In case you ever wondered what the most prestigious mathematical competition for high school students looks like, you get to solve 6 problems, in two sets of three problems. For each set, you get 4.5 hours of uninterrupted work, with all the scratch paper you might need, you bring your standard geometry drawing tools (ruler and compass) and you go.

For the list of problems, the IMO organization has a list of the official problems. Some are quite entertaining to do. This year, after being piqued a bit, I worked problem one in about half an hour to 45 minutes (I still got it). I used to compete in these types of competitions when I was in high school.

In any case, the problems are *elementary: they just use algebra, trigonometry, basic plane geometry and elementary number theory*. Here is problem one:

Let n be a positive integer and let

be distinct integers in the set {1, . . . , n} such that n divides for i = 1, . . . , k − 1. Prove that n does not divide .

If you ever have tried to design these problems, you will notice that the conditions are somewhat contrived. n appears in two places: both in the size of the original set, and in the division condition. Could one do without the first?

The answer is no. If all of the are multiples of n, then the problem does not follow. But having all of them small must be important. The next thing one needs to do is understand the relation of divisibility better so that one can solve the problem. I’ll let you figure the rest out…

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