I recently read this article http://blog.mathfights.com/once-upon-a-time-on-imo/ where the author discusses an IMO problem from 2006 that only about 20 participants out of 600 were able to solve. So this got me wondering, have there ever been any IMO problems where no contestant was able to solve it.If not does any one know of any other problems with only a few number of correct solutions?
[Math] Unsolved/Least Solved IMO Questions
contest-mathsoft-question
Related Solutions
1, 2) Telescoping is one of the ideas behind modern algorithms to automatically prove hypergeometric identities. These algorithms allow you, for example, to automatically prove binomial coefficient identities. The standard reference here is Petkovsek, Wilf, and Zeilberger's A=B.
3) The name comes from the process of collapsing a telescope, which is analogous to the collapsing of a telescoping sum.
Philosophically telescoping is the same as "discrete integration": telescoping a sum $\sum f(n)$ is the same as finding $g(n)$ such that $f(n) = g(n+1) - g(n)$. In that sense it is part of the theory of finite differences, although people probably don't call it "telescoping" in this context. The context in which I hear the term "telescoping" being used is high school math competitions. It's one of those basic ideas that everyone has in the back of their head, I suppose. It's elementary and effective when it applies, but usually there are more sophisticated methods available.
Edit: Some specific examples. The ur-example of a telescoping sum is probably
$$\sum_{k=1}^n \frac{1}{k(k+1)} = \sum_{k=1}^{n} \left( \frac{1}{k} - \frac{1}{k+1} \right) = 1 - \frac{1}{n+1}$$
and many people have seen this application, but probably far fewer have seen its generalization:
$$\sum_{k=1}^n \frac{1}{k(k+1)...(k+r)} = \frac{1}{r} \sum_{k=1}^n \left( \frac{1}{k(k+1)...(k+r-1)} - \frac{1}{(k+1)...(k+r)} \right) = \frac{1}{r} \left( \frac{1}{r!} - \frac{n!}{(n+r)!} \right).$$
The other classic example I remember from my competition days is
$$\sum_{k=1}^n \frac{k}{k^4 - k^2 + 1} = \frac{1}{2} \sum_{k=1}^n \left( \frac{1}{k^2 - k + 1} - \frac{1}{(k+1)^2 - (k+1) + 1} \right) = \frac{1}{2} \left( 1 - \frac{1}{n^2 + n + 1} \right)$$
although I have to admit I always found it a little contrived. Finally, telescoping was put to good use to solve this math.SE question I posed.
Here is a collection of answers from comments, so that this question can be put to bed.
J.M. says:
You could start with looking at this and this. If memory serves, apart from the Babylonians, the Chinese and Hindus also knew about (a special form of) the quadratic formula, but I don't have my references right now.
Robert Israel says:
Also look at http://www-history.mcs.st-and.ac.uk/HistTopics/Quadratic_etc_equations.html
Andre Nicolas says:
For the "completing the square part" see al-Khwarizmi, who was completing an honest to goodness square. For example, to explain solution of $x^2+2bx=d$ ($b$, $d$ positive) imagine a square house of side $x$, with porches of width $b$, length $x$ on north and east. Complete this to a square by adding $b\times b$ square. Big square has side $x+b$, area $d+b^2$, so $x=\sqrt{d+b^2}−b$.
The teacher may be interested in the following simpler derivation of the quadratic formula. The main pedagogical advantage is that one ends up avoiding division until the very end. The trick is very old, going back to Sridhara, but is in my opinion insufficiently used.
Dave L. Renfro says:
I'll send you some .pdf files by email shortly, but the following (freely available on the internet) will probably be more useful: Solving Quadratic Equations By analytic and graphic methods; Including several methods you may never have seen by Pat Ballew (2007).
Best Answer
Migrated from a comment.
Note the IMO website under results allows you to click on the year where you will find statistics as to how well participants did on the questions in that year.
Also see problem 6 in 2009 - anyone who has seen an answer to this problem will find it hard to see what the difficulty was.
It is normal that the sixth problem in the IMO is the most difficult of the six, and that the third is very difficult too. However the judgment of the examiners is not perfect e.g. see 1996 when problem 5 turned out to be hardest, and problem $3$ in 2007 was also very difficult.
The Art of Problem Solving website has a section devoted to Olympiad problems.
For $2009/6$ (spoiler)