As we talked about here, there was plans afoot to group-solve online an interesting and problem from the IMO. The online discussion going on right now (unless, of course, you’re reading this later). They decided to work on Question 3. Here it is:
Problem 3. The liar’s guessing game is a game played between two players and . The rules of the game depend on two positive integers and which are known to both players.
At the start of the game, chooses two integers and with . Player keeps secret, and truthfully tells to player . Player now tries to obtain information about by asking player A questions as follows. Each question consists of specifying an arbitrary set of positive integers (possibly one specified in a previous question), and asking whether belongs to . Player may ask as many such questions as he wishes. After each question, player must immediately answer it with yes or no, but is allowed to lie as many times as she wishes; the only restriction is that, among any consecutive answers, at least one answer must be truthful.
After has asked as many questions as he wants, he must specify a set of at most positive integers. If belongs to , then wins; otherwise, he loses. Prove that:
If , then can guarantee a win.
For all sufficiently large , there exists an integer such that cannot guarantee a win.
In the meanwhile, a full scale polymath project has started up to try and prove the “Hot Spots Conjecture for Acute Triangles”. It was proposed by Chris Evans and they are working on it here. It is a more geometric/analytic/physics based question this time. Dr. Evans describes the problem like this:
Suppose a flat piece of metal, represented by a two-dimensional bounded connected domain, is given an initial heat distribution which then flows throughout the metal. Assuming the metal is insulated (i.e. no heat escapes from the piece of metal), then given enough time, the hottest point on the metal will lie on its boundary.
– from Chris Evans’s Polymath proposal
A picture of the Hot Spots Conjecture (have they tried applying talcum powder to the problem?)
Also on the polymathprojects.org website you can find several other important unsolved problems in math which might tickle your fancy. Check it out!
As last year, Terence Tao is hosting a mini polymath group. That’s a online group discussion to solve one of the IMO problems. It started yesterday at 2pm Oklahoma time and already has 130+ comments! If you’re interested in joining in, go here. This year’s question is geometric:
Problem 2. Let be a finite set of at least two points in the plane. Assume that no three points of are collinear. A windmill is a process that starts with a line going through a single point . The line rotates clockwise about the pivot until the first time that the line meets some other point belonging to . This point takes over as the new pivot, and the line now rotates clockwise about , until it next meets a point of . This process continues indefinitely.
Show that we can choose a point in and a line going through such that the resulting windmill uses each point of as a pivot infinitely many times.
Some time ago we posted about an online math research collaboration project initiated by Timothy Gowers which you, your dog, and everyone else on the Internet was welcome to participate in. It turned out to be remarkably successful. Not only was the original problem solved but various other results were obtained as well.
They are currently working on writing up the results of that project as a research paper for publication. The author of the paper is D. H. J. Polymath. Timothy Gowers is planning to run another polymath project this fall (keep an eye out for it!).
Problem 6. Let be distinct positive integers and let be a set of positive integers not containing . A grasshopper is to jump along the real axis, starting at the point and making jumps to the right with lengths in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in .
The good news is that the IMO is an international competition for high school students. This means the above problem is meant to be solvable by students of this age and, in particular, is known to have a solution (which doesn’t happen when you’re doing research!). The bad news is that the IMO is for the best of the best of high school math students from around the world and only 6 or so out of 500 were able to solve it.
If you’re feeling good about your math kung-fu, take a crack at the problem. If not so much, then swing on by Terence Tao’s blog and join in with the crowd working on solving it right now.