The International Mathematical Olympiad is the largest and most prestigious mathematics competition in the world. It is held each July, and the host city changes from year to year. It has existed since 1959. Originally it was a competition between students from a small group of communist countries, but by the late 1960s, social-democratic nations were starting to send
teams. Over the years the enthusiasm for this competition has built up so much that very soon (I write in 2008) there will be an IMO with students participating from over 100 countries. In recent years, the format has become stable. Each nation can send a team of up to six students. The students compete as individuals, and must try to solve 6 problems in 9 hours of examination time, spread over two days. The nations which do consistently well at this competition must have at least one (and probably at least two) of the following attributes:
(a) A large population.
(b) A significant proportion of its population in receipt of a good education.
(c) A well-organized training infrastructure to support mathematics competitions.
(d) A culture which values intellectual achievement.
Alternatively, you need a cloning facility and a relaxed regulatory framework. Mathematics competitions began in the Austro-Hungarian Empire in the 19th century, and the IMO has stimulated people into organizing many other related regional and world competitions. Thus there are quite a few opportunities to take part in international mathematics competitions other than the IMO. The issue arises as to where talented students can get help while they prepare themselves for these competitions. In some countries the students are lucky, and there is a well-developed training regime. Leaving aside the coaching, one of the most important features of these regimes is that they put talented young mathematicians together. This is very important, not just because of the resulting exchanges of ideas, but also for mutual encouragment in a world where interest in mathematics is not always widely understood. There are some very good books available, and a wealth of resources on the internet, including this excellent book Infinity. The principal author of Infinity is Hojoo Lee of Korea. He is the creator of many beautiful problems, and IMO juries have found his style most alluring. Since 2001 they have chosen 8 of his problems for IMO papers. He has some way to go to catch up with the sage of Scotland, David Monk, who has had 14 problems on IMO papers. These two gentlemen are reciprocal Nemeses, dragging themselves out of bed every morning to face the possibility that the other has just had a good idea. What they each need is a framed picture of the other, hung in their respective studies. I will organize this.
numbers 
from th first hundred natural numbers (from 1 to 100) so that the sum of any two distinct numbers is a multiple of 6. What is the largest possible number 
and 
are the side lengths of a traingle. Compare 
ang 
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