there are two age categories (under 18 years and over 18 years)."

I think you can estimate "level" best when having a look at previous problems: https://iymc.info/en/training. Note that all participants are given the same problem sets, but the over 18 year olds are required to achieve more points in order to qualify for the next round.

There is no ranking of all participants, but for the final round the winners (without score) are published: https://iymc.info/en/announcement2023

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