# Sample Problems

## IMO Problem #1

IMO problems are usually quite short, and the jury goes to some length to ensure that the language is unambiguous. Often this involves the use of technical terms, mathematical jargon which hinders non-mathematicians who might wish to understand the question. The jury does this for a good reason. Mathematical language is very precise, and the jury uses it to make sure that the problem is explained in a very clear way to the contestants.

Every now and again a question is used which does not need any jargon. For example, the following question was used at IMO 2001 in Washington DC, and was a “problem 3″. This means that the jury thought it was a hard problem to solve. Note, however, that the problem is very easy to understand.

Twenty-one girls and twenty-one boys took part in a mathematical contest.

(i) Each contestant solved at most six problems.

(ii) For each girl and each boy, at least one problem was solved by both of them.

Prove that there was a problem that was solved by at least three girls and at least three boys.

At an IMO, the students are set six problems, so there is an engaging self-referential quality to this problem. However, there is a trap for the unwary, because in the mathematical contest in the question, it does not say that there were only six questions on the papers. Rather it says that each student managed to solve at most six of them. It is safe to assume that there were a finite number of questions in the mathematical contest, but we are not told that number.

We will make a solution available to you, but you are urged to think hard about the problem yourself, before looking at our suggested answer. This attitude typifies the ethos of practising for mathematics competitions. Reading a solution only does you a little bit of good (if ever you are asked this question again, you will know the answer!). The real benefit is only obtained when you struggle to find the solution yourself, even if you do not solve the problem. This is because in playing with the problem in your mind, you will have ideas, some of which will be better than others. Thinking gets easier and you do it better if you practise a lot. In this respect, thinking is no different to many other human activities.

Incidentally, this problem was proposed by Germany, and its author was Christian Bey.

## IMO Problem #2

You can view IMO problems on the official IMO website. Here is a problem from the 2014 paper which is quite easy to understand (though not that easy to answer). This was question 2 on day one of the competition.

Let $n$ be a positive integer. Consider an $n\times n$ chessboard consisting of $n^2$ unit squares.

A configuration of $n$ rooks on this board is peaceful if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that, for each peaceful configuration of $n$ rooks, there is a $k\times k$ square which does not contain a rook on any of its $k^2$ unit squares.