Imo 2011

This is the wiki page for the mini-polymath3 project, which seeks solutions to Question 2 from the 2011 International Mathematical Olympiad.

The project will start at July 19 8pm UTC, and will be hosted at the polymath blog. A discussion thread will be hosted at Terry Tao's blog.

Rules

This project will follow the usual polymath rules. In particular:

• Everyone is welcome to participate, though people who have already seen an external solution to the problem should refrain from giving spoilers throughout the experiment.
• This is a team effort, not a race between individuals. Rather than work for extended periods of time in isolation from the rest of the project, the idea is to come up with short observations (or to carry an observation of another participant further) and then report back what one gets to the rest of the team. Partial results or even failures can be worth reporting.
• Participants are encouraged to update the wiki, or to summarise progress within threads, for the benefit of others. (In particular, linking between the wiki and specific comments on the blogs is highly encouraged.)

Threads

Planning:

• Mini-polymath 3: 2011 IMO question, June 9 2011.

Discussion:

• Mini-polymath3 discussion thread, July 19 2011.

Research:

• Minipolymath3 project: 2011 IMO, July 19 2011

The question

Problem 2. Let [math]\displaystyle{ S }[/math] be a finite set of at least two points in the plane. Assume that no three points of [math]\displaystyle{ S }[/math] are collinear. A windmill is a process that starts with a line [math]\displaystyle{ \ell }[/math] going through a single point [math]\displaystyle{ P \in S }[/math]. The line rotates clockwise about the pivot [math]\displaystyle{ P }[/math] until the first time that the line meets some other point [math]\displaystyle{ Q }[/math] belonging to [math]\displaystyle{ S }[/math]. This point [math]\displaystyle{ Q }[/math] takes over as the new pivot, and the line now rotates clockwise about [math]\displaystyle{ Q }[/math], until it next meets a point of [math]\displaystyle{ S }[/math]. This process continues indefinitely.

Show that we can choose a point [math]\displaystyle{ P }[/math] in [math]\displaystyle{ S }[/math] and a line [math]\displaystyle{ \ell }[/math] going through [math]\displaystyle{ P }[/math] such that the resulting windmill uses each point of [math]\displaystyle{ S }[/math] as a pivot infinitely many times.

Observations and partial results

• Clarification: the rotating line extends out from a point in both directions, not just one direction (i.e. it is a full line, not a half line)
• Trivial observation: If we start with a point on the convex hull of S and a line l that is “tangent” to the convex hull then we will only iterate over the points from the convex hull.
• One can start with any point (since every point of S should be pivot infinitely often), the direction of line that one starts with however matters!
• As no three points are collinear, one can always find a line which splits the points into two sets whose number of elements differ at most one.
• The process is time reversible, thus there are no “rho” processes with an initial segment that doesn't repeat. In particular, it’s enough to find a cycle that visits each vertex at least once.

Examples

• If the points form a convex polygon, it is easy.

• Say there are four points: an equilateral triangle, and then one point in the center of the triangle. No three points are collinear. If you start in M, you first hit say A, then C, then M, then B, then A.

Possible strategies

• Maybe the strategy should be to take out the convex hull of S from consideration; follow it up by induction on removing successive convex hulls.
• A possible line of enquiry might be to consider how adding additional points effects an already established “windmill” this might lead to an inductive solution if adding new points can be related to already established cases.
• It might be fun to use projective duality: the space of unoriented lines in the plane is diffeomorphic to the projective plane minus a point, i.e. the open Möbius strip, which I’ll call M. Every point p of S corresponds to a line m_p in M, and every line l passing through p in the plane corresponds to a point n_l on m_p in M. So the projective dual windmill process tracks the position of a point on the arrangement of lines in M dual to the points of S. The process begins with n_l on a line m_p and I suppose the clockwise direction tells us which direction we progress along m_p. I think the goal is then to show that there exists a starting point on the arrangement such that the trace of the projective dual windmill “curve” will hit every single line in this arrangement…?
• It seems that a good place to start is with the set, A, of ordered pairs of points, with each ordered pair representing a “transition” from one pivot point to the next. Each member of A would be a vertex in a directed graph containing an edge from each “transition” to its successor “transition”. One would then need to show that a cycle exists in this graph for which every point participates in at least one (actually two) of the vertices.

Completed solutions

• None yet.